It is clear from Figure 12 and Figure 13 that the waveforms which caused by sudden stop of the slide motion after they reach the characteristic length L′ at time t��� = t∗ + (L′/v) have higher amplitude than stopping of the slide with slow motion at time t��� = t∗ + (2L′/v) .This agrees with the mathematical relation between the wavelength and the wave height where the wave energy is proportional to both the length of the wave and the height

time t��� = t∗ + (2L′/v) with different slide length L and width W and for v=v�.

squared.

t��� = t∗ + (L′/v) with different slide length L and width W and for v=v�.

#### *3.2.2. Displaced block moving with linear decreasing velocity with time t*

The velocity of the movable slide is uniform and equal v(t) up to time t��� as shown in Figure 4a, followed by a decelerating phase in which the velocity is given by

$$\mathbf{v}(\mathbf{t}) = \mathbf{v} - (\mathbf{t} - \mathbf{t}^\*)\mathbf{a} \text{ / for } \mathbf{t}^\* \le \mathbf{t} \le \mathbf{t}\_{\text{max}}.$$

where v=v� = 0.14 km/sec and α is the deceleration of the moving block slide. The block slide moves in the positive x − direction with time t∗ ≤ t ≤ t��� where t��� = t∗ + 2L′/v is the maximum time that the slide takes to stop after reaching the characteristic length L′ = 150 km with minimum deceleration ��. Figure 11 shows the leading tsunami wave propagating in the positive x-direction during time evolution t=t∗ , t∗ + 0.2(2L′/v), t∗ + 0.4(2L′/v), t∗ + +0.6(2L′/v), t∗ + 0.8(2L′/v), t∗ + (2L′/v) min in case t� = t∗ (i.e. minimum magnitude of α ).

**Figure 11.** Normalized tsunami waveforms η/ζ� and their corresponding moving slide ζ/ζ� with variable velocity v(t) along y = 0, at time t∗ ≤ t ≤ t∗ + 2L′/v at H = 2 km, L′ = 150 km and W = 100 km.

It was clear from Figure 10 and Figure 11 that at the instant the slide stops, the peak amplitude in case of sudden stop is higher than that of soft stop.

Figure 12 and Figure 13 show the effect of the water depth at L′ = W = 10, 50,100 km and L′ = 150 , W = 100 km on the normalized peak tsunami amplitude η�,���/ζ� when the slide stops moving at length L′ instantaneously at t��� = t∗ + L′/v with infinite deceleration and stops moving softly at L′ at the time t��� = t∗ + 2L′/v with minimum deceleration for v=v�.

magnitude of α ).

v=v�.

the free surface depends on the characteristic volume of the source model

*3.2.2. Displaced block moving with linear decreasing velocity with time t*

Figure 4a, followed by a decelerating phase in which the velocity is given by

which becomes an important factor in the modeling of the tsunami generation. This was clear from the singular points removed from the block slide model, where the finite limit of

The velocity of the movable slide is uniform and equal v(t) up to time t��� as shown in

v(t) =v− (t−t∗)α , for t∗ ≤ t ≤ t���.

where v=v� = 0.14 km/sec and α is the deceleration of the moving block slide. The block slide moves in the positive x − direction with time t∗ ≤ t ≤ t��� where t��� = t∗ + 2L′/v is the maximum time that the slide takes to stop after reaching the characteristic length L′ = 150 km with minimum deceleration ��. Figure 11 shows the leading tsunami wave propagating in the positive x-direction during time evolution t=t∗ , t∗ + 0.2(2L′/v), t∗ + 0.4(2L′/v), t∗ + +0.6(2L′/v), t∗ + 0.8(2L′/v), t∗ + (2L′/v) min in case t� = t∗ (i.e. minimum

**Figure 11.** Normalized tsunami waveforms η/ζ� and their corresponding moving slide ζ/ζ� with variable velocity v(t) along y = 0, at time t∗ ≤ t ≤ t∗ + 2L′/v at H = 2 km, L′ = 150 km and W = 100 km.

amplitude in case of sudden stop is higher than that of soft stop.

It was clear from Figure 10 and Figure 11 that at the instant the slide stops, the peak

Figure 12 and Figure 13 show the effect of the water depth at L′ = W = 10, 50,100 km and L′ = 150 , W = 100 km on the normalized peak tsunami amplitude η�,���/ζ� when the slide stops moving at length L′ instantaneously at t��� = t∗ + L′/v with infinite deceleration and stops moving softly at L′ at the time t��� = t∗ + 2L′/v with minimum deceleration for

**Figure 12.** Normalized maximum tsunami amplitudes η�����/ζ� when the slide stops suddenly at time t��� = t∗ + (L′/v) with different slide length L and width W and for v=v�.

**Figure 13.** Normalized maximum tsunami amplitudes η�����/ζ� when the slide stops softly at time t��� = t∗ + (2L′/v) with different slide length L and width W and for v=v�.

It is clear from Figure 12 and Figure 13 that the waveforms which caused by sudden stop of the slide motion after they reach the characteristic length L′ at time t��� = t∗ + (L′/v) have higher amplitude than stopping of the slide with slow motion at time t��� = t∗ + (2L′/v) .This agrees with the mathematical relation between the wavelength and the wave height where the wave energy is proportional to both the length of the wave and the height squared.

### *3.2.3. Displaced block moving with constant velocity v followed by variable velocity* �(�)

Generation and Propagation of Tsunami

/v) =

by a Moving Realistic Curvilinear Slide Shape with Variable Velocitie 103

Figure 15 shows the shape of the tsunami propagation waveform at t��� = t∗ + (2L�

stops).

59.51 min (curves in black) for different deceleration α and time t� (time at which the slide

**Figure 15.** Normalized tsunami propagation waveforms η/ζ� along the axis of the symmetry at y=0 at

t����. = 0, 3.56, 7.14, 10.70, 14.28, 17.85 min with L′ = 150 km and W = 100 km.

In this section, we studied the generation of the tsunami waveforms when the block moves a significant distance with constant velocity v=v� then continue moving with variable velocity v(t) with constant deceleration until it stops at the characteristic length L′ = 150 km. Figure 14 shows the tsunami waveforms at the times calculated in Table 2 when the slide reaches the characteristic length L′ = 150 km.

**Figure 14.** Normalized tsunami waveforms η/ζ� along y = 0 at t=t� calculated in Table 7.2 with L′ = 150 km.

In Figure 14, the first waveform from the left indicates the shape of the wave at the time t� = t��� = t∗ + (L� /v) = 41.65 in the limiting case when the slide stops moving suddenly. The last waveform indicates the wave in the other limiting case at the time t� = t��� = t∗ + (2L� /v) when the slide stop moving with minimum deceleration at the distance L′. In between the two limiting cases, the slide begins moving with constant velocity a significant distance followed by a decelerating movement until it stop at the characteristic length L′ = 150 km at the time t��� ≤ t� ≤ t��� , see Table 2. It is seen how the peak amplitudes of the leading waves decreases gradually from 5.261 to 3.894.

#### **3.3. Tsunami propagation waveforms**

In order to compare shape and maximum height of tsunami wave at certain time for different deceleration α, we choose the time t=t���. We study the case when the propagating waveforms resulting at the time difference between t� and t��� (i.e. propagating of tsunami away from the slide) when the slide stops moving at the length L′.

For the limiting case ��, there is no free propagation, while for the other limiting case " the sudden stop", there is free propagation between time t��� and t���. For the cases between the two limiting cases, the propagation time is t���� = t��� − t�.

Figure 15 shows the shape of the tsunami propagation waveform at t��� = t∗ + (2L� /v) = 59.51 min (curves in black) for different deceleration α and time t� (time at which the slide stops).

102 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

reaches the characteristic length L′ = 150 km.

L′ = 150 km.

(2L�

t� = t��� = t∗ + (L�

*3.2.3. Displaced block moving with constant velocity v followed by variable velocity* �(�)

**Figure 14.** Normalized tsunami waveforms η/ζ� along y = 0 at t=t� calculated in Table 7.2 with

the leading waves decreases gradually from 5.261 to 3.894.

the two limiting cases, the propagation time is t���� = t��� − t�.

**3.3. Tsunami propagation waveforms**

In Figure 14, the first waveform from the left indicates the shape of the wave at the time

The last waveform indicates the wave in the other limiting case at the time t� = t��� = t∗ +

In order to compare shape and maximum height of tsunami wave at certain time for different deceleration α, we choose the time t=t���. We study the case when the propagating waveforms resulting at the time difference between t� and t��� (i.e. propagating of tsunami away from the slide) when the slide stops moving at the length L′.

For the limiting case ��, there is no free propagation, while for the other limiting case " the sudden stop", there is free propagation between time t��� and t���. For the cases between

/v) when the slide stop moving with minimum deceleration at the distance L′. In between the two limiting cases, the slide begins moving with constant velocity a significant distance followed by a decelerating movement until it stop at the characteristic length L′ = 150 km at the time t��� ≤ t� ≤ t��� , see Table 2. It is seen how the peak amplitudes of

/v) = 41.65 in the limiting case when the slide stops moving suddenly.

In this section, we studied the generation of the tsunami waveforms when the block moves a significant distance with constant velocity v=v� then continue moving with variable velocity v(t) with constant deceleration until it stops at the characteristic length L′ = 150 km. Figure 14 shows the tsunami waveforms at the times calculated in Table 2 when the slide

**Figure 15.** Normalized tsunami propagation waveforms η/ζ� along the axis of the symmetry at y=0 at t����. = 0, 3.56, 7.14, 10.70, 14.28, 17.85 min with L′ = 150 km and W = 100 km.

As the wave propagates, the wave height decreases and the slope of the front of the wave become smaller, causing a train of small wave forms behind the main wave. The maximum wave amplitude decreases with time, due to the geometric spreading and also due to the dispersion.

Generation and Propagation of Tsunami

by a Moving Realistic Curvilinear Slide Shape with Variable Velocitie 105

velocities of submarine block slide on the tsunami generation in the limiting cases, in which the slide moves with constant velocity and stops suddenly (with infinite deceleration) and the case in which the slide stops softly at the same place with minimum deceleration. It is seen that the leading tsunami amplitudes are reduced in both cases due to the geometric spreading and also due to the dispersion. We observed that the peak tsunami amplitudes increase with the decrease in the sliding source area and the water depth. We also investigated the more realistic case in which the block slide moves a significant distance with constant velocity *v* then continue moving with time dependence velocity v(t) and different constant deceleration until it stops at the characteristic length. It is seen how the peak amplitudes of the leading waves decrease gradually with time between the two limiting cases. In this case we demonstrated also the shape of tsunami propagated wave at certain time max *t* (time at which the slide stops with minimum deceleration). The results show that the wave height decreases due to dispersion and the slope of the front of the wave becomes smaller, causing a train of small wave forms behind the main wave. It can be observed that just a slight variation in the maximum and the minimum tsunami propagated amplitudes after the block slide stops moving with different deceleration α and time t�, see **Figure 16**. The presented analysis suggests that some abnormally large tsunamis could be explained in part by variable speeds of submarine landslides. Our results should help to enable quantitative tsunami forecasts and warnings based on recoverable seismic data and to increase the possibilities for the use of tsunami data to study earthquakes, particularly historical events for which

adequate seismic data do not exist.

33, No. 1, 2009, pp. 451-466.

*Department of Basic and Applied Science, College of Engineering and Technology,*

*Computation*, Vol. 177, No. 2, 2006, pp. 785-806.

*Computation*, Vol. 174, No. 2, 2006, pp. 795-809.

*Arab Academy for Science, Technology and Maritime Transport, Abu Quir Campus, Alexandria,*

[1] M. S. Abou-Dina and F. M. Hassan, "Generation and Propagation of Nonlinear Tsunamis in Shallow Water by a Moving Topography," *Applied Mathematics and*

[2] F. M. Hassan, "Boundary Integral Method Applied to the Propagation of Non-linear Gravity Waves Generated by a Moving Bottom," *Applied Mathematical Modelling*, Vol.

[3] N. Zahibo, E. Pelinovsky, T. Talipova , A. Kozelkov and A. Kurkin, "Analytical and Numerical Study of Nonlinear Effects at Tsunami Modeling," *Applied Mathematics and*

**Author details**

Khaled T. Ramadan

**5. References**

*Egypt*

Figure 16 represents the normalized peak tsunami amplitudes |η|���/ζ� and η���/ζ� of the leading propagating wave in the far-field at time t=t��� for the different deceleration α and time t� (time at which the slide stops) chosen in Figure 7.15.

**Figure 16.** The normalized peak tsunami amplitudes |η|���/ζ� and η���/ζ� at time t=t��� for the different deceleration α and time t�.

It can be seen in Figure 16 that the absolute minimum peak amplitudes of the leading propagated waves at time t��� = t∗ + (2L� /v), after the block slide stops moving with different deceleration α and time t�, decreases gradually, while the maximum peak amplitudes increases progressively.

### **4. Conclusion**

In this paper, we presented a review of the main physical characteristics of the tsunami generation caused by realistic curvilinear submarine slumps and slides in the near-field. It is seen that the tsunami waveform has two large peaks of comparable amplitudes, one in the front of the block due to forward sliding of the block, and the other one behind the block due to spreading of the depletion zone. The negative peak wave amplitudes are approximately equal to the positive peak amplitudes. We studied the effect of variable velocities of submarine block slide on the tsunami generation in the limiting cases, in which the slide moves with constant velocity and stops suddenly (with infinite deceleration) and the case in which the slide stops softly at the same place with minimum deceleration. It is seen that the leading tsunami amplitudes are reduced in both cases due to the geometric spreading and also due to the dispersion. We observed that the peak tsunami amplitudes increase with the decrease in the sliding source area and the water depth. We also investigated the more realistic case in which the block slide moves a significant distance with constant velocity *v* then continue moving with time dependence velocity v(t) and different constant deceleration until it stops at the characteristic length. It is seen how the peak amplitudes of the leading waves decrease gradually with time between the two limiting cases. In this case we demonstrated also the shape of tsunami propagated wave at certain time max *t* (time at which the slide stops with minimum deceleration). The results show that the wave height decreases due to dispersion and the slope of the front of the wave becomes smaller, causing a train of small wave forms behind the main wave. It can be observed that just a slight variation in the maximum and the minimum tsunami propagated amplitudes after the block slide stops moving with different deceleration α and time t�, see **Figure 16**. The presented analysis suggests that some abnormally large tsunamis could be explained in part by variable speeds of submarine landslides. Our results should help to enable quantitative tsunami forecasts and warnings based on recoverable seismic data and to increase the possibilities for the use of tsunami data to study earthquakes, particularly historical events for which adequate seismic data do not exist.

### **Author details**

104 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

and time t� (time at which the slide stops) chosen in Figure 7.15.

dispersion.

different deceleration α and time t�.

amplitudes increases progressively.

**4. Conclusion**

propagated waves at time t��� = t∗ + (2L�

As the wave propagates, the wave height decreases and the slope of the front of the wave become smaller, causing a train of small wave forms behind the main wave. The maximum wave amplitude decreases with time, due to the geometric spreading and also due to the

Figure 16 represents the normalized peak tsunami amplitudes |η|���/ζ� and η���/ζ� of the leading propagating wave in the far-field at time t=t��� for the different deceleration α

**Figure 16.** The normalized peak tsunami amplitudes |η|���/ζ� and η���/ζ� at time t=t��� for the

It can be seen in Figure 16 that the absolute minimum peak amplitudes of the leading

different deceleration α and time t�, decreases gradually, while the maximum peak

In this paper, we presented a review of the main physical characteristics of the tsunami generation caused by realistic curvilinear submarine slumps and slides in the near-field. It is seen that the tsunami waveform has two large peaks of comparable amplitudes, one in the front of the block due to forward sliding of the block, and the other one behind the block due to spreading of the depletion zone. The negative peak wave amplitudes are approximately equal to the positive peak amplitudes. We studied the effect of variable

/v), after the block slide stops moving with

Khaled T. Ramadan

*Department of Basic and Applied Science, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport, Abu Quir Campus, Alexandria, Egypt*

### **5. References**

	- [4] Ben-Menahem and M. Rosenman, "Amplitude Patterns of Tsunami Waves from Submarine Earthquakes," *Journal of Geophysical Research*, Vol. 77, No. 17, 1972, pp. 3097- 3128.

**Chapter 6**

© 2012 Tan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Tsunami Propagation Models**

A. Tan, A.K. Chilvery, M. Dokhanian and S.H. Crutcher

tsunamis and warning the populace in the potentially vulnerable areas.

and are, by and large, outside the reach of the broader audience.

Tsunamis are ocean waves generated by the displacement of a large volume of water due to earthquakes, volcanic eruptions, landslides or other causes above or below the ocean floor (e.g., Karling, 2005; Parker, 2012). The great Indian Ocean tsunami of December 2004 will be remembered for its ferocity, devastation and unprecedented loss of life for a long time (Stewart, 2005; The Indian Ocean Tsunami, 2011). It is also the same tsunami which has galvanized the international community to set up warning systems and undertake preventive measures against the onslaught of future tsunamis in the vulnerable regions around the globe. A surge of scientific studies on all aspects of the tsunami is in evidence in the literature. And a volume entitled "*The Tsunami Threat – Research and Technology*" (Mörner ed., 2011) has been brought out. The current volume entitled "*Tsunami*" (Lopez, ed., 2012) is a sequel to the above in a continued effort to promote understanding and predicting future

There are three distinct stages of a tsunami event: (1) Generation; (2) Propagation; and (3) Inundation/landfall (cf. Cecioni & Belloti, 2011). The generation stage is the most complex and most difficult to analyze, since each tsunami is different and no single mechanism can account for all tsunamis. The inundation stage is also different for different areas affected, and again, no single scenario can describe all affected areas. The propagation stage covers the most extensive area, and is the only one that can be attacked by simple theory and analysis, even though detailed numerical models are found in the literature (see, for example, Imteaz, et al., 2011, and the references therein). These models consist of solving hydrodynamic equations with suitable boundary conditions that necessarily involve tedious numerical integrations. Such models, unfortunately, fall within the realm of the specialists,

**Based on First Principles**

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50508

**1. Introduction**

[5] Kjetil Braathen Haugen, Finn Løvholt and Carl B. Harbitz , " Fundamental mechanisms for tsunami generation by submarine mass flows in idealised geometries", Marine and Petroleum Geology, Vol. 22, Issues 1–2, 2005, pp 209–217.

**Chapter 6**

## **Tsunami Propagation Models Based on First Principles**

A. Tan, A.K. Chilvery, M. Dokhanian and S.H. Crutcher

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50508

### **1. Introduction**

106 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

Petroleum Geology, Vol. 22, Issues 1–2, 2005, pp 209–217.

3128.

[4] Ben-Menahem and M. Rosenman, "Amplitude Patterns of Tsunami Waves from Submarine Earthquakes," *Journal of Geophysical Research*, Vol. 77, No. 17, 1972, pp. 3097-

[5] Kjetil Braathen Haugen, Finn Løvholt and Carl B. Harbitz , " Fundamental mechanisms for tsunami generation by submarine mass flows in idealised geometries", Marine and

> Tsunamis are ocean waves generated by the displacement of a large volume of water due to earthquakes, volcanic eruptions, landslides or other causes above or below the ocean floor (e.g., Karling, 2005; Parker, 2012). The great Indian Ocean tsunami of December 2004 will be remembered for its ferocity, devastation and unprecedented loss of life for a long time (Stewart, 2005; The Indian Ocean Tsunami, 2011). It is also the same tsunami which has galvanized the international community to set up warning systems and undertake preventive measures against the onslaught of future tsunamis in the vulnerable regions around the globe. A surge of scientific studies on all aspects of the tsunami is in evidence in the literature. And a volume entitled "*The Tsunami Threat – Research and Technology*" (Mörner ed., 2011) has been brought out. The current volume entitled "*Tsunami*" (Lopez, ed., 2012) is a sequel to the above in a continued effort to promote understanding and predicting future tsunamis and warning the populace in the potentially vulnerable areas.

> There are three distinct stages of a tsunami event: (1) Generation; (2) Propagation; and (3) Inundation/landfall (cf. Cecioni & Belloti, 2011). The generation stage is the most complex and most difficult to analyze, since each tsunami is different and no single mechanism can account for all tsunamis. The inundation stage is also different for different areas affected, and again, no single scenario can describe all affected areas. The propagation stage covers the most extensive area, and is the only one that can be attacked by simple theory and analysis, even though detailed numerical models are found in the literature (see, for example, Imteaz, et al., 2011, and the references therein). These models consist of solving hydrodynamic equations with suitable boundary conditions that necessarily involve tedious numerical integrations. Such models, unfortunately, fall within the realm of the specialists, and are, by and large, outside the reach of the broader audience.

© 2012 Tan et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This chapter takes an alternative approach to the study of tsunami propagation in the open ocean. It commences with the theory of water wave propagation in general and applies them to tsunami propagation in particular, using analytical models. It is based upon first principles of physics and avoids numerical analysis, which is thus accessible to the broader scientific community. It only requires the knowledge of general science and basic calculusbased physics. The derivations of the relevant equations are relegated to the appendices for quick reference so that the need to search for them outside this article is kept to a minimum.

### **2. Theory of water waves and tsunamis**

The theory of water waves is well-documented in the literature (e.g., Coulson, 1955; Sharman, 1963; Towne, 1967; Elmore & Heald, 1969). The wave velocity of waves on water of density *ρ* under the action of gravity and surface tension *T* is obtained from a linear wave equation by ignoring the non-linear term of ½*v*2 in Bernoulli's equation (cf. Coulson, 1955; Elmore & Heald, 1969):

$$\psi = \sqrt{\frac{g}{k} + \frac{Tk}{\rho}} \tanh kh\tag{1}$$

Tsunami Propagation Models Based on First Principles 109

*dk ρ* (6)

*k k* (7)

*v gh* (9)

*dω u gh v dk* (10)

*k* (11)

*dk k* (12)

3 3 2 2

*dω Tk u u*

The group velocity is faster than the wave velocity! The individual wave-crests fall behind the group while new crests build up at the forward edge of the group (cf. Towne, 1967). Such waves are called *capillary waves* or *ripples*. They are severely attenuated by viscous

For longer wavelengths (*λ* > *λ*c), the gravity term dominates. We have (vide Appendix A):

tanh *ω g v kh*

*ω gk kh* tanh (8)

Such waves are called *gravity waves*. Customarily, two kinds of gravity waves are recognized depending upon the wavelength *λ* = 2*π*/*k*: (1) If *λ* > *h*, the waves are called *long waves in*

Owing to the equality of the wave velocity and the group velocity, such waves are non-

2 2 *dω g u v*

In such waves, the individual waves travel faster than the group and rapidly diminish in

Tsunami waves in the deep ocean typically have wavelengths of about 200 km and velocity of 700 kilometers per hour. They maintain their form in dispersion-less propagation for long

*shallow water*; (2) If *λ* < *h*, the waves are called *short waves in deep water*.

In the first case, tanh*kh kh* and the wave velocity

*g v*

1 1

amplitude as they cross the group (Towne, 1967).

The group velocity is, from Eq. (8):

In the second case, tanh*kh* 1 . Then

effects.

and

dispersive.

and from Eq. (8):

Here *h* is the depth of water and *k* the wave number. In terms of the wavelength *λ* = 2*π*/*k*, we have:

$$v = \sqrt{\left(\frac{g\lambda}{2\pi} + \frac{2\pi T}{\lambda\rho}\right) \tanh\frac{2\pi h}{\lambda}}\tag{2}$$

When the gravity and surface tension terms are equal, *λ* is called the critical wavelength *λ*c:

$$
\lambda\_c = 2\pi \sqrt{\frac{T}{\rho g}}\tag{3}
$$

For water waves, *λ*c = 1.73 cm (cf. Towne, 1967; Elmore & Heald, 1969). The corresponding velocity *v*c = 27 cm/s (Sharman, 1963).

For waves of *λ* < *λ*c < *h*, the surface tension term dominates and

$$
\omega = \frac{\omega}{k} = \sqrt{\frac{Tk}{\rho}}\tag{4}
$$

Thus

$$
\omega = \sqrt{\frac{T}{\rho}} k^{3/2} \tag{5}
$$

and the group velocity

Tsunami Propagation Models Based on First Principles 109

$$
\mu = \frac{d\omega}{dk} = \frac{3}{2} \sqrt{\frac{Tk}{\rho}} = \frac{3}{2}\mu \tag{6}
$$

The group velocity is faster than the wave velocity! The individual wave-crests fall behind the group while new crests build up at the forward edge of the group (cf. Towne, 1967). Such waves are called *capillary waves* or *ripples*. They are severely attenuated by viscous effects.

For longer wavelengths (*λ* > *λ*c), the gravity term dominates. We have (vide Appendix A):

$$
\omega = \frac{\omega}{k} = \sqrt{\frac{g}{k} \tanh kh} \tag{7}
$$

and

108 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

**2. Theory of water waves and tsunamis**

2 2

2 *c*

velocity *v*c = 27 cm/s (Sharman, 1963).

*v*

For waves of *λ* < *λ*c < *h*, the surface tension term dominates and

minimum.

have:

Thus

and the group velocity

Elmore & Heald, 1969):

This chapter takes an alternative approach to the study of tsunami propagation in the open ocean. It commences with the theory of water wave propagation in general and applies them to tsunami propagation in particular, using analytical models. It is based upon first principles of physics and avoids numerical analysis, which is thus accessible to the broader scientific community. It only requires the knowledge of general science and basic calculusbased physics. The derivations of the relevant equations are relegated to the appendices for quick reference so that the need to search for them outside this article is kept to a

The theory of water waves is well-documented in the literature (e.g., Coulson, 1955; Sharman, 1963; Towne, 1967; Elmore & Heald, 1969). The wave velocity of waves on water of density *ρ* under the action of gravity and surface tension *T* is obtained from a linear wave equation by ignoring the non-linear term of ½*v*2 in Bernoulli's equation (cf. Coulson, 1955;

> tanh *g Tk v kh k ρ*

Here *h* is the depth of water and *k* the wave number. In terms of the wavelength *λ* = 2*π*/*k*, we

When the gravity and surface tension terms are equal, *λ* is called the critical wavelength *λ*c:

*T λ π*

For water waves, *λ*c = 1.73 cm (cf. Towne, 1967; Elmore & Heald, 1969). The corresponding

*ω Tk*

*T* 3 2/ *ω k*

*v*

tanh *gλ πT πh*

*π λρ λ*

2

(1)

(2)

*ρg* (3)

*k ρ* (4)

*ρ* (5)

$$
\omega = \sqrt{gk \tanh kh} \tag{8}
$$

Such waves are called *gravity waves*. Customarily, two kinds of gravity waves are recognized depending upon the wavelength *λ* = 2*π*/*k*: (1) If *λ* > *h*, the waves are called *long waves in shallow water*; (2) If *λ* < *h*, the waves are called *short waves in deep water*.

In the first case, tanh*kh kh* and the wave velocity

$$
v = \sqrt{gh} \tag{9}$$

The group velocity is, from Eq. (8):

$$
\mu = \frac{d\omega}{dk} = \sqrt{gh} = \upsilon
\tag{10}
$$

Owing to the equality of the wave velocity and the group velocity, such waves are nondispersive.

In the second case, tanh*kh* 1 . Then

$$
v = \sqrt{\frac{g}{k}}\tag{11}$$

and from Eq. (8):

$$
\mu = \frac{d\omega}{dk} = \frac{1}{2}\sqrt{\frac{g}{k}} = \frac{1}{2}\upsilon \tag{12}
$$

In such waves, the individual waves travel faster than the group and rapidly diminish in amplitude as they cross the group (Towne, 1967).

Tsunami waves in the deep ocean typically have wavelengths of about 200 km and velocity of 700 kilometers per hour. They maintain their form in dispersion-less propagation for long

distances with little dissipation of energy. They are thus identified as 'long waves in shallow water' even as they travel above the deepest parts of the ocean, the 'long' and 'shallow' adjectives referring to the relative magnitudes of the wavelength and depth of water. Many of the properties of the tsunamis, such as velocity and dispersion-less propagation follow from this description (e.g., Margaritondo, 2005; Helene & Yamashita, 2006; Tan & Lyatskaya, 2009). However, the linearized theory of water waves is only an approximation to the true situation and a rigorous theory calls for the inclusion of the non-linear term in Bernoulli's equation. For example, the form of these waves having a narrow crest and very wide trough can only be accounted for by a complete non-linear theory (cf. Elmore & Heald, 1969).

Tsunami Propagation Models Based on First Principles 111

*h* (14)

*ρ* (15)

*ρ* (16)

*θ* (17)

*θ* (18)

4

characteristics as they approach land.

the Boxing day tsunami of December 2004.

inversely as the distance from the epicenter *ρ*:

varies as (vide Appendix C):

1 *ψModelB ModelB I*

1

1

north pole. Hence the wave amplitude is a function of the polar angle:

1 *A*

A 1 m high wave at a depth of 1000 m would become 5.62 m at a depth of 1 m! That goes to illustrate the devastating effects of tsunamis as they make landfall. Helene & Yamashita (2006) have shown that the Eq. (14) holds true when the depth of the ocean floor varies gradually instead of having abrupt steps, which helps the tsunamis to maintain their

Helene & Yamashita (2006) have further shown how a tsunami will bend dramatically around an obstacle and strike land in the shadow regions. Since *v h* according to Eq. (9), the velocity decreases nearer the coast as the depth decreases. The wave-fronts are able to bend dramatically towards the coast in a diffraction pattern and to strike land in the shadow region. This explains the damage caused in the western coasts of Sri Lanka and India during

Useful as they are, the one-dimensional models of are not appropriate when the tsunami propagates in the open ocean from a well-defined epicenter since the waves will spread out in concentric circles, which calls for two-dimensional models. In **Model B** (Tan & Lyatskaya, 2009), waves propagate outwards on a flat two-dimensional ocean from the epicenter. When the energy conservation principle is imposed, the energy density of the wave falls off

*ModelB I*

The wave amplitude therefore falls off inversely as the square-root of the radial distance *ρ*:

The formal derivation of the results (15) and (16) are given in Appendix B where it is

For long distance propagation, the curvature of the Earth must be taken into consideration. This is incorporated in **Model C** (Tan & Lyatskaya, 2009) which analyzes tsunami propagation on a spherical oceanic surface. In this model, the energy density of the wave

sin *ModelC I*

sin *ψModelC ModelC I*

where *θ* is the zenith angle measured from the center of the Earth with the epicenter at the

mentioned that strictly speaking, they apply at distances away from the epicenter.

1

Tsunamis also exhibit the characteristics of 'canal waves', first observed by Scott Russell in Scotland in 1834 (Russell, 1844). These disturbances travel like a single wave over long distances and maintain their shapes with little loss of energy. The mathematical theory of such waves, now-a-days called 'solitary waves' or 'solitons', was developed by Boussinesq (1871), Korteweg & de Vries (1895) and Lord Rayleigh (1914). Their analyses show that in a dispersive medium, the non-linear effect can exactly cancel out the dispersive effect to preserve the form of the wave (cf. Stoker, 1957; Lamb, 1993).

### **3. Tsunami propagation models**

One-dimensional propagation models are the easiest to construct and analyze. Even though they may not represent the real situation for tsunami propagation in the open ocean, valuable results can come out of these models. The 'long waves in shallow water' model noted above represents a one-dimensional model (**Model A**) in which the displacement is in the vertical direction and propagation takes place in the horizontal direction (see Appendix A). The Models of Margaritondo (2005) and Helene & Yamashita (2006) are examples of this model. This model correctly predicts the speed of the tsunami by means of Eq. (9). In fact, this equation is widely used in bathymetry. It is also used to calculate the travel times of tsunamis.

In the 'long waves in shallow water' model, the group velocity is equal to the wave velocity according to Eqs. (9) and (10). This helps to explain the dispersion-less propagation of the tsunamis. An alternative derivation of these equations is provided by Margaritondo (2005). By assuming that the vertical displacement of water is proportional to distance from the bottom, Margaritondo (2005) arrives at the same result by applying the energy conservation law. As stated earlier, however, a rigorous derivation would call for the solitary wave solution of non-linear differential equation obtained from Navier-Stokes equation (cf. Stoker, 1957).

Eq. (9) has profound consequences in the inundation phase as the tsunami makes landfall. The wave energy density *E* within a wavelength *λ* is

$$E \propto A^2 \lambda \propto A^2 v \propto A^2 \sqrt{h} \tag{13}$$

where *A* is the amplitude of the wave. Assuming *E* to remain constant, we have

Tsunami Propagation Models Based on First Principles 111

$$A \propto \frac{1}{\sqrt[4]{\hbar}}\tag{14}$$

A 1 m high wave at a depth of 1000 m would become 5.62 m at a depth of 1 m! That goes to illustrate the devastating effects of tsunamis as they make landfall. Helene & Yamashita (2006) have shown that the Eq. (14) holds true when the depth of the ocean floor varies gradually instead of having abrupt steps, which helps the tsunamis to maintain their characteristics as they approach land.

110 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

preserve the form of the wave (cf. Stoker, 1957; Lamb, 1993).

The wave energy density *E* within a wavelength *λ* is

**3. Tsunami propagation models**

tsunamis.

Stoker, 1957).

distances with little dissipation of energy. They are thus identified as 'long waves in shallow water' even as they travel above the deepest parts of the ocean, the 'long' and 'shallow' adjectives referring to the relative magnitudes of the wavelength and depth of water. Many of the properties of the tsunamis, such as velocity and dispersion-less propagation follow from this description (e.g., Margaritondo, 2005; Helene & Yamashita, 2006; Tan & Lyatskaya, 2009). However, the linearized theory of water waves is only an approximation to the true situation and a rigorous theory calls for the inclusion of the non-linear term in Bernoulli's equation. For example, the form of these waves having a narrow crest and very wide trough can only be accounted for by a complete non-linear theory (cf. Elmore & Heald, 1969).

Tsunamis also exhibit the characteristics of 'canal waves', first observed by Scott Russell in Scotland in 1834 (Russell, 1844). These disturbances travel like a single wave over long distances and maintain their shapes with little loss of energy. The mathematical theory of such waves, now-a-days called 'solitary waves' or 'solitons', was developed by Boussinesq (1871), Korteweg & de Vries (1895) and Lord Rayleigh (1914). Their analyses show that in a dispersive medium, the non-linear effect can exactly cancel out the dispersive effect to

One-dimensional propagation models are the easiest to construct and analyze. Even though they may not represent the real situation for tsunami propagation in the open ocean, valuable results can come out of these models. The 'long waves in shallow water' model noted above represents a one-dimensional model (**Model A**) in which the displacement is in the vertical direction and propagation takes place in the horizontal direction (see Appendix A). The Models of Margaritondo (2005) and Helene & Yamashita (2006) are examples of this model. This model correctly predicts the speed of the tsunami by means of Eq. (9). In fact, this equation is widely used in bathymetry. It is also used to calculate the travel times of

In the 'long waves in shallow water' model, the group velocity is equal to the wave velocity according to Eqs. (9) and (10). This helps to explain the dispersion-less propagation of the tsunamis. An alternative derivation of these equations is provided by Margaritondo (2005). By assuming that the vertical displacement of water is proportional to distance from the bottom, Margaritondo (2005) arrives at the same result by applying the energy conservation law. As stated earlier, however, a rigorous derivation would call for the solitary wave solution of non-linear differential equation obtained from Navier-Stokes equation (cf.

Eq. (9) has profound consequences in the inundation phase as the tsunami makes landfall.

2 22 *E A λ Av A h* (13)

where *A* is the amplitude of the wave. Assuming *E* to remain constant, we have

Helene & Yamashita (2006) have further shown how a tsunami will bend dramatically around an obstacle and strike land in the shadow regions. Since *v h* according to Eq. (9), the velocity decreases nearer the coast as the depth decreases. The wave-fronts are able to bend dramatically towards the coast in a diffraction pattern and to strike land in the shadow region. This explains the damage caused in the western coasts of Sri Lanka and India during the Boxing day tsunami of December 2004.

Useful as they are, the one-dimensional models of are not appropriate when the tsunami propagates in the open ocean from a well-defined epicenter since the waves will spread out in concentric circles, which calls for two-dimensional models. In **Model B** (Tan & Lyatskaya, 2009), waves propagate outwards on a flat two-dimensional ocean from the epicenter. When the energy conservation principle is imposed, the energy density of the wave falls off inversely as the distance from the epicenter *ρ*:

$$I\_{ModelB} \propto \frac{1}{\rho} \tag{15}$$

The wave amplitude therefore falls off inversely as the square-root of the radial distance *ρ*:

$$
\psi\_{\text{ModelB}} \propto \sqrt{I\_{\text{ModelB}}} \propto \frac{1}{\sqrt{\rho}} \tag{16}
$$

The formal derivation of the results (15) and (16) are given in Appendix B where it is mentioned that strictly speaking, they apply at distances away from the epicenter.

For long distance propagation, the curvature of the Earth must be taken into consideration. This is incorporated in **Model C** (Tan & Lyatskaya, 2009) which analyzes tsunami propagation on a spherical oceanic surface. In this model, the energy density of the wave varies as (vide Appendix C):

$$I\_{Model\mathbb{C}} \propto \frac{1}{\sin \theta} \tag{17}$$

where *θ* is the zenith angle measured from the center of the Earth with the epicenter at the north pole. Hence the wave amplitude is a function of the polar angle:

$$
\psi\_{Model\mathbb{C}} \propto \sqrt{I\_{Model\mathbb{C}}} \propto \frac{1}{\sqrt{\sin \theta}} \tag{18}
$$

The amplitude variation depends solely on the polar angle. Since the epicenter is considered to be located at the north pole (*θ* = 0), the wave amplitude gradually falls until it reaches the equator (*θ* = *π*/2), 10,000 km away, and then starts to rise again as the convergence effect due to the curvature of the Earth comes into play and finally regains its original value at the south pole (*θ* = *π*). Thus, if the Earth were entirely ocean, barring any losses, the wave amplitude would regain its original value at the anti-podal point.

The comparison of wave amplitudes in Model C (true) and Model B (approximate) is given in Appendix C. We have:

$$\frac{\psi\_{\text{Model}\mathbb{C}}}{\psi\_{\text{Model}\mathbb{B}}} = \sqrt{\frac{\Theta}{\sin \Theta}}\tag{19}$$

Tsunami Propagation Models Based on First Principles 113

and Indian oceans, which comprise 51.5%, 25.6% and 22.9% of the water surface, respectively (we disregard 'Southern Ocean' as a separate entity). The Pacific ocean is thus larger than the two other oceans put together. It alone covers 34% or just over a full onethird of the Earth's surface and is comfortably larger than all the landmasses (at 29% of the Earth's surface) put together. The Pacific ocean is thought to be the remnant of 'Panthalassa',

The Pacific ocean provides an ideal venue for tsunami propagation studies for several reasons. First, as stated above, it is the largest body of water, covering a full one-third of the globe. Second, it is bounded by active tectonic plate junctions, studded with volcanoes called the 'Ring of Fire'. Tsunamis produced at these hotspots can traverse the length and breadth of the ocean with relative ease. Third, there are no landmasses or large islands to block or interfere with the propagation of tsunamis formed in the ocean. Fourth, the ocean itself is dotted with small islands which pose little interference with tsunami propagation, but provide valuable platforms for recording tsunami wave amplitudes. Many of these islands are volcanic in origin and are sources of tsunamis themselves. Fifth, the longest stretch of ocean water is found between the Japan archipelago in the north-west and southern Chile on the south-east covering a distance of over 17,000 km or 85% of the distance between the North pole and the South Pole. At both the ends of this diameter lie some of the most active plate tectonic regions and tsunamis from either ends have traversed this favorite racetrack. Last but not least, an astonishing 80% of all tsunamis are recorded in

With this geographical backdrop, we now proceed to study representative tsunami event to illustrate the validity of our propagation models. Model A, even though uni-dimensional, is a valuable tool for all tsunami events, as it correctly furnishes the velocity given the depth of the ocean, or vice-versa. It further predicts the travel times, which are vital for warning purposes. These results are independent of the direction, given the isotropy of space. Model A fails when the amplitude of the wave is to be studied, in which case Model B, C or D is called into consideration. In the following, we provide examples where one of the latter models, in conjuction with Model A, is used to analyze historic tsunami events. The data are taken from the National Oceanic and Atmospheric Administration website at

On 29 November 1975, a magnitude 7.2 earthquake occurred on the southern coast of the island of Hawaii with the epicenter at 19.3oN and 155.0oW at a focal depth of 8 km (cf. Pararas-Carayannis, 1976). The earthquake, the largest local one since 1868, generated a locally damaging submarine landslide tsunami which was recorded at 76 tide gauge stations in Alaska, California, Hawaii, Japan, Galapagos Islands, Peru and Chile. The tsunami caused \$1.5 million damage in Hawaii, 2 deaths and 19 injuries (Dudley & Lee, 1988). From the travel times registered, the tsunami reached Guadalupe Island, Mexico, 3864 km away in 5 h 9 m at an average speed of 750 kph, while it took 6 h 8 m to reach Tofino Island, Canada, 4210 km away, at the average speed of 686 kph. The slower speed in the first case is likely to

www.ngdc.noaa.gov/hazard/tsu\_travel\_time\_events.shtml.

the world ocean, when all the landmasses were joined together as 'Pangaea'.

the Pacific ocean.

**4.1. Hawaii tsunami of 1975**

Table C.1 shows that the difference in the two solutions is slight for small values of *θ*. Even at the equator (*θ* = *π*/2), the enhancement in amplitude due to curvature is only 25%. The enhancement becomes progressively greater until the south pole (*θ* = *π*) is reached, where it becomes infinite.

When the tsunami is caused by an oceanic plate sliding under a continental plate, the subduction zone can be described by a finite line source instead of a localized point source. In that case, **Model D** is appropriate. In this model, the wave-fronts are ellipses with foci at the end-points of the line source and the energy propagates along con-focal hyperbolas (Appendix D). The wave amplitude near the source will be far different from that of a point source (Model B). If *c* is the length of the line source, then at distances *ρb* from the center across the line source, one has (vide Appendix D):

$$\frac{\psi\_{ModelD}}{\psi\_{ModelB}} = \frac{1}{\sqrt{1 + \frac{c^2}{\rho\_b^2}}}\tag{20}$$

The wave amplitude in Model D is finite at the origin as opposed to being infinite in Models B and C. However, away from the source, the solutions for Models B and D rapidly converge, as the wave-fronts become more circular. For *ρb* > *c*, the difference is below 10%, whereas for *ρb* > 10*c*, the two solutions are virtually indistinguishable (Table D.1).

Tsunamis are vast and highly complex geophysical phenomena, each having a character of its own. It is impossible to construct one model for any tsunami even with a high-speed numerical code. Further, each stage of the tsunami – generation, propagation and inundation, has to be modeled and studied separately. Nonetheless, simplified models based on first principles are able to explain individual aspects of this very complex geophysical phenomenon without detailed numerical computations.

### **4. Model applications**

Viewed from space, the Earth is a watery planet with the oceans covering a full 71% of the surface area. The world ocean consists of three inter-connected oceans of the Pacific, Atlantic and Indian oceans, which comprise 51.5%, 25.6% and 22.9% of the water surface, respectively (we disregard 'Southern Ocean' as a separate entity). The Pacific ocean is thus larger than the two other oceans put together. It alone covers 34% or just over a full onethird of the Earth's surface and is comfortably larger than all the landmasses (at 29% of the Earth's surface) put together. The Pacific ocean is thought to be the remnant of 'Panthalassa', the world ocean, when all the landmasses were joined together as 'Pangaea'.

The Pacific ocean provides an ideal venue for tsunami propagation studies for several reasons. First, as stated above, it is the largest body of water, covering a full one-third of the globe. Second, it is bounded by active tectonic plate junctions, studded with volcanoes called the 'Ring of Fire'. Tsunamis produced at these hotspots can traverse the length and breadth of the ocean with relative ease. Third, there are no landmasses or large islands to block or interfere with the propagation of tsunamis formed in the ocean. Fourth, the ocean itself is dotted with small islands which pose little interference with tsunami propagation, but provide valuable platforms for recording tsunami wave amplitudes. Many of these islands are volcanic in origin and are sources of tsunamis themselves. Fifth, the longest stretch of ocean water is found between the Japan archipelago in the north-west and southern Chile on the south-east covering a distance of over 17,000 km or 85% of the distance between the North pole and the South Pole. At both the ends of this diameter lie some of the most active plate tectonic regions and tsunamis from either ends have traversed this favorite racetrack. Last but not least, an astonishing 80% of all tsunamis are recorded in the Pacific ocean.

With this geographical backdrop, we now proceed to study representative tsunami event to illustrate the validity of our propagation models. Model A, even though uni-dimensional, is a valuable tool for all tsunami events, as it correctly furnishes the velocity given the depth of the ocean, or vice-versa. It further predicts the travel times, which are vital for warning purposes. These results are independent of the direction, given the isotropy of space. Model A fails when the amplitude of the wave is to be studied, in which case Model B, C or D is called into consideration. In the following, we provide examples where one of the latter models, in conjuction with Model A, is used to analyze historic tsunami events. The data are taken from the National Oceanic and Atmospheric Administration website at www.ngdc.noaa.gov/hazard/tsu\_travel\_time\_events.shtml.

### **4.1. Hawaii tsunami of 1975**

112 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

amplitude would regain its original value at the anti-podal point.

sin

across the line source, one has (vide Appendix D):

in Appendix C. We have:

becomes infinite.

**4. Model applications**

The amplitude variation depends solely on the polar angle. Since the epicenter is considered to be located at the north pole (*θ* = 0), the wave amplitude gradually falls until it reaches the equator (*θ* = *π*/2), 10,000 km away, and then starts to rise again as the convergence effect due to the curvature of the Earth comes into play and finally regains its original value at the south pole (*θ* = *π*). Thus, if the Earth were entirely ocean, barring any losses, the wave

The comparison of wave amplitudes in Model C (true) and Model B (approximate) is given

*ψ θ*

Table C.1 shows that the difference in the two solutions is slight for small values of *θ*. Even at the equator (*θ* = *π*/2), the enhancement in amplitude due to curvature is only 25%. The enhancement becomes progressively greater until the south pole (*θ* = *π*) is reached, where it

When the tsunami is caused by an oceanic plate sliding under a continental plate, the subduction zone can be described by a finite line source instead of a localized point source. In that case, **Model D** is appropriate. In this model, the wave-fronts are ellipses with foci at the end-points of the line source and the energy propagates along con-focal hyperbolas (Appendix D). The wave amplitude near the source will be far different from that of a point source (Model B). If *c* is the length of the line source, then at distances *ρb* from the center

> 2 4 2

*b*

*ρ*

1

1

*ψ c*

The wave amplitude in Model D is finite at the origin as opposed to being infinite in Models B and C. However, away from the source, the solutions for Models B and D rapidly converge, as the wave-fronts become more circular. For *ρb* > *c*, the difference is below 10%,

Tsunamis are vast and highly complex geophysical phenomena, each having a character of its own. It is impossible to construct one model for any tsunami even with a high-speed numerical code. Further, each stage of the tsunami – generation, propagation and inundation, has to be modeled and studied separately. Nonetheless, simplified models based on first principles are able to explain individual aspects of this very complex

Viewed from space, the Earth is a watery planet with the oceans covering a full 71% of the surface area. The world ocean consists of three inter-connected oceans of the Pacific, Atlantic

*ModelD ModelB*

whereas for *ρb* > 10*c*, the two solutions are virtually indistinguishable (Table D.1).

geophysical phenomenon without detailed numerical computations.

*ψ*

*ψ θ* (19)

(20)

*ModelC ModelB*

> On 29 November 1975, a magnitude 7.2 earthquake occurred on the southern coast of the island of Hawaii with the epicenter at 19.3oN and 155.0oW at a focal depth of 8 km (cf. Pararas-Carayannis, 1976). The earthquake, the largest local one since 1868, generated a locally damaging submarine landslide tsunami which was recorded at 76 tide gauge stations in Alaska, California, Hawaii, Japan, Galapagos Islands, Peru and Chile. The tsunami caused \$1.5 million damage in Hawaii, 2 deaths and 19 injuries (Dudley & Lee, 1988). From the travel times registered, the tsunami reached Guadalupe Island, Mexico, 3864 km away in 5 h 9 m at an average speed of 750 kph, while it took 6 h 8 m to reach Tofino Island, Canada, 4210 km away, at the average speed of 686 kph. The slower speed in the first case is likely to

be due to the fact that the tsunami had to bend considerably before heading towards its destination (cf. Helene & Yamashita, 2006).

Fig. 1 is a scatter plot of the wave amplitude versus the propagation distance. Since the maximum distance was under 8000 km, the flat space approximation holds (vide Appendix C) and Model B is applicable. The variation of the wave amplitude in this model is given by Eq. (16):

$$
\psi\_{ModelB} = \frac{A}{\sqrt{\rho}}\tag{21}
$$

Tsunami Propagation Models Based on First Principles 115

*ρ* (29)

*βρ ψObserved ModelB e ψ* (30)

**Figure 1.** Observed and Model wave amplitudes of Hawaii tsunami of 1975.

variation of the wave amplitude with distance is then expressed as:

6137

**0**

**2**

**4**

**6**

**8**

**Wave Amplitude, m**

**10**

**12**

**14**

**16**

(vide Appendix B)

*A* is then obtained from Eq. (23). The results give: *A* = 22.644; and *α* = -.6137. The actual

**Model B**

**Observed**

**0 2000 4000 6000 8000 Distance from Epicenter, km**

. *ψObserved*

The actual wave amplitude falls off slightly faster than that predicted by Model B, which is based on conservation of energy. The difference may be assumed to represent the loss of energy due to as yet unidentified causes. The prime candidate appears to be the generation of atmospheric internal gravity waves by tsunamis, which can transport energy and momentum vertically through the atmosphere and produce travelling ionospheric disturbances (cf. Hickey, 2011). Assuming an exponential attenuation factor, we can write

22 644 .

where *A* is the amplitude constant to be determined by regression analysis. Summing over the data points, one obtains:

$$A = \frac{\sum \psi\_{ModelB}}{\sum \frac{1}{\sqrt{\rho}}} \tag{22}$$

The current tsunami data yield: *A* = 20.5265. The model equation is shown in Fig. 1. Also shown in the figure is the actual variation of the wave amplitude in accordance with the data. This is determined by first assuming a functional variation of the form

$$
\psi\_{observed} = A\rho^a \tag{23}
$$

where *A* and *α* are two constants to be determined from regression analysis. By taking natural logarithms of both sides first and then multiplying both sides by *ρ*, we get the two normal equations required to find the constants:

$$
\log \psi = \log A + a \log \rho \tag{24}
$$

and

$$
\rho \log \psi = \rho \log A + \alpha \rho \log \rho \tag{25}
$$

By summing Eqs. (24) and (25) over the *n* data points, we get:

$$
\sum \log \psi = n \log A + a \sum \log \rho \tag{26}
$$

and

$$
\sum \rho \log \psi = \log A \sum \rho + a \sum \rho \log \rho \tag{27}
$$

By eliminating *A* between Eqs. (26) and (27), we have

$$\alpha = \frac{\sum \rho \sum \log \psi - n \sum \rho \log \psi}{\sum \rho \sum \log \rho - n \sum \rho \log \rho} \tag{28}$$

**Figure 1.** Observed and Model wave amplitudes of Hawaii tsunami of 1975.

114 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

destination (cf. Helene & Yamashita, 2006).

1

normal equations required to find the constants:

By summing Eqs. (24) and (25) over the *n* data points, we get:

By eliminating *A* between Eqs. (26) and (27), we have

*α*

Eq. (16):

and

and

the data points, one obtains:

be due to the fact that the tsunami had to bend considerably before heading towards its

Fig. 1 is a scatter plot of the wave amplitude versus the propagation distance. Since the maximum distance was under 8000 km, the flat space approximation holds (vide Appendix C) and Model B is applicable. The variation of the wave amplitude in this model is given by

> *ModelB A ψ*

where *A* is the amplitude constant to be determined by regression analysis. Summing over

*ψModelB A*

The current tsunami data yield: *A* = 20.5265. The model equation is shown in Fig. 1. Also shown in the figure is the actual variation of the wave amplitude in accordance with the

*α ψobserved Aρ* (23)

where *A* and *α* are two constants to be determined from regression analysis.  This is determined by first assuming a functional variation of the form

*ρ*

*ρ* (21)

(22)

*A* is then obtained from Eq. (23). The results give: *A* = 22.644; and *α* = -.6137. The actual variation of the wave amplitude with distance is then expressed as:

$$
\psi\_{\text{Observed}} = \frac{22.644}{\rho^{6137}} \tag{29}
$$

The actual wave amplitude falls off slightly faster than that predicted by Model B, which is based on conservation of energy. The difference may be assumed to represent the loss of energy due to as yet unidentified causes. The prime candidate appears to be the generation of atmospheric internal gravity waves by tsunamis, which can transport energy and momentum vertically through the atmosphere and produce travelling ionospheric disturbances (cf. Hickey, 2011). Assuming an exponential attenuation factor, we can write (vide Appendix B)

$$
\psi\_{Observed} = e^{-\beta \rho} \psi\_{ModelB} \tag{30}
$$

The attenuation factor then follows as

$$e^{-\beta \rho} = \frac{\psi\_{Cbsored}}{\psi\_{ModelB}} = \rho^{\mathbf{S}-\alpha} \tag{31}$$

Tsunami Propagation Models Based on First Principles 117

translate to linear distances of 5,000 km, 10,000 km, and 15,000 km, respectively, from the epicenter. For constant propagation speeds, the circles of latitude define circular wavefronts. The 10,000 km great circle marks the 'equator' (*θ* = *π*/2), past which the waves begin to converge according to Model C. A tsunami propagating in this lune does not encounter any continental landmass until after a distance of 17,000 km, which is 85% of the distance

**Figure 2.** Propagation geometry of 2011 Japan tsunami in a lune of angle 90o with wavefronts at

south Atlantic Ocean, and barring losses, the original wave amplitude restored.

The 2011 Japan tsunami was felt throughout the Pacific Ocean. Wave amplitudes were recorded at over 293 stations scattered in and around the Pacific. Fig. 3 is a scatter plot of the wave amplitude versus the distance from the epicenter. The highest amplitudes were recorded near the epicenter. Amidst a considerable scatter, a well-defined trend in the wave amplitudes emerges from the figure. The wave amplitude diminished rapidly as a function of the distance from the source, becoming nearly constant around the 10,000 km mark, and showing a discernible rise thereafter in accord with Model C. But for the intervention of the South American landmass, the waves would have converged at the anti-podal point in

between the poles.

(adapted from worldatlas.com).

intervals of 5,000 km

The attenuation factor and the attenuation coefficient are calculated for this event as functions of the distance from the epicenter and entered in Table 1. It shows that the amplitude loss is most rapid near the epicenter with almost 50% of it dissipated in the first 400 km. The dissipation slows down dramatically thereafter and at 10,000 km, it is increases only to 65%. Consequently, the attenuation coefficient is not a constant but a function of the distance from the epicenter. By all accounts, the attenuation is small, suggesting the validity of models A and B.

**Table 1.** Attenuation factor and attenuation coefficient as functions of distance from epicenter

### **4.2. The Great Japan tsunami of 2011**

The Japan archipelago is one of the most earthquake-prone regions of the world. On 11 March 2011, a 9.0 magnitude earthquake struck on the east coast of Honshu. The epicenter was at 38.322oN latitude and 142.369oE longitude, 72 km east of Oshika peninsula with the hypocenter at a depth of 32 km below sea level (cf. http://www.tsunamiresearchcenter. com/news/earthquake-and-tsunami-strikes-japan; http://itic.ioc-unesco.org/index.php). It was the greatest earthquake to strike Japan and one of the greatest in recorded history. It was comparable to the 2004 Indian Ocean earthquake. There were an estimated 16,000 deaths, 27,000 injured and 3,000 missing (http://www.npa.go.jp/archive/keibi/ higaijokya\_e.pdf) with total property damage of \$235 billion (according to World Bank reports), making it the costliest natural disaster of all time.

Fig. 2 provides a geometrical perspective of the 2011 Japan tsunami. The geodesic lines from the epicenter shown in the figure are great circles with a longitudinal separation of 90o, which define a 'lune' that covers one quarter of the Earth's surface area. Intersecting the great circles are 'circles of latitude' at angular distances of *θ* = *π*/4, *π*/2 and 3*π*/4 which translate to linear distances of 5,000 km, 10,000 km, and 15,000 km, respectively, from the epicenter. For constant propagation speeds, the circles of latitude define circular wavefronts. The 10,000 km great circle marks the 'equator' (*θ* = *π*/2), past which the waves begin to converge according to Model C. A tsunami propagating in this lune does not encounter any continental landmass until after a distance of 17,000 km, which is 85% of the distance between the poles.

(adapted from worldatlas.com).

116 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

*βρ Observed* .5 *α*

*ModelB ψ e ρ ψ*

The attenuation factor and the attenuation coefficient are calculated for this event as functions of the distance from the epicenter and entered in Table 1. It shows that the amplitude loss is most rapid near the epicenter with almost 50% of it dissipated in the first 400 km. The dissipation slows down dramatically thereafter and at 10,000 km, it is increases only to 65%. Consequently, the attenuation coefficient is not a constant but a function of the distance from the epicenter. By all accounts, the attenuation is small, suggesting the validity

Distance from epicenter *ρ* Attenuation factor *ρ*-.1137 Attenuation coefficient *β* 200 km .5475 .00301/km 400 km .5060 .00170/km 600 km .4832 .00121/km 800 km .4676 .000950/km 1,000 km .4559 .000785/km 2,000 km .4214 .000432/km 4,000 km .3894 .000236/km 6,000 km .3719 .000165/km 8,000 km .3599 .000128/km 10,000 km .3509 .000105/km

**Table 1.** Attenuation factor and attenuation coefficient as functions of distance from epicenter

The Japan archipelago is one of the most earthquake-prone regions of the world. On 11 March 2011, a 9.0 magnitude earthquake struck on the east coast of Honshu. The epicenter was at 38.322oN latitude and 142.369oE longitude, 72 km east of Oshika peninsula with the hypocenter at a depth of 32 km below sea level (cf. http://www.tsunamiresearchcenter. com/news/earthquake-and-tsunami-strikes-japan; http://itic.ioc-unesco.org/index.php). It was the greatest earthquake to strike Japan and one of the greatest in recorded history. It was comparable to the 2004 Indian Ocean earthquake. There were an estimated 16,000 deaths, 27,000 injured and 3,000 missing (http://www.npa.go.jp/archive/keibi/ higaijokya\_e.pdf) with total property damage of \$235 billion (according to World Bank

Fig. 2 provides a geometrical perspective of the 2011 Japan tsunami. The geodesic lines from the epicenter shown in the figure are great circles with a longitudinal separation of 90o, which define a 'lune' that covers one quarter of the Earth's surface area. Intersecting the great circles are 'circles of latitude' at angular distances of *θ* = *π*/4, *π*/2 and 3*π*/4 which

(31)

The attenuation factor then follows as

**4.2. The Great Japan tsunami of 2011**

reports), making it the costliest natural disaster of all time.

of models A and B.

**Figure 2.** Propagation geometry of 2011 Japan tsunami in a lune of angle 90o with wavefronts at intervals of 5,000 km

The 2011 Japan tsunami was felt throughout the Pacific Ocean. Wave amplitudes were recorded at over 293 stations scattered in and around the Pacific. Fig. 3 is a scatter plot of the wave amplitude versus the distance from the epicenter. The highest amplitudes were recorded near the epicenter. Amidst a considerable scatter, a well-defined trend in the wave amplitudes emerges from the figure. The wave amplitude diminished rapidly as a function of the distance from the source, becoming nearly constant around the 10,000 km mark, and showing a discernible rise thereafter in accord with Model C. But for the intervention of the South American landmass, the waves would have converged at the anti-podal point in south Atlantic Ocean, and barring losses, the original wave amplitude restored.

**Figure 3.** Observed wave amplitudes of 2011 Japan tsunami with least-squares error lines according to Models B and C**.**

The variation of the wave amplitude according to Model C can be written as [from Eq. (18)]:

$$
\psi\_{ModelC} = \frac{A}{\sqrt{\sin \theta}} \tag{32}
$$

Tsunami Propagation Models Based on First Principles 119

*n* (34)

*αρ* (35)

*g* (36)

2 *ψ αρ ModelC* sin

giving *A* = .4896 m. The least-square error line according to Model C is then given by Eqs.

sin *ModelB ModelC αρ ψ ψ*

The regression lines according to Models B and C are superimposed on the data points in Fig. 3. As is shown in Appendix C, the difference between the two wave amplitudes is slight up to about the equator mark (10,000 km), after which the two curves begin to diverge. The predicted amplitude in Model B continues to fall according to Eq. (21), whereas that in Model C begins to rise according to Eqs. (32) and (33). The actual data clearly supports Model C and validates the convergence effect of the Earth's curvature on the wave amplitude. The same effect was earlier observed in the 8.3 magnitude Kuril islands

Finally, Eq. (9) of Model A furnishes a means to determine the average depth of the ocean

2 *v d*

From the observed travel times and the geodesic distances from the epicenter to 13 destinations along the coast of Chile, the average travel speeds were calculated from which the average depth of the ocean along the travel path determined (Table 2). The mean average speed of 739 kph yielded a mean average depth of 4303 m for the Pacific Ocean along these paths, which compares favorably with various estimates found in the literature: e.g., 4282 m (Herring & Clarke, 1971), 4190 m (Smith & Demopoulos, 2003), 4267 m

The average depth of the Pacific Ocean (4300 m) is considerably grater than those of the Atlantic Ocean (3600 m) and Indian Ocean (3500 m) (cf. Herring & Clarke, 1971). Further, the smaller north-western half of the Pacific Ocean is substantially deeper than the larger south-eastern remainder, even though separate depth figures are hard to find in the literature. In order to estimate the average depth of north-western Pacific Ocean, we consider the travel times of the tsunami to reach various destinations on the coasts of the Hawaiian islands, which lie entirely in that region (Table 3). Travel time data from the epicenter to 8 destinations in the Hawaiian islands yield a mean average speed of 798 kph for a mean average depth of 5016 m for north-western Pacific Ocean along these paths. This confirms the fact that north-western Pacific Ocean is considerably deeper than the southeastern remainder. These travel time studies further re-affirm the validity of the 'long wave

along the travel path given the distance and the travel time to the destination:

(http://oceanservice.noaa.gov) , and 4080 m (britannica online encyclopedia).

in shallow water' approximation for tsunami propagation.

*A*

(33) and (34). The same in Model B is obtained by Eq. (19):

earthquake of 2006 (Tan & Lyatskaya, 2009).

where *A* is a constant to be determined from the data. In terms of the radial distance *ρ*, we have

$$
\psi\_{ModelC} = \frac{A}{\sqrt{a \sin \rho}}\tag{33}
$$

where *ρ* is in kilometers and *α* = .009 deg/km or .00015708/km if *θ* is expressed in radian measure.

The constant *A* is determined from a single normal equation obtained by summing over the *n* observed data points

Tsunami Propagation Models Based on First Principles 119

$$A = \sqrt{\frac{\sum \psi\_{Model} \,^2 \sin \alpha \rho}{n}} \tag{34}$$

giving *A* = .4896 m. The least-square error line according to Model C is then given by Eqs. (33) and (34). The same in Model B is obtained by Eq. (19):

118 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

**Figure 3.** Observed wave amplitudes of 2011 Japan tsunami with least-squares error lines according to

0 5000 10000 15000 20000 **Distance from Epicenter, km**

**Model B**

The variation of the wave amplitude according to Model C can be written as [from Eq. (18)]:

sin *ModelC*

where *A* is a constant to be determined from the data. In terms of the radial distance *ρ*, we

*A ψ*

where *ρ* is in kilometers and *α* = .009 deg/km or .00015708/km if *θ* is expressed in radian

The constant *A* is determined from a single normal equation obtained by summing over the

sin *ModelC*

0

2

4

6

8

**Wave Amplitude, m**

10

12

14

16

*A ψ θ* (32)

**Model C**

*α ρ* (33)

Models B and C**.**

have

measure.

*n* observed data points

$$
\psi\_{ModelB} = \sqrt{\frac{\sin \alpha \rho}{\alpha \rho}} \psi\_{ModelC} \tag{35}
$$

The regression lines according to Models B and C are superimposed on the data points in Fig. 3. As is shown in Appendix C, the difference between the two wave amplitudes is slight up to about the equator mark (10,000 km), after which the two curves begin to diverge. The predicted amplitude in Model B continues to fall according to Eq. (21), whereas that in Model C begins to rise according to Eqs. (32) and (33). The actual data clearly supports Model C and validates the convergence effect of the Earth's curvature on the wave amplitude. The same effect was earlier observed in the 8.3 magnitude Kuril islands earthquake of 2006 (Tan & Lyatskaya, 2009).

Finally, Eq. (9) of Model A furnishes a means to determine the average depth of the ocean along the travel path given the distance and the travel time to the destination:

$$d = \frac{v^2}{g} \tag{36}$$

From the observed travel times and the geodesic distances from the epicenter to 13 destinations along the coast of Chile, the average travel speeds were calculated from which the average depth of the ocean along the travel path determined (Table 2). The mean average speed of 739 kph yielded a mean average depth of 4303 m for the Pacific Ocean along these paths, which compares favorably with various estimates found in the literature: e.g., 4282 m (Herring & Clarke, 1971), 4190 m (Smith & Demopoulos, 2003), 4267 m (http://oceanservice.noaa.gov) , and 4080 m (britannica online encyclopedia).

The average depth of the Pacific Ocean (4300 m) is considerably grater than those of the Atlantic Ocean (3600 m) and Indian Ocean (3500 m) (cf. Herring & Clarke, 1971). Further, the smaller north-western half of the Pacific Ocean is substantially deeper than the larger south-eastern remainder, even though separate depth figures are hard to find in the literature. In order to estimate the average depth of north-western Pacific Ocean, we consider the travel times of the tsunami to reach various destinations on the coasts of the Hawaiian islands, which lie entirely in that region (Table 3). Travel time data from the epicenter to 8 destinations in the Hawaiian islands yield a mean average speed of 798 kph for a mean average depth of 5016 m for north-western Pacific Ocean along these paths. This confirms the fact that north-western Pacific Ocean is considerably deeper than the southeastern remainder. These travel time studies further re-affirm the validity of the 'long wave in shallow water' approximation for tsunami propagation.

120 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

Tsunami Propagation Models Based on First Principles 121

(http://neic.usgs.gov/neis/eq\_depot/world/1960\_05\_22articles.html). The resulting tsunami raced across the Pacific Ocean causing the death of 61 people in Hawaii and 200 others in Japan and elsewhere (USGS reports). The estimated damage costs were near half a billion

**Figure 4.** Propagation geometry of 1960 Chilean tsunami in a lune of angle 90o with wavefronts at

(http://www.soest.hawaii.edu/GG/ASK/chile-tsunami.html).

The Great Chilean tsunami of 1960 is similar to the Great Japan tsunami of 2011 coming from the opposite direction, only having greater amplitude. Both of these tsunamis, as well as many other analogous ones, traversed the longest stretch of continuous water covering over 85% of the distance between the poles. Thus the amplitude variations with distance of both of these tsunamis were similar. The strength of the Great Chilean tsunami was such that reflected waves from the Asian coasts were detectable

Fig. 4 provides the geometrical perspective of the 1960 Chilean tsunami. As in the earlier example, the geodesic lines from the epicenter shown in the figure are great circles with a longitudinal separation of 90o, which define a 'lune' that covers one quarter of the Earth's surface area. Intersecting the great circles are 'circles of latitude' at angular distances of *θ* = *π*/4, *π*/2 and 3*π*/4 which translate to linear distances of 5,000 km, 10,000 km, and 15,000 km,

dollars.

(adapted from worldatlas.com).

intervals of 5,000 km

**Table 2.** Travel Times, Average Speed and Average Depth of Pacific Ocean.

**Table 3.** Travel Times, Average Speed and Average Depth of north-western Pacific Ocean.

### **4.3. The Great Chilean tsunami of 1960**

On Sunday, May 22, 1960, at 19:11 GMT (15.11 LT), a super-massive earthquake occurred off the coast of south central Chile, with epicenter at 39.5oS latitude, 74.5oW longitude and focal depth of 33 km (cf. http://earthquake.usgs.gov/earthquakes/world/events/ 1960\_05\_22\_tsunami.php). It happened when a piece of the Nazca Plate of the Pacific Ocean subducted beneath the South American Plate. The magnitude of the earthquake of 9.5 makes it the most powerful earthquake in recorded history (Kanamori, 2010). The tsunami generated by the earthquake, along with coastal subsidence and flooding, caused tremendous damage along the Chilean coast where an estimated 2,000 people lost their lives (http://neic.usgs.gov/neis/eq\_depot/world/1960\_05\_22articles.html). The resulting tsunami raced across the Pacific Ocean causing the death of 61 people in Hawaii and 200 others in Japan and elsewhere (USGS reports). The estimated damage costs were near half a billion dollars.

(adapted from worldatlas.com).

120 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

18.467oS Arica 16,166 km 21 h 26 m 754 kph

Distance from Epicenter

**Table 2.** Travel Times, Average Speed and Average Depth of Pacific Ocean.

 Travel Times, Average Speed and Average Depth of north-western Pacific Ocean.

On Sunday, May 22, 1960, at 19:11 GMT (15.11 LT), a super-massive earthquake occurred off the coast of south central Chile, with epicenter at 39.5oS latitude, 74.5oW longitude and focal depth of 33 km (cf. http://earthquake.usgs.gov/earthquakes/world/events/ 1960\_05\_22\_tsunami.php). It happened when a piece of the Nazca Plate of the Pacific Ocean subducted beneath the South American Plate. The magnitude of the earthquake of 9.5 makes it the most powerful earthquake in recorded history (Kanamori, 2010).  The Great Chilean tsunami of 1960**

Destination Destination

**Figure 4.** Propagation geometry of 1960 Chilean tsunami in a lune of angle 90o with wavefronts at intervals of 5,000 km

The Great Chilean tsunami of 1960 is similar to the Great Japan tsunami of 2011 coming from the opposite direction, only having greater amplitude. Both of these tsunamis, as well as many other analogous ones, traversed the longest stretch of continuous water covering over 85% of the distance between the poles. Thus the amplitude variations with distance of both of these tsunamis were similar. The strength of the Great Chilean tsunami was such that reflected waves from the Asian coasts were detectable (http://www.soest.hawaii.edu/GG/ASK/chile-tsunami.html).

Fig. 4 provides the geometrical perspective of the 1960 Chilean tsunami. As in the earlier example, the geodesic lines from the epicenter shown in the figure are great circles with a longitudinal separation of 90o, which define a 'lune' that covers one quarter of the Earth's surface area. Intersecting the great circles are 'circles of latitude' at angular distances of *θ* = *π*/4, *π*/2 and 3*π*/4 which translate to linear distances of 5,000 km, 10,000 km, and 15,000 km, respectively, from the epicenter. For constant propagation speeds, the circles of latitude define circular wavefronts. The 10,000 km great circle marks the 'equator' (*θ* = *π*/2), past which the waves begin to converge according to Model C.

Tsunami Propagation Models Based on First Principles 123

indicates that the energy density of the 1960 Chilean tsunami was 15.84 times that of the 2011 Japan tsunami, thus illustrating the power of the most powerful earthquake in recorded history. Normally, a 9.5 magnitude earthquake releases about 5.5 times the energy

From the observed travel times and distances from the epicenter to destinations on the eastern coast of Japan, the average travel speeds were calculated and the average depth of the ocean determined along these paths (table 4). The mean average depth of 4,231 m compares favorably with the value of 4,303 m obtained from the 2011 Japan tsunami (Table 2). Also calculated were the travel speeds from the epicenter to locations on the Hawaiian Islands and the average depth of the ocean (Table 5).  Travel Times, Average Speed and Average Depth of south-eastern Pacific Ocean

The great 2004 Indian Ocean earthquake occurred off the west coast of Sumatra, Indonesia, on Boxing Day, December 26, 2004. Its revised magnitude of 9.2 makes it the second largest earthquake in recorded history, after the 9.5 magnitude Chilean earthquake of 1960 (cf. http://walrus.usgs.gov/tsunami/sumatraEQ/; Lay, et al., 2005). In terms of human casualties, however, it was the greatest natural disaster in recorded history, by far.

Distance from Epicenter

41.783oN Hakodate 17,058 km 23 h 27 m 727 kph

**Table 4.** Travel Times, Average Speed and Average Depth of Pacific Ocean

Distance from Epicenter

19.730oN Coconut I. 10,621 km 15 h 30 m 685 kph

**4.4. The Great Indian Ocean tsunami of 2004**

Destination

Destination

Latitude of Destinatio n

Latitude of Destinatio n

The Great Chilean tsunami of 1960 was felt throughout the Pacific Ocean. Over 1,000 measurements of wave amplitudes were recorded at 815 stations scattered in and around the Pacific. Fig. 5 is a scatter plot of the wave amplitude versus the distance from the epicenter. The highest amplitudes were recorded near the Chilean coast and at the diametrically opposite end, mostly on the Japanese coasts, where hundreds of data points were clustered. There is a second cluster past the 10,000 km mark at the Hawaiian Islands, where numerous measurements were taken. The over-all trend of the data points closely agrees with that predicted by the 'spherical ocean' Model C.

**Figure 5.** Observed wave amplitudes of 1960 Chilean tsunami with least-squares error lines according to Models B and C**.**

The least-squares regression lines of the data points as obtained from Eqs. (32) - (35) are shown in Fig. 5, with the amplitude constant *A* = 1.9488 m. This is 3.98 times that of the 2011 Japan tsunami constant. Since the energy density of the waves is proportional to *A*2, this indicates that the energy density of the 1960 Chilean tsunami was 15.84 times that of the 2011 Japan tsunami, thus illustrating the power of the most powerful earthquake in recorded history. Normally, a 9.5 magnitude earthquake releases about 5.5 times the energy an a 9.0 earthquake (cf. Kanamori, 1977).

From the observed travel times and distances from the epicenter to destinations on the eastern coast of Japan, the average travel speeds were calculated and the average depth of the ocean determined along these paths (table 4). The mean average depth of 4,231 m compares favorably with the value of 4,303 m obtained from the 2011 Japan tsunami (Table 2). Also calculated were the travel speeds from the epicenter to locations on the Hawaiian Islands and the average depth of the ocean (Table 5). The mean average depth of 3,972 m reaffirms the fact that the north-western Pacific Ocean, at 5,016 m (Table 3) is substantially deeper than its south-eastern compliment.

**Table 4.** Travel Times, Average Speed and Average Depth of Pacific Ocean

122 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

which the waves begin to converge according to Model C.

agrees with that predicted by the 'spherical ocean' Model C.

respectively, from the epicenter. For constant propagation speeds, the circles of latitude define circular wavefronts. The 10,000 km great circle marks the 'equator' (*θ* = *π*/2), past

The Great Chilean tsunami of 1960 was felt throughout the Pacific Ocean. Over 1,000 measurements of wave amplitudes were recorded at 815 stations scattered in and around the Pacific. Fig. 5 is a scatter plot of the wave amplitude versus the distance from the epicenter. The highest amplitudes were recorded near the Chilean coast and at the diametrically opposite end, mostly on the Japanese coasts, where hundreds of data points were clustered. There is a second cluster past the 10,000 km mark at the Hawaiian Islands, where numerous measurements were taken. The over-all trend of the data points closely

**Figure 5.** Observed wave amplitudes of 1960 Chilean tsunami with least-squares error lines according

0 5000 10000 15000 20000 **Distance from Epicenter, km**

**Model B**

**Model C**

The least-squares regression lines of the data points as obtained from Eqs. (32) - (35) are shown in Fig. 5, with the amplitude constant *A* = 1.9488 m. This is 3.98 times that of the 2011 Japan tsunami constant. Since the energy density of the waves is proportional to *A*2, this

to Models B and C**.**

0

10

20

30

**Wave Amplitude, m**

40

50

60

**Table 5.** Travel Times, Average Speed and Average Depth of south-eastern Pacific Ocean

### **4.4. The Great Indian Ocean tsunami of 2004**

The great 2004 Indian Ocean earthquake occurred off the west coast of Sumatra, Indonesia, on Boxing Day, December 26, 2004. Its revised magnitude of 9.2 makes it the second largest earthquake in recorded history, after the 9.5 magnitude Chilean earthquake of 1960 (cf. http://walrus.usgs.gov/tsunami/sumatraEQ/; Lay, et al., 2005). In terms of human casualties, however, it was the greatest natural disaster in recorded history, by far. The earthquake generated a super-massive tsunami that took the lives of an estimated 230,000

people in Indonesia, Sri Lanka, India, Thailand and elsewhere (Mörner, 2010). More than 1,000,000 people were displaced in the aftermath following the tsunami, which was eventually registered at every coast of the world ocean.

Tsunami Propagation Models Based on First Principles 125

Africa, and entering the Atlantic Ocean. The northern segment of the fault-line (henceforth called the Andaman-Nicobar fault-line) produced tsunami which affected greater areas of landmass. Phuket lied near the perpendicular bisector of this fault and took a direct hit from the tsunami as did other coastal locations of Thailand and Myanmar. To the west, the east coasts of Sri Lanka and southern India were greatly affected. The tsunami rolled over the Maldive Islands and reached the eastern coast of Africa (Somalia, in particular), causing damage there. In consolation, Bangladesh, a densely populated area to the north, was

**Figure 7.** Observed and Model wave amplitudes of 2004 Indian Ocean earthquake due to the

**Model D**

**Sri Lanka**

**Maldives**

**Model B**

**India**

Assuming that the tsunami from the Sumatra fault-line was largely intercepted by the island (vide Fig. 6), we proceed to analyze the tsunami propagation from the Andaman-Nicobar fault-line based on Models B and D. For this, we have to consider the mid-point of the faultline (approximately 9oN latitude and 92.5oE longitude) as the origin of the tsunami instead of the epicenter which lay on the Sumatra fault-line. Distances are now reckoned from this new center. If the coordinates (latitude, longitude) of the source and destination points be

0 1000 2000 3000 4000 5000 6000 **Distance from Center, km**

**Seychelles**

**Somalia**

spared the devastation.

Andaman-Nicobar fault-line.

0

**Andaman**

5

10

15

**Thailand**

**Wave Amplitude, m**

20

25

30

**Figure 6.** Propagation geometry of 2004 Indian Ocean tsunami showing the two major faultiness and propagation along their perpendicular bisectors (adapted from phuket.news.com).

This great earthquake was caused by the subduction of the Indo-Australian plate under the Eurasian plate near the Andaman and Nicobar Islands chain and its extension southwards under the Bay of Bengal (Fig. 6). Its hypocenter is listed at 3.295oN altitude and 95.982oW longitude. However, its fault-line was 1000 km long, which roughly consisted of two linear segments (cf. Kowalik, et al., 2005): (1) a 700 km long section off the west coasts of Andaman and Nicobar Islands; and (2) a 300 km section west of Aceh province of Sumatra, Indonesia (Fig. 6). The alignments of both the segments were generally north-south, with the southern segment titled slightly towards the south-easterly direction. The shorter southern segment had the more intense earthquake and generated the greater tsunami. Consequently, the epicenter lied on this segment of the fault-line. The earthquake is also variously referred to as the Sumatra-Andaman earthquake or the Boxing Day earthquake.

It is evident from Fig. 6 that the eastward tsunami from the southern segment of the faultline (henceforth referred to as the Sumatra fault-line) had a direct impact on the Aceh province in northern Sumatra, where the highest waves of over 50 m were registered. More than half of all casualties were reported there. The westward tsunami from this fault-line, on the other hand, passed harmlessly over the open ocean, reaching the east coast of South

Africa, and entering the Atlantic Ocean. The northern segment of the fault-line (henceforth called the Andaman-Nicobar fault-line) produced tsunami which affected greater areas of landmass. Phuket lied near the perpendicular bisector of this fault and took a direct hit from the tsunami as did other coastal locations of Thailand and Myanmar. To the west, the east coasts of Sri Lanka and southern India were greatly affected. The tsunami rolled over the Maldive Islands and reached the eastern coast of Africa (Somalia, in particular), causing damage there. In consolation, Bangladesh, a densely populated area to the north, was spared the devastation.

124 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

eventually registered at every coast of the world ocean.

people in Indonesia, Sri Lanka, India, Thailand and elsewhere (Mörner, 2010). More than 1,000,000 people were displaced in the aftermath following the tsunami, which was

**Figure 6.** Propagation geometry of 2004 Indian Ocean tsunami showing the two major faultiness and

This great earthquake was caused by the subduction of the Indo-Australian plate under the Eurasian plate near the Andaman and Nicobar Islands chain and its extension southwards under the Bay of Bengal (Fig. 6). Its hypocenter is listed at 3.295oN altitude and 95.982oW longitude. However, its fault-line was 1000 km long, which roughly consisted of two linear segments (cf. Kowalik, et al., 2005): (1) a 700 km long section off the west coasts of Andaman and Nicobar Islands; and (2) a 300 km section west of Aceh province of Sumatra, Indonesia (Fig. 6). The alignments of both the segments were generally north-south, with the southern segment titled slightly towards the south-easterly direction. The shorter southern segment had the more intense earthquake and generated the greater tsunami. Consequently, the epicenter lied on this segment of the fault-line. The earthquake is also variously referred to

It is evident from Fig. 6 that the eastward tsunami from the southern segment of the faultline (henceforth referred to as the Sumatra fault-line) had a direct impact on the Aceh province in northern Sumatra, where the highest waves of over 50 m were registered. More than half of all casualties were reported there. The westward tsunami from this fault-line, on the other hand, passed harmlessly over the open ocean, reaching the east coast of South

propagation along their perpendicular bisectors (adapted from phuket.news.com).

as the Sumatra-Andaman earthquake or the Boxing Day earthquake.

**Figure 7.** Observed and Model wave amplitudes of 2004 Indian Ocean earthquake due to the Andaman-Nicobar fault-line.

Assuming that the tsunami from the Sumatra fault-line was largely intercepted by the island (vide Fig. 6), we proceed to analyze the tsunami propagation from the Andaman-Nicobar fault-line based on Models B and D. For this, we have to consider the mid-point of the faultline (approximately 9oN latitude and 92.5oE longitude) as the origin of the tsunami instead of the epicenter which lay on the Sumatra fault-line. Distances are now reckoned from this new center. If the coordinates (latitude, longitude) of the source and destination points be

(*λs*, *φs*) and (*λd*, *φd*), respectively, the angle subtended at the center of the Earth by the two locations is, from spherical trigonometry,

$$\theta = \cos^{-1}\left[\sin\lambda\_s \sin\lambda\_d + \cos\lambda\_s \cos\lambda\_d \cos(\phi\_d - \phi\_s)\right] \tag{37}$$

Tsunami Propagation Models Based on First Principles 127

Average Depth

3,795 m

Mean Average Depth

3,840 m

speed of the tsunami was calculated, and the average depth of the Indian Ocean determined (Table 6). The average speed of 700 kph translates to an average depth of 3,840 m which is quite consistent with the reference figure of 3,963 m found in the literature (Herring & Clarke, 1971). Once again, the tsunami speed and depth of the ocean predicted by the 'long

> Average Speed

Mean Average Speed

700 kph

Travel Time

33.027oS East London 8,193 km 11 h 29 m 713 kph 4,008 m 33.958oS Pt. Elizabeth 8,423 km 12 h 13 m 698 kph 3,743 m 34.178oS Mossel Bay 8,743 km 13 h 02 m 671 kph 3,543 m 34.188oS Simons Bay 9,078 km 12 h 53 m 705 kph 3,909 m 69.007oS 39.584oE 9,088 km 12 h 41 m 717 kph 4,042 m

Tsunamis are complex geophysical phenomena not easily amenable to theoretical rendering. The formal theory based on solitary wave model is not easily accessible to the great majority of the scientific readers. This chapter has demonstrated that the 'long waves in shallow water' approximation of the tsunami explains many facets of tsunami propagation in the open ocean. The one-dimensional model provides accurate assessments of the general properties such as dispersion-less propagation, speed of propagation, bending of tsunamis around obstacles and depth of the ocean, among others. Two-dimensional models on flat and spherical ocean substantially account for the wave amplitudes for far-reaching tsunami propagation. Finally, the finite line-source model satisfactorily predicts the wave amplitudes

A wave is a transfer of energy from one part of a medium to another, the medium itself not being transported in which process (e.g., Coulson, 1955). The individual particles of the medium execute simple harmonic motions in one or two dimensions, depending upon the nature of the wave. For sound waves in air, the oscillations are parallel with the direction of propagation. Such waves are called *longitudinal waves*. For waves along a stretched string, the oscillations are perpendicular to the direction of propagation. Such waves are called *transverse waves*. Longitudinal waves can propagate through solids and fluid media (liquid or gas), whereas transverse waves can propagate through solids only. Longitudinal waves

**Table 6.** Travel Times, Average Speed and Average Depth of Central Indian Ocean

near the source when the tsunami is caused by a long subduction zone.

**A. Propagation of water waves in one dimension**

waves in shallow water' model turns out to be reliable.

Distance from Epicenter

28.800oS Richards Bay 7,683 km 11 h 04 m 694 kph

Destination

Latitude of Destinatio n

**5. Conclusion**

**Appendix**

The linear distance between the two points on the surface of the Earth is then *e d θr* , where *re* = 6371 km, is the volumetric radius of the Earth prescribed by the International Union of Geodesy and Geophysics.

Fig. 7 is a scatter plot of the wave amplitudes as functions of the distance from the center of the Andaman-Nicobar fault-line. The data betray distinct clumps for Thailand, Sri Lanka, India and Maldives. At the far end are data for the Somalia coast. Also shown are the isolated data points for Andaman and Seychelles islands. Superimposed on the data points are the model amplitudes according to Models B and D. The model amplitudes are virtually indistinguishable for distances upwards of 1000 km (cf. Appendix D). In comparison with the Hawaii and Japan earthquakes (Figs. 1 and 3), the wave amplitudes are significantly higher, which is indicative of the magnitude of the earthquake. Unlike Hawaii and Japan earthquakes, however, there is a dearth of data points with high amplitudes near the origin. This is because the earthquake occurred beneath the ocean and there was no land close to it. Nevertheless, the lack of high wave amplitudes close to the fault-line appears to support the validity of Model D.

As before, the wave amplitudes according to Model B is given by (vide Eq. D.12):

$$
\psi\_{ModelB} = \frac{A}{\sqrt{\rho\_b}} \tag{38}
$$

where *A* is a constant to be determined from regression analysis of the data points. By summing over the data, we have

$$A = \frac{\sum \psi}{\sum \frac{1}{\sqrt{\rho\_b}}} \tag{39}$$

The data yield a value of *A* = 146.296 m. The wave amplitudes in accordance with Model D then follows (vide Eq. D.8):

$$
\psi\_{ModelID} = \frac{\psi\_{ModelB}}{\sqrt{1 + \frac{c^2}{2\rho\_b^2}}} \tag{40}
$$

Here *c* = 350 km.

From the travel times and the travel distances from the epicenter on the Sumatra fault-line to five destinations on the South African coast and one on the Antarctic coast, the average speed of the tsunami was calculated, and the average depth of the Indian Ocean determined (Table 6). The average speed of 700 kph translates to an average depth of 3,840 m which is quite consistent with the reference figure of 3,963 m found in the literature (Herring & Clarke, 1971). Once again, the tsunami speed and depth of the ocean predicted by the 'long waves in shallow water' model turns out to be reliable.

**Table 6.** Travel Times, Average Speed and Average Depth of Central Indian Ocean

### **5. Conclusion**

126 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

locations is, from spherical trigonometry,

*ModelB*

1

2

summing over the data, we have

then follows (vide Eq. D.8):

Here *c* = 350 km.

Geodesy and Geophysics.

(*λs*, *φs*) and (*λd*, *φd*), respectively, the angle subtended at the center of the Earth by the two

1 cos sin sin cos cos cos *s d s d ds θ λ λ λ λ φφ* (37)

The linear distance between the two points on the surface of the Earth is then *e d θr* , where *re* = 6371 km, is the volumetric radius of the Earth prescribed by the International Union of

Fig. 7 is a scatter plot of the wave amplitudes as functions of the distance from the center of the Andaman-Nicobar fault-line. The data betray distinct clumps for Thailand, Sri Lanka, India and Maldives. At the far end are data for the Somalia coast. Also shown are the isolated data points for Andaman and Seychelles islands. Superimposed on the data points are the model amplitudes according to Models B and D. The model amplitudes are virtually indistinguishable for distances upwards of 1000 km (cf. Appendix D). In comparison with the Hawaii and Japan earthquakes (Figs. 1 and 3), the wave amplitudes are significantly higher, which is indicative of the magnitude of the earthquake. Unlike Hawaii and Japan earthquakes, however, there is a dearth of data points with high amplitudes near the origin. This is because the earthquake occurred beneath the ocean and there was no land close to it. Nevertheless, the lack of high wave amplitudes close to the fault-line appears to support the validity of Model D.

As before, the wave amplitudes according to Model B is given by (vide Eq. D.12):

*A ψ*

where *A* is a constant to be determined from regression analysis of the data points. By

*ψ A*

The data yield a value of *A* = 146.296 m. The wave amplitudes in accordance with Model D

*ModelD*

*ψ ψ*

From the travel times and the travel distances from the epicenter on the Sumatra fault-line to five destinations on the South African coast and one on the Antarctic coast, the average

*b*

*b*

4 2 1 2

*b*

*ModelB*

*c ρ*

*ρ*

*ρ* (38)

(39)

(40)

Tsunamis are complex geophysical phenomena not easily amenable to theoretical rendering. The formal theory based on solitary wave model is not easily accessible to the great majority of the scientific readers. This chapter has demonstrated that the 'long waves in shallow water' approximation of the tsunami explains many facets of tsunami propagation in the open ocean. The one-dimensional model provides accurate assessments of the general properties such as dispersion-less propagation, speed of propagation, bending of tsunamis around obstacles and depth of the ocean, among others. Two-dimensional models on flat and spherical ocean substantially account for the wave amplitudes for far-reaching tsunami propagation. Finally, the finite line-source model satisfactorily predicts the wave amplitudes near the source when the tsunami is caused by a long subduction zone.

### **Appendix**

### **A. Propagation of water waves in one dimension**

A wave is a transfer of energy from one part of a medium to another, the medium itself not being transported in which process (e.g., Coulson, 1955). The individual particles of the medium execute simple harmonic motions in one or two dimensions, depending upon the nature of the wave. For sound waves in air, the oscillations are parallel with the direction of propagation. Such waves are called *longitudinal waves*. For waves along a stretched string, the oscillations are perpendicular to the direction of propagation. Such waves are called *transverse waves*. Longitudinal waves can propagate through solids and fluid media (liquid or gas), whereas transverse waves can propagate through solids only. Longitudinal waves

are sustained by compression forces, while transverse waves are due to shear forces. The sources of all waves are vibrations of some kinds.

In an earthquake, as many as four kinds of waves are produced, of which two are *body waves* and the other two are *surface waves*. The body waves are categorized as: (1) Longitudinal waves (called the *primary waves*); and (2) Transverse waves (called *secondary waves*). The surface waves, too, fall into two categories: (1) Transverse waves called *Love waves*, where the oscillations are parallel with the surface but perpendicular to the direction of propagation; and (2) *Rayleigh waves*, in which the oscillations take place in directions both parallel with and perpendicular to the direction of propagation, the latter taking place in the vertical direction. In Rayleigh waves, the particles below the surface undergo elliptical motions in the vertical plane.

Waves on the surface of water are similar to the Rayleigh waves. They are influenced by three physical factors: (1) *Gravitation*, which acts to return the disturbed surface back to the equilibrium configuration; (2) *Surface Tension*, since the pressure under a curved surface is different from that beneath a flat surface; and (3) *Viscosity*, which causes dissipation of energy. Of these, gravity is the dominant force except in the very short wavelength regions. Water waves controlled by gravity are called *gravity waves*. When surface tension dominates, the waves are called *capillary waves* or *ripples*. Such waves have wavelengths shorter than 1.7 cm and are quickly damped out by viscous forces (cf. Towne, 1967; Elmore & Heald, 1969).

The theory of gravity wave propagation in one dimension is well documented in the literature. A particularly elegant treatment is found in Elmore & Heald (1969). Conventionally, *x* is taken as the direction of propagation and *y* is taken to be the vertical direction. The conditions are assumed to be uniform in the *z* direction, which then disappears from the view of analysis. As water is an incompressible fluid, its density *ρ* is constant. The equation of continuity gives

$$
\vec{\nabla} \circ \vec{v} = \mathbf{0} \tag{A.1}
$$

Tsunami Propagation Models Based on First Principles 129

(A.5)

(A.6)

(A.7)

(A.9)

*φxyt X xY yT t* , , (A.10)

*X Y dx dy* (A.11)

(A.12)

*dy* (A.13)

2

(A.8)

*x*

and

and

separately, Eq. (A.11) gives

*y*

term ½*v*2 in Bernoulli's equation, we can write

At the bottom, the vertical component of velocity must vanish:

., *φ P ρg h η ρ const y h t*

0 0 , *y*

*ξ φ v t x*

*η φ v t y*

*φ v y y*

*y*

the two boundary conditions (A.7) and (A.9). We assume a trial solution

and apply the method of separation of variables. From Eq.  Omitting the non-linear

where *h* is the depth of water and *g* the acceleration due to gravity. Eq. (A.8) constitutes the relation between the variation of pressure due to the vertical displacement of water and the resulting change in the velocity potential. Differentiating Eq. (A.8) with respect to *t*, we get

> 2 2

1

*η φ v t g t*

Our task now is to find the velocity potential which satisfies Laplace's equation (A.4) and

2 2

2

*dx*

2

2 2 1 1 *dX dY k*

where the separation constant *k* is chosen to insure that *X* is a periodic function of *x*. Written

2 2 0 *d X k X*

2 2 0 *d Y k Y*

where ˆ ˆ *x y v vx vy* represents the velocity. Also, for irrotational motion, we have

$$
\vec{\nabla} \times \vec{v} = \vec{0} \tag{A.2}
$$

According to the potential theory, *v* is obtained from a velocity potential:

$$
\vec{v} = -\vec{\nabla}\phi\tag{A.3}
$$

where *φ* is a scalar function which satisfies Laplace's equation:

$$\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0 \tag{A.4}$$

If *ξ* and *η* represent the displacements in the horizontal and vertical directions, respectively, then

Tsunami Propagation Models Based on First Principles 129

$$
\sigma\_{\chi} = \frac{\partial \xi}{\partial t} = -\frac{\partial \phi}{\partial \chi} \tag{A.5}
$$

and

128 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

sources of all waves are vibrations of some kinds.

motions in the vertical plane.

constant. The equation of continuity gives

According to the potential theory, *v*

then

are sustained by compression forces, while transverse waves are due to shear forces. The

In an earthquake, as many as four kinds of waves are produced, of which two are *body waves* and the other two are *surface waves*. The body waves are categorized as: (1) Longitudinal waves (called the *primary waves*); and (2) Transverse waves (called *secondary waves*). The surface waves, too, fall into two categories: (1) Transverse waves called *Love waves*, where the oscillations are parallel with the surface but perpendicular to the direction of propagation; and (2) *Rayleigh waves*, in which the oscillations take place in directions both parallel with and perpendicular to the direction of propagation, the latter taking place in the vertical direction. In Rayleigh waves, the particles below the surface undergo elliptical

Waves on the surface of water are similar to the Rayleigh waves. They are influenced by three physical factors: (1) *Gravitation*, which acts to return the disturbed surface back to the equilibrium configuration; (2) *Surface Tension*, since the pressure under a curved surface is different from that beneath a flat surface; and (3) *Viscosity*, which causes dissipation of energy. Of these, gravity is the dominant force except in the very short wavelength regions. Water waves controlled by gravity are called *gravity waves*. When surface tension dominates, the waves are called *capillary waves* or *ripples*. Such waves have wavelengths shorter than 1.7 cm and are quickly damped out by viscous forces (cf. Towne, 1967; Elmore & Heald, 1969). The theory of gravity wave propagation in one dimension is well documented in the literature. A particularly elegant treatment is found in Elmore & Heald (1969). Conventionally, *x* is taken as the direction of propagation and *y* is taken to be the vertical direction. The conditions are assumed to be uniform in the *z* direction, which then disappears from the view of analysis. As water is an incompressible fluid, its density *ρ* is

*v* 0 (A.1)

*v* 0 (A.2)

*v φ* (A.3)

2 2

2 2 0 *φ φ φ x y*

If *ξ* and *η* represent the displacements in the horizontal and vertical directions, respectively,

is obtained from a velocity potential:

(A.4)

where ˆ ˆ *x y v vx vy* represents the velocity. Also, for irrotational motion, we have

where *φ* is a scalar function which satisfies Laplace's equation:

2

$$
\upsilon\_y = \frac{\partial \eta}{\partial t} = -\frac{\partial \phi}{\partial y} \tag{A.6}
$$

At the bottom, the vertical component of velocity must vanish:

$$
\sigma\_y = -\frac{\partial \phi}{\partial y} = 0, \qquad y = 0 \tag{A.7}
$$

At the surface of water, the pressure is atmospheric pressure *P*. Omitting the non-linear term ½*v*2 in Bernoulli's equation, we can write

$$P + \rho g \left( h + \eta \right) = \rho \frac{\partial \phi}{\partial t} + \text{const.} \tag{A.8}$$

where *h* is the depth of water and *g* the acceleration due to gravity. Eq. (A.8) constitutes the relation between the variation of pressure due to the vertical displacement of water and the resulting change in the velocity potential. Differentiating Eq. (A.8) with respect to *t*, we get

$$
\sigma\_y = \frac{\partial \eta}{\partial t} = \frac{1}{g} \frac{\partial^2 \phi}{\partial t^2} \tag{A.9}
$$

Our task now is to find the velocity potential which satisfies Laplace's equation (A.4) and the two boundary conditions (A.7) and (A.9). We assume a trial solution

$$\phi(\mathbf{x}, y, t) = \mathbf{X}(\mathbf{x})\mathbf{Y}(y)T(t) \tag{A.10}$$

and apply the method of separation of variables. From Eq. (C.4), we get

$$\frac{1}{X}\frac{d^2X}{dx^2} = -\frac{1}{Y}\frac{d^2Y}{dy^2} = -k^2\tag{A.11}$$

where the separation constant *k* is chosen to insure that *X* is a periodic function of *x*. Written separately, Eq. (A.11) gives

$$\frac{d^2X}{dx^2} + k^2X = 0\tag{A.12}$$

and

$$\frac{d^2Y}{dy^2} - k^2Y = 0\tag{A.13}$$

The general solutions to Eqs. (A.12) and (A.13) are, respectively

$$X(\infty) = Ae^{i\mathbf{k}\cdot\mathbf{x}} + Be^{-i\mathbf{k}\cdot\mathbf{x}} \tag{A.14}$$

Tsunami Propagation Models Based on First Principles 131

(B.1)

*k k* (A.24)

(B.2)

(B.4)

2

(B.5)

(B.6)

The wave velocity of the gravity wave is thus

Zatzkis, 1960):

we get:

Then Eq. (B.2) becomes

(- *k*2), we get

The *t*-equation is

whose solution is

The *ρ*-equation is

**B. Wave propagation on two-dimensional flat surface**

To apply the method of separation of variables, let

tanh *ω g v kh*

The equation of a wave emanating from a point source in a two-dimensional plane is conveniently expressed in plane polar coordinates (*ρ*, *φ*) with the source at the origin (e.g.,

> 2 22 2 2 2 22 *ψψ ψ ψ* 11 1 *ρ ρφ ρ ρ v t*

where *v* is the velocity of the wave. Assuming circular symmetry (i.e., *ψ* independent of *φ*),

2 2 2 22 *ψψ ψ* 1 1 *ρ ρ ρ v t*

*ψ ρ* ,*t P ρT t* (B.3)

2 2 2 22 *d P T dP P d T T dρ ρ dρ v dt*

Dividing both sides by *ψ*, separating the variables, and letting each side equal to a constant

2 2

2

*dt*

2 22 111 *d P dP d T k P P dρ ρ dρ v T dt*

> 2 2 2 0 *d T kvT*

*ikvt T e* (B.7)

and

$$Y(y) = \mathbb{C}e^{k\mathbf{x}} + De^{-k\mathbf{x}} \tag{A.15}$$

For a wave travelling in the forward direction, *B* = 0. Further, from Eq. (A.10):

$$\frac{\partial \phi}{\partial y} = XT \frac{dY}{dy} \tag{A.16}$$

The boundary condition at the bottom (A.7) dictates that *C* = *D*. Thus

$$Y(y) = 2C \cosh ky\tag{A.17}$$

and

$$\phi(\mathbf{x}, y, t) = A \cosh kye^{i\mathbf{k}\cdot\mathbf{x}} T(t) \tag{A.18}$$

where *A* is a new arbitrary constant. Now from Eqs. (A.6) and (A.9), we get:

$$
\Delta v\_y = -\frac{\partial \phi}{\partial y} = -kA\sinh\chi ke^{ik\eta}T\left(t\right) \tag{A.19}
$$

and

$$\sigma\_y = \frac{1}{\mathcal{S}} \frac{\hat{\sigma}^2 \phi}{\hat{\sigma} t^2} = \frac{1}{\mathcal{S}} A \cosh ky e^{i\mathbf{k}\mathbf{x}} \frac{d^2 T(t)}{dt^2} \tag{A.20}$$

Eqs. (A.19) and (A.20) yield:

$$\frac{d^2T}{dt^2} + \left(gk \tanh kh\right)T = 0\tag{A.21}$$

Hence, the velocity potential has a simple harmonic time-dependence with angular frequency

$$
\omega = \sqrt{gk \tanh kh} \tag{A.22}
$$

If we choose *iωt T e* , the velocity potential of a gravity wave propagating in the forward *x* direction over water of depth *h* becomes

$$\phi = A \cosh kye^{\mathrm{i}(kx - \omega t)}\tag{A.23}$$

The wave velocity of the gravity wave is thus

130 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

The general solutions to Eqs. (A.12) and (A.13) are, respectively

*φ dY XT*

For a wave travelling in the forward direction, *B* = 0. Further, from Eq. (A.10):

The boundary condition at the bottom (A.7) dictates that *C* = *D*. Thus

where *A* is a new arbitrary constant. Now from Eqs. (A.6) and (A.9), we get:

*y*

*y*

*y*

*y dy*

, , cosh *ikx φ x y t A kye T t* (A.18)

*φ v kA kye T t*

1 1 cosh *ikx*

*g g t dt*

Hence, the velocity potential has a simple harmonic time-dependence with angular

*ω gk kh* tanh (A.22)

cosh *i kx ωt φ A kye* (A.23)

If we choose *iωt T e* , the velocity potential of a gravity wave propagating in the forward *x*

2 tanh 0 *d T gk kh T dt*

*v A kye*

2

sinh *ikn*

 2 2 2 2

*φ dTt*

and

and

and

frequency

Eqs. (A.19) and (A.20) yield:

direction over water of depth *h* becomes

*ikx ikx X x Ae Be* (A.14)

*kx kx Y y Ce De* (A.15)

(A.16)

*Y y C ky* 2 cosh (A.17)

(A.19)

(A.21)

(A.20)

$$
\omega = \frac{\omega}{k} = \sqrt{\frac{g}{k} \tanh kh} \tag{A.24}
$$

#### **B. Wave propagation on two-dimensional flat surface**

The equation of a wave emanating from a point source in a two-dimensional plane is conveniently expressed in plane polar coordinates (*ρ*, *φ*) with the source at the origin (e.g., Zatzkis, 1960):

$$\frac{\partial^2 \psi}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial \psi}{\partial \rho} + \frac{1}{\rho^2} \frac{\partial^2 \psi}{\partial \phi^2} = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} \tag{\text{B.1}}$$

where *v* is the velocity of the wave. Assuming circular symmetry (i.e., *ψ* independent of *φ*), we get:

$$\frac{\partial^2 \psi}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial \psi}{\partial \rho} = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} \tag{B.2}$$

To apply the method of separation of variables, let

$$
\psi(\rho, t) = P(\rho)T(t) \tag{B.3}
$$

Then Eq. (B.2) becomes

$$T\frac{d^2P}{d\rho^2} + \frac{T}{\rho}\frac{dP}{d\rho} = \frac{P}{v^2}\frac{d^2T}{dt^2} \tag{\text{B.4}}$$

Dividing both sides by *ψ*, separating the variables, and letting each side equal to a constant (- *k*2), we get

$$\frac{1}{P}\frac{d^2P}{d\rho^2} + \frac{1}{P\rho}\frac{dP}{d\rho} = \frac{1}{v^2T}\frac{d^2T}{dt^2} = -k^2\tag{\text{B.5}}$$

The *t*-equation is

$$\frac{d^2T}{dt^2} + k^2v^2T = 0\tag{\text{B.6}}$$

whose solution is

$$T \propto e^{\pm ikvt} \tag{\text{B.7}}$$

The *ρ*-equation is

$$
\rho^2 \frac{d^2 P}{d\rho^2} + \rho \frac{dP}{d\rho} + k^2 \rho^2 P = 0 \tag{\text{B.8}}
$$

Tsunami Propagation Models Based on First Principles 133

(B.17)

*ρ* (B.18)

*ρ* (B.19)

(B.20)

 0 ,

The amplitude of the wave falls off inversely as the square-root of the distance from the

1 *ψModelB*

The intensity of the wave (i.e., the energy density) is thus inversely proportional to the

There is a simple alternative procedure to obtain Eq. (B.18) from Eq. (B.19) (Tan & Lyatskaya, 2009). Assuming conservation of energy, the energy spreads out in concentric circles of radius *ρ*, so that the intensity of the wave varies inversely as the radial distance

In Eq. (B.20), *e*-*βρ* gives the attenuation factor when loss processes are present, with *β*

For long-distance propagation of waves on a spherical surface, the curvature of the surface must be taken into account (Tan & Lyatskaya, 2009). Use spherical coordinates (*r*, *θ*, *φ*) with the center of the sphere as the origin and place the source of the wave at the north pole (*θ* = 0) (Fig. C.1). Assume azimuthal symmetry, i.e., *ψ* independent of *φ*. Then circular waves will propagate on the surface of the sphere (*r* = *a*) outwards in the direction of increasing zenith angle *θ*. The wave-fronts will be small circles on the sphere having radii *a*sin*θ* and circumferences 2*πa*sin*θ*. The energy conservation principle now requires that the intensity

Hence the amplitude of the wave varies inversely as the square-root of sin*θ*:

*ModelC* *βρ i kρ ωt*

*ρ*

2 1 *IModelC* (C.1)

(C.2)

sin

sin

0 ,

*e e ψ ρ t ψ*

*ModelB ModelB I ψ*

whence the wave amplitude varies according to Eq. (B.18).

representing the attenuation coefficient.

of the wave varies inversely as sin*θ*:

1

When *k* is complex, we can write *k* = *κ* + *iβ* ., Eq. (B.16) then takes the form

**C. Wave propagation on two-dimensional spherical surface**

*e ψ ρ t ψ*

source:

distance from the source:

*i kρ ωt*

*ρ*

2 1

or

$$
\rho \frac{d}{d\rho} \left(\rho \frac{dP}{d\rho}\right) + k^2 \rho^2 P = 0 \tag{\text{B.9}}
$$

This is Bessel's equation of the zeroth order. A novel technique to solve this equation is found in Irving & Mullineaux (1959). Let

$$P = \frac{R}{\sqrt{\rho}}\tag{B.10}$$

Then Eq. (B.9) assumes the form

$$\left[\frac{d^2R}{d\rho^2} + k^2\right] \left[1 + \frac{1}{4k^2\rho^2}\right] R = 0\tag{B.11}$$

For large *ρ* (i.e., away from the source), the second term within the square bracket may be neglected. Thus

$$\frac{d^2R}{d\rho^2} + k^2R = 0\tag{\text{B.12}}$$

giving the solution

$$R \propto e^{i k \rho} \tag{\mathbb{B}.13}$$

Hence, from Eq. (B.10):

$$P \propto \frac{e^{ik\rho}}{\sqrt{\rho}}\tag{\text{B.14}}$$

and

$$\psi\left(\rho, t\right) = P\left(\rho\right)T\left(t\right) = \psi\_0 \frac{e^{i\mathbf{k}\left(\rho \pm vt\right)}}{\sqrt{\rho}}\tag{\text{B.15}}$$

Since the wave is propagating outwards, we retain the + sign only, giving

$$
\psi(\rho, t) = \psi\_0 \frac{e^{ik(\rho - vt)}}{\sqrt{\rho}} \tag{B.16}
$$

In terms of the wave number k,

Tsunami Propagation Models Based on First Principles 133

$$
\psi(\rho, t) = \psi\_0 \frac{e^{i(k\rho - \omega t)}}{\sqrt{\rho}} \tag{\text{B.17}}
$$

The amplitude of the wave falls off inversely as the square-root of the distance from the source:

$$
\psi\_{ModelB} \propto \frac{1}{\sqrt{\rho}} \tag{B.18}
$$

The intensity of the wave (i.e., the energy density) is thus inversely proportional to the distance from the source:

$$I\_{ModelB} \propto \psi\_{ModelB} \frac{1}{\rho} \propto \frac{1}{\rho} \tag{B.19}$$

There is a simple alternative procedure to obtain Eq. (B.18) from Eq. (B.19) (Tan & Lyatskaya, 2009). Assuming conservation of energy, the energy spreads out in concentric circles of radius *ρ*, so that the intensity of the wave varies inversely as the radial distance whence the wave amplitude varies according to Eq. (B.18).

When *k* is complex, we can write *k* = *κ* + *iβ* ., Eq. (B.16) then takes the form

132 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

found in Irving & Mullineaux (1959). Let

Then Eq. (B.9) assumes the form

neglected. Thus

giving the solution

and

Hence, from Eq. (B.10):

In terms of the wave number k,

*R P*

2

2

Since the wave is propagating outwards, we retain the + sign only, giving

2 2 2 2

2

*dρ*

1 1 0

4 *d R k R dρ k ρ*

For large *ρ* (i.e., away from the source), the second term within the square bracket may be

2 2 0 *d R k R*

*ikρ e P*

0 ,

0 ,

*e ψ ρ t ψ*

*e ψ ρ t P ρ T t ψ*

2 2 2 2 0 *d P dP ρ ρ k ρ P dρ dρ*

2 2 0 *d dP ρ ρ k ρ P dρ dρ*

This is Bessel's equation of the zeroth order. A novel technique to solve this equation is

(B.8)

*ρ* (B.10)

(B.12)

*ikρ R e* (B.13)

*ρ* (B.14)

(B.15)

(B.16)

*ik ρ vt*

*ρ*

*ik ρ vt*

*ρ*

(B.9)

(B.11)

or

$$
\psi\left(\rho, t\right) = \psi\_0 \frac{e^{-\beta \rho} e^{i\left(k\rho - \omega t\right)}}{\sqrt{\rho}}\tag{B.20}
$$

In Eq. (B.20), *e*-*βρ* gives the attenuation factor when loss processes are present, with *β* representing the attenuation coefficient.

#### **C. Wave propagation on two-dimensional spherical surface**

For long-distance propagation of waves on a spherical surface, the curvature of the surface must be taken into account (Tan & Lyatskaya, 2009). Use spherical coordinates (*r*, *θ*, *φ*) with the center of the sphere as the origin and place the source of the wave at the north pole (*θ* = 0) (Fig. C.1). Assume azimuthal symmetry, i.e., *ψ* independent of *φ*. Then circular waves will propagate on the surface of the sphere (*r* = *a*) outwards in the direction of increasing zenith angle *θ*. The wave-fronts will be small circles on the sphere having radii *a*sin*θ* and circumferences 2*πa*sin*θ*. The energy conservation principle now requires that the intensity of the wave varies inversely as sin*θ*:

$$I\_{ModelC} \propto \varphi^2 \propto \frac{1}{\sin \theta} \tag{C.1}$$

Hence the amplitude of the wave varies inversely as the square-root of sin*θ*:

$$\nu\_{\text{Model}\mathbb{C}} \propto \frac{1}{\sqrt{\sin \theta}} \tag{C.2}$$

The amplitude and intensity of the wave decrease from the north pole (*θ* = 0) to the equator (*θ* = *π*/2), but increase thereafter until the wave converges at the south pole (*θ* = *π*), where the original amplitude and intensity are restored.

Tsunami Propagation Models Based on First Principles 135

sin

*xyTt* , (D.1)

2 *a b* (D.3)

(D.2)

sin

0 0 0 1 1 15 .2618 .2588 1.0115 1.0057 30 .5236 .5000 1.0472 1.0233 45 .7854 .7071 1.1107 1.0539 60 1.0472 .8660 1.2092 1.0996 75 1.3090 .9659 1.3552 1.1641 90 1.5708 1.0000 1.5708 1.2533 105 1.8326 .9659 1.8972 1.3774 120 2.0944 .8660 2.4184 1.5551 135 2.3562 .7071 3.3322 1.8254 150 2.6180 .5000 5.2360 2.2882 165 2.8798 .2588 11.1267 3.3357 180 3.1426 0 ∞ ∞

**Table C.1.** Intensity and Amplitude Ratios in Spherical and Flat Surface Solutions.

**D. Wave propagation from finite line source in two dimensions**

Consider a finite line source of strength *q* and length 2*c* in the *x* - *y* plane, whose end-points

are located at (- *c*, 0) and (*c*, 0) (Fig. D.1). The velocity potential *ψ* can be expressed as

2 2 2 2 0 *x y*

The potential due to a line source constitutes a well-known problem in electrostatics (cf. Abraham & Becker, 1950). The equi-potential lines on which *φ* (*x*, *y*) is constant are con-focal ellipses with their foci located at the end-points of the line source. The lines of force are confocal hyperbolas perpendicular to the ellipses (cf. Morse & Feshbach, 1953). In the case of a tsunami from a linear subduction zone, the ellipses represent the wave-fronts, whereas the

The amplitude of the wave can be obtained by a practical approach similar to that of Tan & Lyatskaya (2009). An approximate expression for the perimeter of an ellipse with semi-

*x*, *y* satisfies the two-dimensional Laplace's equation

hyperbolas are the lines along which the enrrgy is transferred (Fig. D.1).

2 2 *L*

major axis *a* and semi-minor axis *b* is found in the literature (cf. Weisstein, 2003):

where

*θ*, deg *θ*, rad sin*θ*

**Figure C.1.** Spherical surface with source at north pole (*θ* = 0).

In order to compare the spherical surface solution with that of the flat surface solution (cf. Bhatnagar, et al., 2006), we notice that the corresponding circular wave-fronts on the flat surface will have radii of *aθ* and circumferences 2*πaθ* (Fig. C.1). Thus the ratio of the intensities of the waves will be

$$\frac{I\_{\text{ModelC}}}{I\_{\text{ModelB}}} = \frac{I\_{\text{spherical}}}{I\_{\text{flat}}} = \frac{\theta}{\sin \theta} \tag{C.3}$$

Hence, the ratio of the amplitudes of the waves is

$$\frac{\left\|\nu\_{\text{Model}\gets}\right\|\_{\text{M}\text{\textquotedblleft}\text{C}}}{\left\|\nu\_{\text{M}\text{\textquotedblleft}\text{B}}\right\|\_{\text{flat}}}=\frac{\left\|\nu\_{\text{spherical}}\right\|}{\left\|\text{s}\text{\textquotedblright}\right\|}=\sqrt{\frac{\left\|\theta\right\|}{\text{s}\text{\textquotedblleft}\text{B}}}\tag{C.4}$$

Table C.1 shows the ratios of the wave intensities and amplitudes in the spherical and flat surface solutions. The values are independent of the radius of the sphere and are therefore applicable to all spherical surfaces. The departures of the spherical surface solutions from those of the flat surface solutions are slight for small values of *θ*. Even at the equator (*θ* = *π*/2), the enhancements of the wave intensity and amplitude are merely 57% and 25% respectively. The enhancement becomes progressively greater until the south pole (*θ* = *π*) is reached, where it becomes infinite.

**Table C.1.** Intensity and Amplitude Ratios in Spherical and Flat Surface Solutions.

### **D. Wave propagation from finite line source in two dimensions**

Consider a finite line source of strength *q* and length 2*c* in the *x* - *y* plane, whose end-points are located at (- *c*, 0) and (*c*, 0) (Fig. D.1). The velocity potential *ψ* can be expressed as

$$
\psi = \varphi(\mathbf{x}, y) T(t) \tag{\mathbb{D}.1}
$$

where *x*, *y* satisfies the two-dimensional Laplace's equation

134 Tsunami – Analysis of a Hazard – From Physical Interpretation to Human Impact

the original amplitude and intensity are restored.