2.10: Dynamic Mechanical Analysis
Dynamic mechanical analysis (DMA), also known as forced oscillatory measurements and dynamic rheology, is a basic tool used to measure the viscoelastic properties of materials (particularly polymers). To do so, DMA instrument applies an oscillating force to a material and measures its response; from such experiments, the viscosity (the tendency to flow) and stiffness of the sample can be calculated. These viscoelastic properties can be related to temperature, time, or frequency. As a result, DMA can also provide information on the transitions of materials and characterize bulk properties that are important to material performance. DMA can be applied to determine the glass transition of polymers or the response of a material to application and removal of a load, as a few common examples. The usefulness of DMA comes from its ability to mimic operating conditions of the material, which allows researchers to predict how the material will perform.
A Brief History
Oscillatory experiments have appeared in published literature since the early 1900s and began with rudimentary experimental setups to analyze the deformation of metals. In an initial study, the material in question was hung from a support, and torsional strain was applied using a turntable. Early instruments of the 1950s from manufacturers Weissenberg and Rheovibron exclusively measured torsional stress, where force is applied in a twisting motion.
Due to its usefulness in determining polymer molecular structure and stiffness, DMA became more popular in parallel with the increasing research on polymers. The method became integral in the analysis of polymer properties by 1961. In 1966, the revolutionary torsional braid analysis was developed; because this technique used a fine glass substrate imbued with the material of analysis, scientists were no longer limited to materials that could provide their own support. Using torsional braid analysis, the transition temperatures of polymers could be determined through temperature programming. Within two decades, commercial instruments became more accessible, and the technique became less specialized. In the early 1980s, one of the first DMAs using axial geometries (linear rather than torsional force) was introduced.
Since the 1980s, DMA has become much more user-friendly, faster, and less costly due to competition between vendors. Additionally, the developments in computer technology have allowed easier and more efficient data processing. Today, DMA is offered by most vendors, and the modern instrument is detailed in the Instrumentation section.
Basic Principles of DMA
DMA is based on two important concepts of stress and strain. Stress (σ) provides a measure of force ( F ) applied to area ( A ), \ref{1} .
\[ \sigma \ =\ F/A \label{1} \]
Stress to a material causes strain (γ), the deformation of the sample. Strain can be calculated by dividing the change in sample dimensions (∆ Y ) by the sample’s original dimensions ( Y ) (\ref{2} ). This value is often given as a percentage of strain.
\[ \gamma \ =\ \Delta Y/Y \label{2} \]
The modulus ( E ), a measure of stiffness, can be calculated from the slope of the stress-strain plot, Figure \(\PageIndex{1}\), as displayed in \label{3} . This modulus is dependent on temperature and applied stress. The change of this modulus as a function of a specified variable is key to DMA and determination of viscoelastic properties. Viscoelastic materials such as polymers display both elastic properties characteristic of solid materials and viscous properties characteristic of liquids; as a result, the viscoelastic properties are often a compromise between the two extremes. Ideal elastic properties can be related to Hooke’s spring, while viscous behavior is often modeled using a dashpot, or a motion-resisting damper.
\[ E \ =\ \sigma /y \label{3} \]
Creep-recovery
Creep-recovery testing is not a true dynamic analysis because the applied stress or strain is held constant; however, most modern DMA instruments have the ability to run this analysis. Creep-recovery tests the deformation of a material that occurs when load applied and removed. In the “creep” portion of this analysis, the material is placed under immediate, constant stress until the sample equilibrates. “Recovery” then measures the stress relaxation after the stress is removed. The stress and strain are measured as functions of time. From this method of analysis, equilibrium values for viscosity, modulus, and compliance (willingness of materials to deform; inverse of modulus) can be determined; however, such calculations are beyond the scope of this review.
Creep-recovery tests are useful in testing materials under anticipated operation conditions and long test times. As an example, multiple creep-recovery cycles can be applied to a sample to determine the behavior and change in properties of a material after several cycles of stress.
Dynamic Testing
DMA instruments apply sinusoidally oscillating stress to samples and causes sinusoidal deformation. The relationship between the oscillating stress and strain becomes important in determining viscoelastic properties of the material. To begin, the stress applied can be described by a sine function where σ o is the maximum stress applied, ω is the frequency of applied stress, and t is time. Stress and strain can be expressed with the following \ref{4} .
\[ \sigma \ = \ \sigma_{0} sin(\omega t + \delta);\ y=y_{0} cos(\omega t) \label{4} \]
The strain of a system undergoing sinusoidally oscillating stress is also sinuisoidal, but the phase difference between strain and stress is entirely dependent on the balance between viscous and elastic properties of the material in question. For ideal elastic systems, the strain and stress are completely in phase, and the phase angle (δ) is equal to 0. For viscous systems, the applied stress leads the strain by 90 o . The phase angle of viscoelastic materials is somewhere in between (Figure \(\PageIndex{2}\) ).
In essence, the phase angle between the stress and strain tells us a great deal about the viscoelasticity of the material. For one, a small phase angle indicates that the material is highly elastic; a large phase angle indicates the material is highly viscous. Furthermore, separating the properties of modulus, viscosity, compliance, or strain into two separate terms allows the analysis of the elasticity or the viscosity of a material. The elastic response of the material is analogous to storage of energy in a spring, while the viscosity of material can be thought of as the source of energy loss.
A few key viscoelastic terms can be calculated from dynamic analysis; their equations and significance are detailed in Table \(\PageIndex{1}\).
| Term | Equation | Significance |
| Complex modulus ( E *) | E* = E’ + iE” | Overall modulus representing stiffness of material; combined elastic and viscous components |
| Elastic modulus ( E’ ) | E’ = ( σ o /γ o ) cosδ | Storage modulus; measures stored energy and represents elastic portion |
| Viscous modulus ( E” ) | E” = ( σ o /γ o ) sinδ | Loss modulus; contribution of viscous component on polymer that flows under stress |
| Loss tangent ( tan δ) | Tan δ = E”/E’ | Damping or index of viscoelasticity; compares viscous and elastic moduli |
Types of Dynamic Experiments
A temperature sweep is the most common DMA test used on solid materials. In this experiment, the frequency and amplitude of oscillating stress is held constant while the temperature is increased. The temperature can be raised in a stepwise fashion, where the sample temperature is increased by larger intervals (e.g., 5 o C) and allowed to equilibrate before measurements are taken. Continuous heating routines can also be used (1-2 o C/minute). Typically, the results of temperature sweeps are displayed as storage and loss moduli as well as tan delta as a function of temperature. For polymers, these results are highly indicative of polymer structure. An example of a thermal sweep of a polymer is detailed later in this module.
In time scans, the temperature of the sample is held constant, and properties are measured as functions of time, gas changes, or other parameters. This experiment is commonly used when studying curing of thermosets, materials that change chemically upon heating. Data is presented graphically using modulus as a function of time; curing profiles can be derived from this information.
Frequency scans test a range of frequencies at a constant temperature to analyze the effect of change in frequency on temperature-driven changes in material. This type of experiment is typically run on fluids or polymer melts. The results of frequency scans are displayed as modulus and viscosity as functions of log frequency.
Instrumentation
The most common instrument for DMA is the forced resonance analyzer, which is ideal for measuring material response to temperature sweeps. The analyzer controls deformation, temperature, sample geometry, and sample environment.
Figure \(\PageIndex{3}\) displays the important components of the DMA, including the motor and driveshaft used to apply torsional stress as well as the linear variable differential transformer (LVDT) used to measure linear displacement. The carriage contains the sample and is typically enveloped by a furnace and heat sink.
The DMA should be ideally selected to analyze the material at hand. The DMA can be either stress or strain controlled: strain-controlled analyzers move the probe a certain distance and measure the stress applied; strain-controlled analyzers provide a constant deformation of the sample (Figure \(\PageIndex{4}\) ) Although the two techniques are nearly equivalent when the stress-strain plot (Figure \(\PageIndex{1}\) ) is linear, stress-controlled analyzers provide more accurate results.
DMA analyzers can also apply stress or strain in two manners—axial and torsional deformation (Figure \(\PageIndex{5}\) ) Axial deformation applies a linear force to the sample and is typically used for solid and semisolid materials to test flex, tensile strength, and compression. Torsional analyzers apply force in a twisting motion; this type of analysis is used for liquids and polymer melts but can also be applied to solids. Although both types of analyzers have wide analysis range and can be used for similar samples, the axial instrument should not be used for fluid samples with viscosities below 500 Pa-s, and torsional analyzers cannot handle materials with high modulus.
Different fixtures can be used to hold the samples in place and should be chosen according to the type of samples analyzed. The sample geometry affects both stress and strain and must be factored into the modulus calculations through a geometry factor. The fixture systems are specific to the type of stress application. Axial analyzers have a greater number of fixture options; one of the most commonly used fixtures is extension/tensile geometry used for thin films or fibers. In this method, the sample is held both vertically and lengthwise by top and bottom clamps, and stress is applied upwards
For torsional analyzers, the simplest geometry is the use of parallel plates. The plates are separated by a distance determined by the viscosity of the sample. Because the movement of the sample depends on its radius from the center of the plate, the stress applied is uneven; the measured strain is an average value.
DMA of the glass transition polymers
As the temperature of a polymer increases, the material goes through a number of minor transitions ( T γ and T β ) due to expansion; at these transitions, the modulus also undergoes changes. The glass transition of polymers ( T g ) occurs with the abrupt change of physical properties within 140-160 o C; at some temperature within this range, the storage (elastic) modulus of the polymer drops dramatically. As the temperature rises above the glass transition point, the material loses its structure and becomes rubbery before finally melting. The idealized modulus transition is pictured in Figure \(\PageIndex{6}\).
The glass transition temperature can be determined using either the storage modulus, complex modulus, or tan δ (vs temperature) depending on context and instrument; because these methods result in such a range of values (Figure \(\PageIndex{6}\) ), the method of calculation should be noted. When using the storage modulus, the temperature at which E’ begins to decline is used as the T g . Tan δ and loss modulus E” show peaks at the glass transition; either onset or peak values can be used in determining T g . These different methods of measurement are depicted graphically in Figure \(\PageIndex{7}\).
Advantages and limitations of DMA
Dynamic mechanical analysis is an essential analytical technique for determining the viscoelastic properties of polymers. Unlike many comparable methods, DMA can provide information on major and minor transitions of materials; it is also more sensitive to changes after the glass transition temperature of polymers. Due to its use of oscillating stress, this method is able to quickly scan and calculate the modulus for a range of temperatures. As a result, it is the only technique that can determine the basic structure of a polymer system while providing data on the modulus as a function of temperature. Finally, the environment of DMA tests can be controlled to mimic real-world operating conditions, so this analytical method is able to accurately predict the performance of materials in use.
DMA does possess limitations that lead to calculation inaccuracies. The modulus value is very dependent on sample dimensions, which means large inaccuracies are introduced if dimensional measurements of samples are slightly inaccurate. Additionally, overcoming the inertia of the instrument used to apply oscillating stress converts mechanical energy to heat and changes the temperature of the sample. Since maintaining exact temperatures is important in temperature scans, this also introduces inaccuracies. Because data processing of DMA is largely automated, the final source of measurement uncertainty comes from computer error.


7.16: Polyprotic Acids
Learning Objectives
- Extend previously introduced equilibrium concepts to acids and bases that may donate or accept more than one proton
We can classify acids by the number of protons per molecule that they can give up in a reaction. Acids such as \(\ce{HCl}\), \(\ce{HNO3}\), and \(\ce{HCN}\) that contain one ionizable hydrogen atom in each molecule are called monoprotic acids . Their reactions with water are:
\[\ce{HCl}(aq)+\ce{H2O}(l)⟶\ce{H3O+}(aq)+\ce{Cl-}(aq)\]
\[\ce{HNO3}(aq)+\ce{H2O}(l)⟶\ce{H3O+}(aq)+\ce{NO3-}(aq)\]
\[\ce{HCN}(aq)+\ce{H2O}(l)⟶\ce{H3O+}(aq)+\ce{CN-}(aq)\]
Even though it contains four hydrogen atoms, acetic acid, \(\ce{CH3CO2H}\), is also monoprotic because only the hydrogen atom from the carboxyl group (\(\ce{-COOH}\)) reacts with bases:
Similarly, monoprotic bases are bases that will accept a single proton.
Diprotic Acids
Diprotic acids contain two ionizable hydrogen atoms per molecule; ionization of such acids occurs in two steps. The first ionization always takes place to a greater extent than the second ionization. For example, sulfuric acid, a strong acid, ionizes as follows:
- The first ionization is
\[ \ce{H2SO4}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HSO4-}(aq)\]
with \(K_{\ce a1} > 10^2;\: {complete\: dissociation}\).
- The second ionization is
\[ \ce{HSO4-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{SO4^{2−}}(aq)\]
with \( K_{\ce a2}=1.2×10^{−2}\).
This stepwise ioniza tion process occurs for all polyprotic acids, meaning that one H + is lost at the time. This process contrasts with the ionization of strong bases such as Ca(OH) 2 in which all the OH- ions are lost in a single dissociation reaction:
Ca(OH) 2 (aq) → Ca 2+ (aq) + 2 OH - (aq)
This is because hydroxides are ionic compounds, while acids are covalent compounds.
When we make a solution of a weak diprotic acid, we get a solution that contains a mixture of acids. Carbonic acid, \(\ce{H2CO3}\), is an example of a weak diprotic acid. The first ionization of carbonic acid yields hydronium ions and bicarbonate ions in small amounts.
- First Ionization
\[\ce{H2CO3}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HCO3-}(aq)\]
with
\[K_{\ce{H2CO3}}=\ce{\dfrac{[H3O+][HCO3- ]}{[H2CO3]}}=4.3×10^{−7}\]
The bicarbonate ion can also act as an acid. It ionizes and forms hydronium ions and carbonate ions in even smaller quantities.
- Second Ionization
\[\ce{HCO3-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CO3^2-}(aq)\]
with
\[ K_{\ce{HCO3-}}=\ce{\dfrac{[H3O+][CO3^2- ]}{[HCO3- ]}}=4.7×10^{−11}\]
\(K_{\ce{H2CO3}}\) is larger than \(K_{\ce{HCO3-}}\) by a factor of 10 4 , so H 2 CO 3 is the dominant producer of hydronium ion in the solution.
Example \(\PageIndex{1}\): Ionization of a Diprotic Acid
When we buy soda water (carbonated water), we are buying a solution of carbon dioxide in water. The solution is acidic because CO 2 reacts with water to form carbonic acid, H 2 CO 3 . What are \(\ce{[H3O+]}\) , \(\ce{[HCO3- ]}\), and \(\ce{[CO3^2- ]}\) in a saturated solution of CO 2 with an initial [H 2 CO 3 ] = 0.033 M ?
\[\ce{H2CO3}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HCO3-}(aq) \hspace{20px} K_{\ce a1}=4.3×10^{−7} \label{step1} \tag{equilibrium step 1}\]
\[\ce{HCO3-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{CO3^2-}(aq) \hspace{20px} K_{\ce a2}=4.7×10^{−11} \label{step2} \tag{equilibrium step 2}\]
Solution
As indicated by the ionization constants, H 2 CO 3 is a much stronger acid than \(\ce{HCO3-}\), so \(\ce{H2CO3}\) is the dominant producer of hydronium ion in solution.
- Using the customary four steps, we determine the concentration of H 3 O + and \(\ce{HCO3-}\) produced by ionization of H 2 CO 3 .
1. First Ionization: Determine the concentrations of \(\ce{H3O+}\) and \(\ce{HCO3-}\).
An abbreviated table of changes and concentrations shows:
| ICE Table | \(\ce{H2CO3}(aq)\) | \(\ce{H2O}(l)\) | \( \ce{H3O+}(aq) \) | \(\ce{HCO3-}(aq)\) |
|---|---|---|---|---|
| I nitial (M) | \(0.033 \:M\) | - | \(0\) | \(0\) |
| C hange (M) | \(- x\) | - | \(+x\) | \(+x\) |
| E quilibrium (M) | \(0.033 \:M - x\) | - | \(x\) | \( x\) |
Substituting the equilibrium concentrations into the equilibrium constant gives us:
\[K_{\ce{H2CO3}}=\ce{\dfrac{[H3O+][HCO3- ]}{[H2CO3]}}=\dfrac{(x)(x)}{0.033−x}=4.3×10^{−7} \nonumber\]
Solving the preceding equation making our standard assumptions gives:
\[x=1.2×10^{−4} \nonumber\]
Thus:
\[\ce{[H2CO3]}=0.033\:M \nonumber\]
\[\ce{[H3O+]}=\ce{[HCO3- ]}=1.2×10^{−4}\:M \nonumber\]
pH=-log[H 3 O + ]=3.92
Triprotic Acids
A triprotic acid is an acid that has three dissociable protons that undergo stepwise ionization: Phosphoric acid is a typical example:
- The first ionization is
\[\ce{H3PO4}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{H2PO4-}(aq) \]
with \(K_{\ce a1}=7.5×10^{−3} \).
- The second ionization is
\[\ce{H2PO4-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{HPO4^2-}(aq)\]
with \( K_{\ce a2}=6.2×10^{−8} \).
- The third ionization is
\[\ce{HPO4^2-}(aq)+\ce{H2O}(l)⇌\ce{H3O+}(aq)+\ce{PO4^3-}(aq) \]
with \( K_{\ce a3}=4.2×10^{−13} \).
As with the diprotic acids, the differences in the ionization constants of these reactions tell us that in each successive step the degree of ionization is significantly weaker. This is a general characteristic of polyprotic acids and successive ionization constants often differ by a factor of about 10 5 to 10 6 . Again, using the first dissociation constant is a good approximation to calculate the pH for a solution of phosphoric acid.
Contributors and Attributions
Summary
An acid that contains more than one ionizable proton is a polyprotic acid. The protons of these acids ionize in steps. The differences in the acid ionization constants for the successive ionizations of the protons in a polyprotic acid usually vary by roughly five orders of magnitude. As long as the difference between the successive values of Ka of the acid is significant, it is a good estimation to only consider the first dissociation constant to calculate pH
Glossary
- diprotic acid
- acid containing two ionizable hydrogen atoms per molecule. A diprotic acid ionizes in two steps
- diprotic base
- base capable of accepting two protons. The protons are accepted in two steps
- monoprotic acid
- acid containing one ionizable hydrogen atom per molecule
- stepwise ionization
- process in which an acid is ionized by losing protons sequentially
- triprotic acid
- acid that contains three ionizable hydrogen atoms per molecule; ionization of triprotic acids occurs in three steps
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


7.2: Bounded Waves
Speech is what separates humans most decisively from animals. No other species can master syntax, and even though chimpanzees can learn a vocabulary of hand signs, there is an unmistakable difference between a human infant and a baby chimp: starting from birth, the human experiments with the production of complex speech sounds.
Since speech sounds are instinctive for us, we seldom think about them consciously. How do we control sound waves so skillfully? Mostly we do it by changing the shape of a connected set of hollow cavities in our chest, throat, and head. Somehow by moving the boundaries of this space in and out, we can produce all the vowel sounds. Up until now, we have been studying only those properties of waves that can be understood as if they existed in an infinite, open space with no boundaries. In this chapter we address what happens when a wave is confined within a certain space, or when a wave pattern encounters the boundary between two different media, such as when a light wave moving through air encounters a glass windowpane.
6.2.1 Reflection, transmission, and absorption
Reflection and transmission
Sound waves can echo back from a cliff, and light waves are reflected from the surface of a pond. We use the word reflection, normally applied only to light waves in ordinary speech, to describe any such case of a wave rebounding from a barrier. Figure (a) shows a circular water wave being reflected from a straight wall. In this chapter, we will concentrate mainly on reflection of waves that move in one dimension, as in figure c /1.
b / Circular water waves are reflected from a boundary on the left. (PSSC Physics)
Wave reflection does not surprise us. After all, a material object such as a rubber ball would bounce back in the same way. But waves are not objects, and there are some surprises in store.
First, only part of the wave is usually reflected. Looking out through a window, we see light waves that passed through it, but a person standing outside would also be able to see her reflection in the glass. A light wave that strikes the glass is partly reflected and partly transmitted (passed) by the glass. The energy of the original wave is split between the two. This is different from the behavior of the rubber ball, which must go one way or the other, not both.
Second, consider what you see if you are swimming underwater and you look up at the surface. You see your own reflection. This is utterly counterintuitive, since we would expect the light waves to burst forth to freedom in the wide-open air. A material projectile shot up toward the surface would never rebound from the water-air boundary!
What is it about the difference between two media that causes waves to be partly reflected at the boundary between them? Is it their density? Their chemical composition? Ultimately all that matters is the speed of the wave in the two media. A wave is partially reflected and partially transmitted at the boundary between media in which it has different speeds. For example, the speed of light waves in window glass is about 30% less than in air, which explains why windows always make reflections. Figure c shows examples of wave pulses being reflected at the boundary between two coil springs of different weights, in which the wave speed is different.
Reflections such as b and c /1, where a wave encounters a massive fixed object, can usually be understood on the same basis as cases like c /2 and c /3 where two media meet. Example c /1, for instance, is like a more extreme version of example c /2. If the heavy coil spring in c /2 was made heavier and heavier, it would end up acting like the fixed wall to which the light spring in c /1 has been attached.
Exercise \(\PageIndex{1}\)
In figure c /1, the reflected pulse is upside-down, but its depth is just as big as the original pulse's height. How does the energy of the reflected pulse compare with that of the original?
(answer in the back of the PDF version of the book)
| Example 8: Fish have internal ears. |
|---|
|
Why don't fish have ear-holes? The speed of sound waves in a fish's body is not much different from their speed in water, so sound waves are not strongly reflected from a fish's skin. They pass right through its body, so fish can have internal ears. |
| Example 9: Whale songs traveling long distances |
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Sound waves travel at drastically different speeds through rock, water, and air. Whale songs are thus strongly reflected both at both the bottom and the surface. The sound waves can travel hundreds of miles, bouncing repeatedly between the bottom and the surface, and still be detectable. Sadly, noise pollution from ships has nearly shut down this cetacean version of the internet. |
| Example 10: Long-distance radio communication |
|---|
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Radio communication can occur between stations on opposite sides of the planet. The mechanism is entirely similar to the one explained in the previous example, but the three media involved are the earth, the atmosphere, and the ionosphere. |
self-check:
Sonar is a method for ships and submarines to detect each other by producing sound waves and listening for echoes. What properties would an underwater object have to have in order to be invisible to sonar?
(answer in the back of the PDF version of the book)
The use of the word “reflection” naturally brings to mind the creation of an image by a mirror, but this might be confusing, because we do not normally refer to “reflection” when we look at surfaces that are not shiny. Nevertheless, reflection is how we see the surfaces of all objects, not just polished ones. When we look at a sidewalk, for example, we are actually seeing the reflecting of the sun from the concrete. The reason we don't see an image of the sun at our feet is simply that the rough surface blurs the image so drastically.
Inverted and uninverted reflections
Notice how the pulse reflected back to the right in example c /2 comes back upside-down, whereas the one reflected back to the left in c /3 returns in its original upright form. This is true for other waves as well. In general, there are two possible types of reflections, a reflection back into a faster medium and a reflection back into a slower medium. One type will always be an inverting reflection and one noninverting.
It's important to realize that when we discuss inverted and uninverted reflections on a string, we are talking about whether the wave is flipped across the direction of motion (i.e., upside-down in these drawings). The reflected pulse will always be reversed front to back, as shown in figures d and e . This is because it is traveling in the other direction. The leading edge of the pulse is what gets reflected first, so it is still ahead when it starts back to the left --- it's just that “ahead” is now in the opposite direction.
Absorption
So far we have tacitly assumed that wave energy remains as wave energy, and is not converted to any other form. If this was true, then the world would become more and more full of sound waves, which could never escape into the vacuum of outer space. In reality, any mechanical wave consists of a traveling pattern of vibrations of some physical medium, and vibrations of matter always produce heat, as when you bend a coathangar back and forth and it becomes hot. We can thus expect that in mechanical waves such as water waves, sound waves, or waves on a string, the wave energy will gradually be converted into heat. This is referred to as absorption. The reduction in the wave's energy can also be described as a reduction in amplitude, the relationship between them being, as with a vibrating object, \(E\propto A^2\).
The wave suffers a decrease in amplitude, as shown in figure f . The decrease in amplitude amounts to the same fractional change for each unit of distance covered. For example, if a wave decreases from amplitude 2 to amplitude 1 over a distance of 1 meter, then after traveling another meter it will have an amplitude of 1/2. That is, the reduction in amplitude is exponential. This can be proved as follows. By the principle of superposition, we know that a wave of amplitude 2 must behave like the superposition of two identical waves of amplitude 1. If a single amplitude-1 wave would die down to amplitude 1/2 over a certain distance, then two amplitude-1 waves superposed on top of one another to make amplitude 1+1=2 must die down to amplitude 1/2+1/2=1 over the same distance.
Note
As a wave undergoes absorption, it loses energy. Does this mean that it slows down?
(answer in the back of the PDF version of the book)
In many cases, this frictional heating effect is quite weak. Sound waves in air, for instance, dissipate into heat extremely slowly, and the sound of church music in a cathedral may reverberate for as much as 3 or 4 seconds before it becomes inaudible. During this time it has traveled over a kilometer! Even this very gradual dissipation of energy occurs mostly as heating of the church's walls and by the leaking of sound to the outside (where it will eventually end up as heat). Under the right conditions (humid air and low frequency), a sound wave in a straight pipe could theoretically travel hundreds of kilometers before being noticeable attenuated.
In general, the absorption of mechanical waves depends a great deal on the chemical composition and microscopic structure of the medium. Ripples on the surface of antifreeze, for instance, die out extremely rapidly compared to ripples on water. For sound waves and surface waves in liquids and gases, what matters is the viscosity of the substance, i.e., whether it flows easily like water or mercury or more sluggishly like molasses or antifreeze. This explains why our intuitive expectation of strong absorption of sound in water is incorrect. Water is a very weak absorber of sound (viz. whale songs and sonar), and our incorrect intuition arises from focusing on the wrong property of the substance: water's high density, which is irrelevant, rather than its low viscosity, which is what matters.
Light is an interesting case, since although it can travel through matter, it is not itself a vibration of any material substance. Thus we can look at the star Sirius, \(10^{14}\) km away from us, and be assured that none of its light was absorbed in the vacuum of outer space during its 9-year journey to us. The Hubble Space Telescope routinely observes light that has been on its way to us since the early history of the universe, billions of years ago. Of course the energy of light can be dissipated if it does pass through matter (and the light from distant galaxies is often absorbed if there happen to be clouds of gas or dust in between).
| Example 11: Soundproofing |
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|
Typical amateur musicians setting out to soundproof their garages tend to think that they should simply cover the walls with the densest possible substance. In fact, sound is not absorbed very strongly even by passing through several inches of wood. A better strategy for soundproofing is to create a sandwich of alternating layers of materials in which the speed of sound is very different, to encourage reflection. The classic design is alternating layers of fiberglass and plywood. The speed of sound in plywood is very high, due to its stiffness, while its speed in fiberglass is essentially the same as its speed in air. Both materials are fairly good sound absorbers, but sound waves passing through a few inches of them are still not going to be absorbed sufficiently. The point of combining them is that a sound wave that tries to get out will be strongly reflected at each of the fiberglass-plywood boundaries, and will bounce back and forth many times like a ping pong ball. Due to all the back-and-forth motion, the sound may end up traveling a total distance equal to ten times the actual thickness of the soundproofing before it escapes. This is the equivalent of having ten times the thickness of sound-absorbing material. |
| Example 12: Radio transmission |
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|
A radio transmitting station must have a length of wire or cable connecting the amplifier to the antenna. The cable and the antenna act as two different media for radio waves, and there will therefore be partial reflection of the waves as they come from the cable to the antenna. If the waves bounce back and forth many times between the amplifier and the antenna, a great deal of their energy will be absorbed. There are two ways to attack the problem. One possibility is to design the antenna so that the speed of the waves in it is as close as possible to the speed of the waves in the cable; this minimizes the amount of reflection. The other method is to connect the amplifier to the antenna using a type of wire or cable that does not strongly absorb the waves. Partial reflection then becomes irrelevant, since all the wave energy will eventually exit through the antenna. |
Discussion Questions
◊ A sound wave that underwent a pressure-inverting reflection would have its compressions converted to expansions and vice versa. How would its energy and frequency compare with those of the original sound? Would it sound any different? What happens if you swap the two wires where they connect to a stereo speaker, resulting in waves that vibrate in the opposite way?
6.2.2 Quantitative treatment of reflection
In this subsection we analyze the reasons why reflections occur at a speed-changing boundary, predict quantitatively the intensities of reflection and transmission, and discuss how to predict for any type of wave which reflections are inverting and which are uninverting.
Why reflection occurs
To understand the fundamental reasons for what does occur at the boundary between media, let's first discuss what doesn't happen. For the sake of concreteness, consider a sinusoidal wave on a string. If the wave progresses from a heavier portion of the string, in which its velocity is low, to a lighter-weight part, in which it is high, then the equation \(v=f\lambda\) tells us that it must change its frequency, or its wavelength, or both. If only the frequency changed, then the parts of the wave in the two different portions of the string would quickly get out of step with each other, producing a discontinuity in the wave, g /1. This is unphysical, so we know that the wavelength must change while the frequency remains constant, g /2.
But there is still something unphysical about figure g /2. The sudden change in the shape of the wave has resulted in a sharp kink at the boundary. This can't really happen, because the medium tends to accelerate in such a way as to eliminate curvature. A sharp kink corresponds to an infinite curvature at one point, which would produce an infinite acceleration, which would not be consistent with the smooth pattern of wave motion envisioned in fig. g /2. Waves can have kinks, but not stationary kinks.
We conclude that without positing partial reflection of the wave, we cannot simultaneously satisfy the requirements of (1) continuity of the wave, and (2) no sudden changes in the slope of the wave. In other words, we assume that both the wave and its derivative are continuous functions.)
Does this amount to a proof that reflection occurs? Not quite. We have only proved that certain types of wave motion are not valid solutions. In the following subsection, we prove that a valid solution can always be found in which a reflection occurs. Now in physics, we normally assume (but seldom prove formally) that the equations of motion have a unique solution, since otherwise a given set of initial conditions could lead to different behavior later on, but the Newtonian universe is supposed to be deterministic. Since the solution must be unique, and we derive below a valid solution involving a reflected pulse, we will have ended up with what amounts to a proof of reflection.
Intensity of reflection
I will now show, in the case of waves on a string, that it is possible to satisfy the physical requirements given above by constructing a reflected wave, and as a bonus this will produce an equation for the proportions of reflection and transmission and a prediction as to which conditions will lead to inverted and which to uninverted reflection. We assume only that the principle of superposition holds, which is a good approximations for waves on a string of sufficiently small amplitude.
Let the unknown amplitudes of the reflected and transmitted waves be \(R\) and \(T\), respectively. An inverted reflection would be represented by a negative value of \(R\). We can without loss of generality take the incident (original) wave to have unit amplitude. Superposition tells us that if, for instance, the incident wave had double this amplitude, we could immediately find a corresponding solution simply by doubling \(R\) and \(T\).
Just to the left of the boundary, the height of the wave is given by the height 1 of the incident wave, plus the height \(R\) of the part of the reflected wave that has just been created and begun heading back, for a total height of \(1+R\). On the right side immediately next to the boundary, the transmitted wave has a height \(T\). To avoid a discontinuity, we must have
Next we turn to the requirement of equal slopes on both sides of the boundary. Let the slope of the incoming wave be s immediately to the left of the junction. If the wave was 100% reflected, and without inversion, then the slope of the reflected wave would be \(-s\), since the wave has been reversed in direction. In general, the slope of the reflected wave equals \(-sR\), and the slopes of the superposed waves on the left side add up to \(s-sR\). On the right, the slope depends on the amplitude, \(T\), but is also changed by the stretching or compression of the wave due to the change in speed. If, for example, the wave speed is twice as great on the right side, then the slope is cut in half by this effect. The slope on the right is therefore \(s(v_1/v_2)T\), where \(v_1\) is the velocity in the original medium and \(v_2\) the velocity in the new medium. Equality of slopes gives \(s-sR = s(v_1/v_2)T\), or
Solving the two equations for the unknowns \(R\) and \(T\) gives
The first equation shows that there is no reflection unless the two wave speeds are different, and that the reflection is inverted in reflection back into a fast medium.
The energies of the transmitted and reflected wavers always add up to the same as the energy of the original wave. There is never any abrupt loss (or gain) in energy when a wave crosses a boundary; conversion of wave energy to heat occurs for many types of waves, but it occurs throughout the medium. The equation for \(T\), surprisingly, allows the amplitude of the transmitted wave to be greater than 1, i.e., greater than that of the incident wave. This does not violate conservation of energy, because this occurs when the second string is less massive, reducing its kinetic energy, and the transmitted pulse is broader and less strongly curved, which lessens its potential energy.
Inverted and uninverted reflections in general (optional)
For waves on a string, reflections back into a faster medium are inverted, while those back into a slower medium are uninverted. Is this true for all types of waves? The rather subtle answer is that it depends on what property of the wave you are discussing.
Let's start by considering wave disturbances of freeway traffic. Anyone who has driven frequently on crowded freeways has observed the phenomenon in which one driver taps the brakes, starting a chain reaction that travels backward down the freeway as each person in turn exercises caution in order to avoid rear-ending anyone. The reason why this type of wave is relevant is that it gives a simple, easily visualized example of how our description of a wave depends on which aspect of the wave we have in mind. In steadily flowing freeway traffic, both the density of cars and their velocity are constant all along the road. Since there is no disturbance in this pattern of constant velocity and density, we say that there is no wave. Now if a wave is touched off by a person tapping the brakes, we can either describe it as a region of high density or as a region of decreasing velocity.
The freeway traffic wave is in fact a good model of a sound wave, and a sound wave can likewise be described either by the density (or pressure) of the air or by its speed. Likewise many other types of waves can be described by either of two functions, one of which is often the derivative of the other with respect to position.
Now let's consider reflections. If we observe the freeway wave in a mirror, the high-density area will still appear high in density, but velocity in the opposite direction will now be described by a negative number. A person observing the mirror image will draw the same density graph, but the velocity graph will be flipped across the \(x\) axis, and its original region of negative slope will now have positive slope. Although I don't know any physical situation that would correspond to the reflection of a traffic wave, we can immediately apply the same reasoning to sound waves, which often do get reflected, and determine that a reflection can either be density-inverting and velocity-uninverting or density-uninverting and velocity-inverting.
This same type of situation will occur over and over as one encounters new types of waves, and to apply the analogy we need only determine which quantities, like velocity, become negated in a mirror image and which, like density, stay the same.
A light wave, for instance, consists of a traveling pattern of electric and magnetic fields. All you need to know in order to analyze the reflection of light waves is how electric and magnetic fields behave under reflection; you don't need to know any of the detailed physics of electricity and magnetism. An electric field can be detected, for example, by the way one's hair stands on end. The direction of the hair indicates the direction of the electric field. In a mirror image, the hair points the other way, so the electric field is apparently reversed in a mirror image. The behavior of magnetic fields, however, is a little tricky. The magnetic properties of a bar magnet, for instance, are caused by the aligned rotation of the outermost orbiting electrons of the atoms. In a mirror image, the direction of rotation is reversed, say from clockwise to counterclockwise, and so the magnetic field is reversed twice: once simply because the whole picture is flipped and once because of the reversed rotation of the electrons. In other words, magnetic fields do not reverse themselves in a mirror image. We can thus predict that there will be two possible types of reflection of light waves. In one, the electric field is inverted and the magnetic field uninverted (example 23 , p. 699). In the other, the electric field is uninverted and the magnetic field inverted.
6.2.3 Interference effects
If you look at the front of a pair of high-quality binoculars, you will notice a greenish-blue coating on the lenses. This is advertised as a coating to prevent reflection. Now reflection is clearly undesirable --- we want the light to go in the binoculars --- but so far I've described reflection as an unalterable fact of nature, depending only on the properties of the two wave media. The coating can't change the speed of light in air or in glass, so how can it work? The key is that the coating itself is a wave medium. In other words, we have a three-layer sandwich of materials: air, coating, and glass. We will analyze the way the coating works, not because optical coatings are an important part of your education but because it provides a good example of the general phenomenon of wave interference effects.
There are two different interfaces between media: an air-coating boundary and a coating-glass boundary. Partial reflection and partial transmission will occur at each boundary. For ease of visualization let's start by considering an equivalent system consisting of three dissimilar pieces of string tied together, and a wave pattern consisting initially of a single pulse.
Figure k /1 shows the incident pulse moving through the heavy rope, in which its velocity is low. When it encounters the lighter-weight rope in the middle, a faster medium, it is partially reflected and partially transmitted. (The transmitted pulse is bigger, but nevertheless has only part of the original energy.) The pulse transmitted by the first interface is then partially reflected and partially transmitted by the second boundary, k /3. In figure k /4, two pulses are on the way back out to the left, and a single pulse is heading off to the right. (There is still a weak pulse caught between the two boundaries, and this will rattle back and forth, rapidly getting too weak to detect as it leaks energy to the outside with each partial reflection.)
Note how, of the two reflected pulses in k /4, one is inverted and one uninverted. One underwent reflection at the first boundary (a reflection back into a slower medium is uninverted), but the other was reflected at the second boundary (reflection back into a faster medium is inverted).
Now let's imagine what would have happened if the incoming wave pattern had been a long sinusoidal wave train instead of a single pulse. The first two waves to reemerge on the left could be in phase, l /1, or out of phase, l /2, or anywhere in between. The amount of lag between them depends entirely on the width of the middle segment of string. If we choose the width of the middle string segment correctly, then we can arrange for destructive interference to occur, l /2, with cancellation resulting in a very weak reflected wave.
This whole analysis applies directly to our original case of optical coatings. Visible light from most sources does consist of a stream of short sinusoidal wave-trains such as the ones drawn above. The only real difference between the waves-on-a-rope example and the case of an optical coating is that the first and third media are air and glass, in which light does not have the same speed. However, the general result is the same as long as the air and the glass have light-wave speeds that are either both greater than the coating's or both less than the coating's.
The business of optical coatings turns out to be a very arcane one, with a plethora of trade secrets and “black magic” techniques handed down from master to apprentice. Nevertheless, the ideas you have learned about waves in general are sufficient to allow you to come to some definite conclusions without any further technical knowledge. The self-check and discussion questions will direct you along these lines of thought.
self-check:
Color corresponds to wavelength of light waves. Is it possible to choose a thickness for an optical coating that will produce destructive interference for all colors of light?
(answer in the back of the PDF version of the book)
This example was typical of a wide variety of wave interference effects. With a little guidance, you are now ready to figure out for yourself other examples such as the rainbow pattern made by a compact disc or by a layer of oil on a puddle.
Discussion Questions
- Is it possible to get complete destructive interference in an optical coating, at least for light of one specific wavelength?
- Sunlight consists of sinusoidal wave-trains containing on the order of a hundred cycles back-to-back, for a length of something like a tenth of a millimeter. What happens if you try to make an optical coating thicker than this?
- Suppose you take two microscope slides and lay one on top of the other so that one of its edges is resting on the corresponding edge of the bottom one. If you insert a sliver of paper or a hair at the opposite end, a wedge-shaped layer of air will exist in the middle, with a thickness that changes gradually from one end to the other. What would you expect to see if the slides were illuminated from above by light of a single color? How would this change if you gradually lifted the lower edge of the top slide until the two slides were finally parallel?
- An observation like the one described in discussion question C was used by Newton as evidence against the wave theory of light! If Newton didn't know about inverting and noninverting reflections, what would have seemed inexplicable to him about the region where the air layer had zero or nearly zero thickness?
6.2.4 Waves bounded on both sides
In the example of the previous section, it was theoretically true that a pulse would be trapped permanently in the middle medium, but that pulse was not central to our discussion, and in any case it was weakening severely with each partial reflection. Now consider a guitar string. At its ends it is tied to the body of the instrument itself, and since the body is very massive, the behavior of the waves when they reach the end of the string can be understood in the same way as if the actual guitar string was attached on the ends to strings that were extremely massive. Reflections are most intense when the two media are very dissimilar. Because the wave speed in the body is so radically different from the speed in the string, we should expect nearly 100% reflection.
Although this may seem like a rather bizarre physical model of the actual guitar string, it already tells us something interesting about the behavior of a guitar that we would not otherwise have understood. The body, far from being a passive frame for attaching the strings to, is actually the exit path for the wave energy in the strings. With every reflection, the wave pattern on the string loses a tiny fraction of its energy, which is then conducted through the body and out into the air. (The string has too little cross-section to make sound waves efficiently by itself.) By changing the properties of the body, moreover, we should expect to have an effect on the manner in which sound escapes from the instrument. This is clearly demonstrated by the electric guitar, which has an extremely massive, solid wooden body. Here the dissimilarity between the two wave media is even more pronounced, with the result that wave energy leaks out of the string even more slowly. This is why an electric guitar with no electric pickup can hardly be heard at all, and it is also the reason why notes on an electric guitar can be sustained for longer than notes on an acoustic guitar.
If we initially create a disturbance on a guitar string, how will the reflections behave? In reality, the finger or pick will give the string a triangular shape before letting it go, and we may think of this triangular shape as a very broad “dent” in the string which will spread out in both directions. For simplicity, however, let's just imagine a wave pattern that initially consists of a single, narrow pulse traveling up the neck, m /1. After reflection from the top end, it is inverted, m /3. Now something interesting happens: figure m /5 is identical to figure m /1. After two reflections, the pulse has been inverted twice and has changed direction twice. It is now back where it started. The motion is periodic. This is why a guitar produces sounds that have a definite sensation of pitch.
self-check:
Notice that from m /1 to m /5, the pulse has passed by every point on the string exactly twice. This means that the total distance it has traveled equals 2 L, where L is the length of the string. Given this fact, what are the period and frequency of the sound it produces, expressed in terms of L and v, the velocity of the wave?
(answer in the back of the PDF version of the book)
Note that if the waves on the string obey the principle of superposition, then the velocity must be independent of amplitude, and the guitar will produce the same pitch regardless of whether it is played loudly or softly. In reality, waves on a string obey the principle of superposition approximately, but not exactly. The guitar, like just about any acoustic instrument, is a little out of tune when played loudly. (The effect is more pronounced for wind instruments than for strings, but wind players are able to compensate for it.)
Now there is only one hole in our reasoning. Suppose we somehow arrange to have an initial setup consisting of two identical pulses heading toward each other, as in figure (o). They will pass through each other, undergo a single inverting reflection, and come back to a configuration in which their positions have been exactly interchanged. This means that the period of vibration is half as long. The frequency is twice as high.
This might seem like a purely academic possibility, since nobody actually plays the guitar with two picks at once! But in fact it is an example of a very general fact about waves that are bounded on both sides. A mathematical theorem called Fourier's theorem states that any wave can be created by superposing sine waves. Figure p shows how even by using only four sine waves with appropriately chosen amplitudes, we can arrive at a sum which is a decent approximation to the realistic triangular shape of a guitar string being plucked. The one-hump wave, in which half a wavelength fits on the string, will behave like the single pulse we originally discussed. We call its frequency \(f_\text{o}\). The two-hump wave, with one whole wavelength, is very much like the two-pulse example. For the reasons discussed above, its frequency is \(2f_\text{o}\). Similarly, the three-hump and four-hump waves have frequencies of \(3f_\text{o}\) and \(4f_\text{o}\).
Theoretically we would need to add together infinitely many such wave patterns to describe the initial triangular shape of the string exactly, although the amplitudes required for the very high frequency parts would be very small, and an excellent approximation could be achieved with as few as ten waves.
We thus arrive at the following very general conclusion. Whenever a wave pattern exists in a medium bounded on both sides by media in which the wave speed is very different, the motion can be broken down into the motion of a (theoretically infinite) series of sine waves, with frequencies \(f_\text{o},\ 2f_\text{o},\ 3f_\text{o},\ ...\) Except for some technical details, to be discussed below, this analysis applies to a vast range of sound-producing systems, including the air column within the human vocal tract. Because sounds composed of this kind of pattern of frequencies are so common, our ear-brain system has evolved so as to perceive them as a single, fused sensation of tone.
Musical applications
Many musicians claim to be able to pick out by ear several of the frequencies \(2f_\text{o}\), \(3f_\text{o}\), ..., called overtones or harmonics of the fundamental \(f_\text{o}\), but they are kidding themselves. In reality, the overtone series has two important roles in music, neither of which depends on this fictitious ability to “hear out” the individual overtones.
First, the relative strengths of the overtones is an important part of the personality of a sound, called its timbre (rhymes with “amber”). The characteristic tone of the brass instruments, for example, is a sound that starts out with a very strong harmonic series extending up to very high frequencies, but whose higher harmonics die down drastically as the attack changes to the sustained portion of the note.
Second, although the ear cannot separate the individual harmonics of a single musical tone, it is very sensitive to clashes between the overtones of notes played simultaneously, i.e., in harmony. We tend to perceive a combination of notes as being dissonant if they have overtones that are close but not the same. Roughly speaking, strong overtones whose frequencies differ by more than 1% and less than 10% cause the notes to sound dissonant. It is important to realize that the term “dissonance” is not a negative one in music. No matter how long you search the radio dial, you will never hear more than three seconds of music without at least one dissonant combination of notes. Dissonance is a necessary ingredient in the creation of a musical cycle of tension and release. Musically knowledgeable people do not usually use the word “dissonant” as a criticism of music, and if they do, what they are really saying is that the dissonance has been used in a clumsy way, or without providing any contrast between dissonance and consonance.
Standing waves
Figure s shows sinusoidal wave patterns made by shaking a rope. I used to enjoy doing this at the bank with the pens on chains, back in the days when people actually went to the bank. You might think that I and the person in the photos had to practice for a long time in order to get such nice sine waves. In fact, a sine wave is the only shape that can create this kind of wave pattern, called a standing wave , which simply vibrates back and forth in one place without moving. The sine wave just creates itself automatically when you find the right frequency, because no other shape is possible.
If you think about it, it's not even obvious that sine waves should be able to do this trick. After all, waves are supposed to travel at a set speed, aren't they? The speed isn't supposed to be zero! Well, we can actually think of a standing wave as a superposition of a moving sine wave with its own reflection, which is moving the opposite way. Sine waves have the unique mathematical property that the sum of sine waves of equal wavelength is simply a new sine wave with the same wavelength. As the two sine waves go back and forth, they always cancel perfectly at the ends, and their sum appears to stand still.
Standing wave patterns are rather important, since atoms are really standing-wave patterns of electron waves.
Standing-wave patterns of air columns
The air column inside a wind instrument behaves very much like the wave-on-a-string example we've been concentrating on so far, the main difference being that we may have either inverting or noninverting reflections at the ends.
Some organ pipes are closed at both ends. The speed of sound is different in metal than in air, so there is a strong reflection at the closed ends, and we can have standing waves. These reflections are both density-noninverting, so we get symmetric standing-wave patterns, such as the one shown in figure u /1.
Figure t shows the sound waves in and around a bamboo Japanese flute called a shakuhachi, which is open at both ends of the air column. We can only have a standing wave pattern if there are reflections at the ends, but that is very counterintuitive --- why is there any reflection at all, if the sound wave is free to emerge into open space, and there is no change in medium? Recall the reason why we got reflections at a change in medium: because the wavelength changes, so the wave has to readjust itself from one pattern to another, and the only way it can do that without developing a kink is if there is a reflection. Something similar is happening here. The only difference is that the wave is adjusting from being a plane wave to being a spherical wave. The reflections at the open ends are density-inverting, u /2, so the wave pattern is pinched off at the ends. Comparing panels 1 and 2 of the figure, we see that although the wave pattens are different, in both cases the wavelength is the same: in the lowest-frequency standing wave, half a wavelength fits inside the tube. Thus, it isn't necessary to memorize which type of reflection is inverting and which is inverting. It's only necessary to know that the tubes are symmetric.
Finally, we can have an asymmetric tube: closed at one end and open at the other. A common example is the pan pipes, v , which are closed at the bottom and open at the top. The standing wave with the lowest frequency is therefore one in which 1/4 of a wavelength fits along the length of the tube, as shown in figure u /3.
Sometimes an instrument's physical appearance can be misleading. A concert flute, w , is closed at the mouth end and open at the other, so we would expect it to behave like an asymmetric air column; in reality, it behaves like a symmetric air column open at both ends, because the embouchure hole (the hole the player blows over) acts like an open end. The clarinet and the saxophone look similar, having a moutpiece and reed at one end and an open end at the other, but they act different. In fact the clarinet's air column has patterns of vibration that are asymmetric, the saxophone symmetric. The discrepancy comes from the difference between the conical tube of the sax and the cylindrical tube of the clarinet. The adjustment of the wave pattern from a plane wave to a spherical wave is more gradual at the flaring bell of the saxophone.
self-check:
Draw a graph of pressure versus position for the first overtone of the air column in a tube open at one end and closed at the other. This will be the next-to-longest possible wavelength that allows for a point of maximum vibration at one end and a point of no vibration at the other. How many times shorter will its wavelength be compared to the frequency of the lowest-frequency standing wave, shown in the figure? Based on this, how many times greater will its frequency be?
(answer in the back of the PDF version of the book)
| Example 13: The speed of sound |
|---|
|
We can get a rough and ready derivation of the equation for the speed of sound by analyzing the standing waves in a cylindrical air column as a special type of Helmholtz resonance (example 25 on page 332), in which the cavity happens to have the same cross-sectional area as the neck. Roughly speaking, the regions of maximum density variation act like the cavity. The regions of minimum density variation, on the other hand, are the places where the velocity of the air is varying the most; these regions throttle back the speed of the vibration, because of the inertia of the moving air. If the cylinder has cross-sectional area \(A\), then the “cavity” and “neck” parts of the wave both have lengths of something like \(\lambda/2\), and the volume of the “cavity” is about \(A\lambda/2\). We get \(v=f\lambda=(...)\sqrt{\gamma P_\text{o}/\rho}\), where the factor \((...)\) represents numerical stuff that we can't possibly hope to have gotten right with such a crude argument. The correct result is in fact \(v=\sqrt{\gamma P_\text{o}/\rho}\). Isaac Newton attempted the same calculation, but didn't understand the thermodynamic effects involved, and therefore got a result that didn't have the correct factor of \(\sqrt{\gamma}\). |
Contributors and Attributions
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


2.6.10: A Direct Proof of the Compactness Theorem
We can prove the Compactness Theorem directly, without appealing to the Completeness Theorem, using the same ideas as in the proof of the completeness theorem. In the proof of the Completeness Theorem we started with a consistent set \(\Gamma\) of sentences, expanded it to a consistent , saturated, and complete set \(\Gamma^*\) of sentences, and then showed that in the term model \(\Struct{M(\Gamma^*)}\) constructed from \(\Gamma^*\) , all sentences of \(\Gamma\) are true, so \(\Gamma\) is satisfiable.
We can use the same method to show that a finitely satisfiable set of sentences is satisfiable. We just have to prove the corresponding versions of the results leading to the truth lemma where we replace “consistent” with “finitely satisfiable.”
 Then:
- \((A \land B) \in \Gamma\) iff both \(A \in \Gamma\) and \(B \in \Gamma\) .
- \((A \lor B) \in \Gamma\) iff either \(A \in \Gamma\) or \(B \in \Gamma\) .
- \((A \lif B) \in \Gamma\) iff either \(A \notin \Gamma\) or \(B \in \Gamma\) .
Prove Proposition \(\PageIndex{1}\) . Avoid the use of \(\Proves\) .
Every finitely satisfiable set \(\Gamma\) can be extended to a saturated finitely satisfiable set \(\Gamma'\) .
Prove Lemma \(\PageIndex{1}\) . (Hint: The crucial step is to show that if \(\Gamma_n\) is finitely satisfiable, so is \(\Gamma_n \cup \{D_n\}\) , without any appeal to derivations or consistency.)
Suppose \(\Gamma\) is complete, finitely satisfiable, and saturated.
- \(\lexists{x}{A(x)} \in \Gamma\) iff \(A(t) \in \Gamma\) for at least one closed term \(t\) .
- \(\lforall{x}{A(x)} \in \Gamma\) iff \(A(t) \in \Gamma\) for all closed terms \(t\) .
Prove Proposition \(\PageIndex{2}\) .
Every finitely satisfiable set \(\Gamma\) can be extended to a complete and finitely satisfiable set \(\Gamma^*\) .
Prove Lemma \(\PageIndex{2}\) . (Hint: the crucial step is to show that if \(\Gamma_n\) is finitely satisfiable, then either \(\Gamma_n \cup \{A_n\}\) or \(\Gamma_n \cup \{\lnot A_n\}\) is finitely satisfiable.)
\(\Gamma\) is satisfiable if and only if it is finitely satisfiable.
Proof. If \(\Gamma\) is satisfiable, then there is a structure \(\Struct{M}\) such that \(\Sat{M}{A}\) for all \(A \in \Gamma\) . Of course, this \(\Struct{M}\) also satisfies every finite subset of \(\Gamma\) , so \(\Gamma\) is finitely satisfiable.
Now suppose that \(\Gamma\) is finitely satisfiable. By Lemma \(\PageIndex{1}\) , there is a finitely satisfiable, saturated set \(\Gamma' \supseteq \Gamma\) . By Lemma \(\PageIndex{2}\) , \({\Gamma'}\) can be extended to a complete and finitely satisfiable set \(\Gamma^*\) , and \(\Gamma^*\) is still saturated . Construct the term model \(\Struct{M(\Gamma^*)}\) as in Definition 10.6.1 . Note that Proposition 10.6.1 did not rely on the fact that \(\Gamma^*\) is consistent (or complete or saturated, for that matter), but just on the fact that \(\Struct{M(\Gamma^*)}\) is covered. The proof of the Truth Lemma ( Lemma 10.6.1 ) goes through if we replace references to Proposition 10.3.1 and Proposition 10.4.2 by references to Proposition \(\PageIndex{1}\) and Proposition \(\PageIndex{2}\) . ◻
Write out the complete proof of the Truth Lemma ( Lemma 10.6.1 ) in the version required for the proof of Theorem \(\PageIndex{1}\) .


4: Cell migration assay
Summary
Cell migration assays are used to compare mobility of cells under different conditions. It tests a cell’s ability to “crawl” across a solid substrate.
Also known as:
Scratch assay, wound-healing assay
Samples needed
A plate of confluent cells in a monolayer
Method
First, cells of interest are grown to confluence in a monolayer. Then, a pipette tip or another object is used to create a scratch through the monolayer, leaving an area of a consistent width open and free of cells. Finally, the “scratch” or “wound” is monitored and photographed to determine if one treatment group refills the gap faster than another.
Controls
The cell migration assay is nearly always used to compare mobility of cells +/- a treatment or intervention. Therefore, the usual mock treatment controls apply.
Interpretation
These microscopy images show that cells with decreased Dsg3 expression “heal the wound” faster by migrating to fill the scratched area. Therefore, the reader can conclude that one normal function of Dsg3 is to suppress cell migration. However, regardless of whether or not Dsg3 is expressed, if p38MAPK is inhibited, wound closure will not occur. Therefore, the migration-suppressing function of Dsg3 is dependent on p38MAPK activity.
Image Descriptions
Figure 1 image description:
Data from a cell migration assay.
Panel a: A grid of microscopy images of cells growing in culture. Columns show images from 0 h, 24 h, and 48 h after scratch was created. Rows from top to bottom are siNT + DMSO, siDsg3+DMSO, siNT+ SB202190, siDg3+SB202190. The top two rows both show the scratch "closing," but the siDsg3 cells close the scratch faster and more completely. With SB202190, no closure occurs.
Panel b: A line graph with time in hours on the x axis and wounding area (fold of time point 0h) on the y axis. This graph quantifies what is seen from the microscopy images in panel a. The siDsg3+DMSO condition reaches wounding area of ~0.25 by 48 hours. The siNT+DMSO condition reaches ~0.5, and both of the SB202190 conditions are still very near 1. The difference between the two DMSO conditions is significant, as is the difference between each pair of samples with DMSO vs. SB202190 with the same siRNA. ↵
Thumbnail
"Dh3-RD.jpg" ↗ by Dhifaf zeki is licensed under CC BY-SA 4.0 ↗.
Description: This is image of Scratch wound healing assay of Rhabdomyosarcoma (RD) cell line (cancer cell line) under inverted phase contrast microscope.
Author
Katherine Mattaini, Tufts University
- Rötzer, V., E. Hartlieb, J. Winkler, E. Walter, A. Schlipp, M. Sardy, V. Spindler, and J. Waschke. 2016. Desmoglein 3-dependent signaling regulates keratinocyte migration and wound healing. Journal of Investigative Dermatology 136:301-310. ↵


8.3: A Nation at Risk
Assignment:
Wrap Up
Contributors and Attributions
CC licensed content, Original
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Assignment:
Read a Nation at Risk:
A nation at risk is printed in the back of this book in Appendix B, or available at the link below:
https://www.edreform.com/wp-content/uploads/2013/02/A_Nation_At_Risk_1983.pdf
As you read A Nation at Risk, complete the guided reading*to help prepare for class discussion.
A Nation at Risk was the report published in 1983 by the commission on excellence in education appointed by President Regan. The report quickly pointed out that the level of student illiteracy was high and that SAT scores had been on a steady decline for the two decades before publication of the report. Mediocrity was a word used to describe the state of education in America. SAT scores actually started to improve beginning in 1980.
The report provided specific recommendations for improving the nation’s educational system.  All K-12 schools and colleges should adopt rigorous standards. To be effective, the standards needed to be measurable. In fact, the report kicked off the standard movement with professional education organizations creating and publishing national standards. Longer school days or years would be needed to study the curricula outlined by the new standards. Many suggestions were given to improve teacher preparation programs, educational leadership, and fiscal support.
Education was directly tied to economic competitiveness, and schools were to be accountable for the quality of student education verified by external testing. The call for more testing began with the report. Comparisions of American schools began to be made to international schools. Scores of America’s top-level students were highest or near the top on international tests.
“Any attempt to isolate developments in the schools from those in society at large turns out to reflect principally the inclination to institutionalize blame for whatever is going wrong: the formal part of the learning process cannot be separated from its societal context” (College Board). In the 1960s and 70s society was tumultuous during a large part of the report. Watching tv for longer periods of time was on the rise among students. More students of poverty and color were taking the SATs. Single-parent families were on the increase. These were all factors that, in general, affect the academic performance of students negatively.
Teachers have increasingly resisted what they see as misplaced blame and narrowing of the curriculum. A large part of the problem is that negative political rhetoric about education increased. One aspect of misplaced blame is the concentration and increase of students living in poverty. The tipping point depends on what the purpose of education is determined to be. Teachers lean to expanding the minds of students and want art and music included in the curriculum.
Past educational events continue to have an impact on educational policy and how American teachers go about educating students. Examining history in context gives reason to much of the current educational landscape. Labaree argues for three goals of education: democratic equality, social efficiency, and social mobility. Viewing history through these lenses allows for a decision to be made on the validity of his claim. A Nation at Risk still holds a strong sway on how education is viewed in this country from a need to provide the capability of citizens to maintain the country’s economic well being to holding teachers solely responsible for the quality of graduates.


9.2.1: How Crowded Is this Neighborhood?
Lesson
Let's see how proportional relationships apply to where people live.
Exercise \(\PageIndex{1}\): Dot Density
The figure shows four squares. Each square encloses an array of dots. Squares A and B have side length 2 inches. Squares C and D have side length 1 inch.
-
Complete the table with information about each square.
square area of the square in square inches number of dots number of dots per square inch A B C D Table \(\PageIndex{1}\) - Compare each square to the others. What is the same and what is different?
Exercise \(\PageIndex{2}\): Dot Density with a Twist
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
- Let \(a\) be the area of the square and \(d\) be the number of dots enclosed by the square. For each square, plot a point that represents its values of \(a\) and \(d\). Start by choosing an appropriate scale for each axis, using the sliders to adjust the intervals.
- Draw lines from \((0,0)\) to each point. For each line, write an equation that represents the proportional relationship.
- What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
Exercise \(\PageIndex{3}\): Housing Density
Here are pictures of two different neighborhoods.
This image depicts an area that is 0.3 kilometers long and 0.2 kilometers wide.
This image depicts an area that is 0.4 kilometers long and 0.2 kilometers wide.
For each neighborhood, find the number of houses per square kilometer.
Exercise \(\PageIndex{4}\): Population Density
- New York City has a population of 8,406 thousand people and covers an area of 1,214 square kilometers.
- Los Angeles has a population of 3,884 thousand people and covers an area of 1,302 square kilometers.
- The points labeled \(A\) and \(B\) each correspond to one of the two cities. Which is which? Label them on the graph.
- Write an equation for the line that passes through \((0,0)\) and \(A\). What is the constant of proportionality?
- Write an equation for the line that passes through \((0,0)\) and \(B\). What is the constant of proportionality?
- What do the constants of proportionality tell you about the crowdedness of these two cities?
Are you ready for more?
-
Predict where these types of regions would be shown on the graph:
- a suburban region where houses are far apart, with big yards
- a neighborhood in an urban area with many high-rise apartment buildings
- a rural state with lots of open land and not many people
- Next, use this data to check your predictions:
| place | description | population | area (km 2 ) |
|---|---|---|---|
| Chalco | a suburb of Omaha, Nebraska | \(10,994\) | \(7.5\) |
| Anoka County | a county in Minnesota, near Minneapolis/ St. Paul | \(339,534\) | \(1,155\) |
| Guttenberg | a city in New Jersey | \(11,176\) | \(0.49\) |
| New York | a state | \(19,746,227\) | \(141,300\) |
| Rhode Island | a state | \(1,055,173\) | \(3,140\) |
| Alaska | a state | \(736,732\) | \(1,717,856\) |
| Tok | a community in Alaska | \(1,258\) | \(342.7\) |


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Frequency
-
the number of times a value of the data occurs
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Frequency Table
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a data representation in which grouped data is displayed along with the corresponding frequencies
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Histogram
-
a graphical representation in
x
-
y
form of the distribution of data in a data set;
x
represents the data and
y
represents the frequency, or relative frequency. The graph consists of contiguous rectangles.
-
Interquartile Range
-
or
IQR
, is the range of the middle 50 percent of the data values; the
IQR
is found by subtracting the first quartile from the third quartile.
-
Mean (arithmetic)
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a number that measures the central tendency of the data; a common name for mean is 'average.' The term 'mean' is a shortened form of 'arithmetic mean.' By definition, the mean for a sample (denoted by \(\overline{x}\)) is \(\overline{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample }}\), and the mean for a population (denoted by
μ
) is \(\boldsymbol{\mu}=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\)
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Median
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a number that separates ordered data into halves; half the values are the same number or smaller than the median and half the values are the same number or larger than the median. The median may or may not be part of the data.
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Midpoint
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the mean of an interval in a frequency table
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Mode
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the value that appears most frequently in a set of data
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Outlier
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an observation that does not fit the rest of the data
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Percentile
-
a number that divides ordered data into hundredths; percentiles may or may not be part of the data. The median of the data is the second quartile and the 50
th
percentile. The first and third quartiles are the 25
th
and the 75
th
percentiles, respectively.
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Quartiles
-
the numbers that separate the data into quarters; quartiles may or may not be part of the data. The second quartile is the median of the data.
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Relative Frequency
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the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes
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Standard Deviation
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a number that is equal to the square root of the variance and measures how far data values are from their mean; notation:
s
for sample standard deviation and σ for population standard deviation.
-
Variance
-
mean of the squared deviations from the mean, or the square of the standard deviation; for a set of data, a deviation can be represented as
x
– \(\overline{x}\) where
x
is a value of the data and \(\overline{x}\) is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and one.


14.3: Determining Evolutionary Relationships
By the end of this section, you will be able to do the following:
- Compare homologous and analogous traits
- Discuss the purpose of cladistics
Scientists collect information that allows them to make evolutionary connections between organisms. Similar to detective work, scientists must use evidence to uncover the facts. In the case of phylogeny, evolutionary investigations focus on two types of evidence: morphologic (form and function) and genetic.
Two Measures of Similarity
Organisms that share similar physical features and genetic sequences tend to be more closely related than those that do not. Features that overlap both morphologically and genetically are referred to as homologous structures; the similarities stem from common evolutionary paths. For example, as shown in Figure \(\PageIndex{1}\), the bones in the wings of bats and birds, the arms of humans, and the foreleg of a horse are homologous structures. Notice the structure is not simply a single bone, but rather a grouping of several bones arranged in a similar way in each organism even though the elements of the structure may have changed shape and size.
Misleading Appearances
Some organisms may be very closely related, even though a minor genetic change caused a major morphological difference to make them look quite different. For example, chimpanzees and humans, the skulls of which are shown in Figure \(\PageIndex{2}\) are very similar genetically, sharing 99 percent 1 of their genes. However, chimpanzees and humans show considerable anatomical differences, including the degree to which the jaw protrudes in the adult and the relative lengths of our arms and legs.
However, unrelated organisms may be distantly related yet appear very much alike, usually because common adaptations to similar environmental conditions evolved in both. An example is the streamlined body shapes, the shapes of fins and appendages, and the shape of the tails in fishes and whales, which are mammals. These structures bear superficial similarity because they are adaptations to moving and maneuvering in the same environment—water. When a characteristic that is similar occurs by adaptive convergence (convergent evolution), and not because of a close evolutionary relationship, it is called an analogous structure. In another example, insects use wings to fly like bats and birds. We call them both wings because they perform the same function and have a superficially similar form, but the embryonic origin of the two wings is completely different. The difference in the development, or embryogenesis, of the wings in each case is a signal that insects and bats or birds do not share a common ancestor that had a wing. The wing structures, shown in Figure \(\PageIndex{3}\) evolved independently in the two lineages.
Similar traits can be either homologous or analogous. Homologous traits share an evolutionary path that led to the development of that trait, and analogous traits do not. Scientists must determine which type of similarity a feature exhibits to decipher the phylogeny of the organisms being studied.
Molecular Comparisons
With the advancement of DNA technology, the area of molecular systematics, which describes the use of information on the molecular level including DNA sequencing, has blossomed. New analysis of molecular characters not only confirms many earlier classifications, but also uncovers previously made errors. Molecular characters can include differences in the amino-acid sequence of a protein, differences in the individual nucleotide sequence of a gene, or differences in the arrangements of genes. Phylogenies based on molecular characters assume that the more similar the sequences are in two organisms, the more closely related they are. Different genes change evolutionarily at different rates and this affects the level at which they are useful at identifying relationships. Rapidly evolving sequences are useful for determining the relationships among closely related species. More slowly evolving sequences are useful for determining the relationships between distantly related species. To determine the relationships between very different species such as Eukarya and Archaea, the genes used must be very ancient, slowly evolving genes that are present in both groups, such as the genes for ribosomal RNA. Comparing phylogenetic trees using different sequences and finding them similar helps to build confidence in the inferred relationships.
Sometimes two segments of DNA in distantly related organisms randomly share a high percentage of bases in the same locations, causing these organisms to appear closely related when they are not. For example, the fruit fly shares 60 percent of its DNA with humans. 2 In this situation, computer-based statistical algorithms have been developed to help identify the actual relationships, and ultimately, the coupled use of both morphologic and molecular information is more effective in determining phylogeny.
EVOLUTION IN ACTION: Why Does Phylogeny Matter?
In addition to enhancing our understanding of the evolutionary history of species, our own included, phylogenetic analysis has numerous practical applications. Two of those applications include understanding the evolution and transmission of disease and making decisions about conservation efforts. A 2010 study 3 of MRSA (methicillin-resistant Staphylococcus aureus ), an antibiotic resistant pathogenic bacterium, traced the origin and spread of the strain throughout the past 40 years. The study uncovered the timing and patterns in which the resistant strain moved from its point of origin in Europe to centers of infection and evolution in South America, Asia, North America, and Australasia. The study suggested that introductions of the bacteria to new populations occurred very few times, perhaps only once, and then spread from that limited number of individuals. This is in contrast to the possibility that many individuals had carried the bacteria from one place to another. This result suggests that public health officials should concentrate on quickly identifying the contacts of individuals infected with a new strain of bacteria to control its spread.
A second area of usefulness for phylogenetic analysis is in conservation. Biologists have argued that it is important to protect species throughout a phylogenetic tree rather than just those from one branch of the tree. Doing this will preserve more of the variation produced by evolution. For example, conservation efforts should focus on a single species without sister species rather than another species that has a cluster of close sister species that recently evolved. If the single evolutionarily distinct species goes extinct a disproportionate amount of variation from the tree will be lost compared to one species in the cluster of closely related species. A study published in 2007 4 made recommendations for conservation of mammal species worldwide based on how evolutionarily distinct and at risk of extinction they are. The study found that their recommendations differed from priorities based on simply the level of extinction threat to the species. The study recommended protecting some threatened and valued large mammals such as the orangutans, the giant and lesser pandas, and the African and Asian elephants. But they also found that some much lesser known species should be protected based on how evolutionary distinct they are. These include a number of rodents, bats, shrews and hedgehogs. In addition there are some critically endangered species that did not rate as very important in evolutionary distinctiveness including species of deer mice and gerbils. While many criteria affect conservation decisions, preserving phylogenetic diversity provides an objective way to protect the full range of diversity generated by evolution.
Building Phylogenetic Trees
How do scientists construct phylogenetic trees? Presently, the most accepted method for constructing phylogenetic trees is a method called cladistics. This method sorts organisms into clades, groups of organisms that are most closely related to each other and the ancestor from which they descended. For example, in Figure \(\PageIndex{4}\), all of the organisms in the shaded region evolved from a single ancestor that had amniotic eggs. Consequently, all of these organisms also have amniotic eggs and make a single clade, also called a monophyletic group. Clades must include the ancestral species and all of the descendants from a branch point.
ART CONNECTION
Which animals in this figure belong to a clade that includes animals with hair? Which evolved first: hair or the amniotic egg?
Clades can vary in size depending on which branch point is being referenced. The important factor is that all of the organisms in the clade or monophyletic group stem from a single point on the tree. This can be remembered because monophyletic breaks down into “mono,” meaning one, and “phyletic,” meaning evolutionary relationship.
Shared Characteristics
Cladistics rests on three assumptions. The first is that living things are related by descent from a common ancestor, which is a general assumption of evolution. The second is that speciation occurs by splits of one species into two, never more than two at a time, and essentially at one point in time. This is somewhat controversial, but is acceptable to most biologists as a simplification. The third assumption is that traits change enough over time to be considered to be in a different state .It is also assumed that one can identify the actual direction of change for a state. In other words, we assume that an amniotic egg is a later character state than non-amniotic eggs. This is called the polarity of the character change. We know this by reference to a group outside the clade: for example, insects have non-amniotic eggs; therefore, this is the older or ancestral character state. Cladistics compares ingroups and outgroups. An ingroup (lizard, rabbit and human in our example) is the group of taxa being analyzed. An outgroup (lancelet, lamprey and fish in our example) is a species or group of species that diverged before the lineage containing the group(s) of interest. By comparing ingroup members to each other and to the outgroup members, we can determine which characteristics are evolutionary modifications determining the branch points of the ingroup’s phylogeny.
If a characteristic is found in all of the members of a group, it is a shared ancestral characterbecause there has been no change in the trait during the descent of each of the members of the clade. Although these traits appear interesting because they unify the clade, in cladistics they are considered not helpful when we are trying to determine the relationships of the members of the clade because every member is the same. In contrast, consider the amniotic egg characteristic of Figure \(\PageIndex{4}\). Only some of the organisms have this trait, and to those that do, it is called a shared derived character because this trait changed at some point during descent. This character does tell us about the relationships among the members of the clade; it tells us that lizards, rabbits, and humans group more closely together than any of these organisms do with fish, lampreys, and lancelets.
A sometimes confusing aspect of “ancestral” and “derived” characters is that these terms are relative. The same trait could be either ancestral or derived depending on the diagram being used and the organisms being compared. Scientists find these terms useful when distinguishing between clades during the building of phylogenetic trees, but it is important to remember that their meaning depends on context.
Choosing the Right Relationships
Constructing a phylogenetic tree, or cladogram, from the character data is a monumental task that is usually left up to a computer. The computer draws a tree such that all of the clades share the same list of derived characters. But there are other decisions to be made, for example, what if a species presence in a clade is supported by all of the shared derived characters for that clade except one? One conclusion is that the trait evolved in the ancestor, but then changed back in that one species. Also a character state that appears in two clades must be assumed to have evolved independently in those clades. These inconsistencies are common in trees drawn from character data and complicate the decision-making process about which tree most closely represents the real relationships among the taxa.
To aid in the tremendous task of choosing the best tree, scientists often use a concept called maximum parsimony, which means that events occurred in the simplest, most obvious way. This means that the “best” tree is the one with the fewest number of character reversals, the fewest number of independent character changes, and the fewest number of character changes throughout the tree. Computer programs search through all of the possible trees to find the small number of trees with the simplest evolutionary pathways. Starting with all of the homologous traits in a group of organisms, scientists can determine the order of evolutionary events of which those traits occurred that is the most obvious and simple.
CONCEPT IN ACTION
Practice Parsimony: Click the link to visit the UC Berkeley Museum of Paleontology wepbage on parsimony (new window) to learn how maximum parsimony is used to create phylogenetic trees (be sure to continue to the second page).
These tools and concepts are only a few of the strategies scientists use to tackle the task of revealing the evolutionary history of life on Earth. Recently, newer technologies have uncovered surprising discoveries with unexpected relationships, such as the fact that people seem to be more closely related to fungi than fungi are to plants. Sound unbelievable? As the information about DNA sequences grows, scientists will become closer to mapping the evolutionary history of all life on Earth.
Art Connections
Figure \(\PageIndex{3}\): Which animals in this figure belong to a clade that includes animals with hair? Which evolved first: hair or the amniotic egg?
- Answer
-
Rabbits and humans belong in the clade that includes animals with hair. The amniotic egg evolved before hair, because the Amniota clade branches off earlier than the clade that encompasses animals with hair.
Footnotes
- 1 Gibbons, A. (2012, June 13). Science Now . Retrieved from news.sciencemag.org/scienceno...sequenced.html
- 2 Background on comparative genomic analysis . (2002, December). Retrieved from http://www.genome.gov/10005835
- 3 Harris, S.R. et al. 2010. Evolution of MRSA during hospital transmission and intercontinental spread. Science 327:469–474.
- 4 Isaac NJ, Turvey ST, Collen B, Waterman C, Baillie JE (2007) Mammals on the EDGE: Conservation Priorities Based on Threat and Phylogeny. PLoS ONE 2(3): e296. doi:10.1371/journal.pone.0000296
Contributors and Attributions
Remixed and/or curated from the following works:
Fowler, S., Roush, R., & Wise, J. (2013). 12.2 Determining Evolutionary Relationships . In Concepts of Biology . OpenStax (CC BY 4.0; Access for free at https://openstax.org/books/concepts-biology/pages/1-introduction ).


8.2: Evaluating Entropy Changes Using Thermochemical Cycles
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As for the standard enthalpy of reaction, we can obtain the standard entropy of reaction at a new temperature by evaluating entropy changes around a suitable thermochemical cycle. To do so, we need the standard entropy change at one temperature. We also need heat capacity data for all of the reactants and products. For the reaction
\[a\ A+b\ B\to c\ C+d\ D \nonumber \]
we can evaluate the entropy change at a second temperature by summing the individual contributions to the change in entropy around the cycle in Figure 5. For this cycle, we have
\[{\Delta }_rS^o\left(T_2\right)={\Delta }_rS^o\left(T_1\right)+c\int^{T_2}_{T_1}{\frac{C_P\left(C\right)}{T}dT} +d\int^{T_2}_{T_1}{\frac{C_P\left(D\right)}{T}dT}-a\int^{T_2}_{T_1}{\frac{C_P\left(A\right)}{T}dT}-b\int^{T_2}_{T_1}{\frac{C_P\left(B\right)}{T}dT} \nonumber \]


3.9: Introduction to Accounting
What you’ll learn to do: define accounting
What we think of as accounting can be divided into two distinct functions: bookkeeping, and accounting. The bookkeeping function, recording transactions, is as old as commerce itself, having arisen as a necessity of recording for taxes, business, and investments. In fact, Luca Pacioli, and Italian mathematician who collaborated with Leonardo da Vinci, mentions double entry booking in his his 1494 “Summa de Arithmetica, Geometria, Proportioni et Proportionalita.”
As the world of business grew and expanded, so did the complexity and variety of accounting practices, and so did the opportunity for profits and losses for investors. Today, accounting is the application of theoretical principles to the books and records of the company in order to produce useful information for decision-making, by following, in the U.S., Generally Accepted Accounting Principles (GAAP).
In this section, we’ll define accounting, examine the financial standards, and describe the importance of ethics in accounting.