# Wigner-Smith Time Delay Matrix for Acoustic Scattering: Theory and
Phenomenology

Utkarsh R. Patel  Yiqian Mao  Eric Michielssen  Department of Electrical
Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan
48109-2122, United States

###### Abstract

The Wigner-Smith (WS) time delay matrix relates a lossless system’s scattering
matrix to its frequency derivative. First proposed in the realm of quantum
mechanics to characterize time delays experienced by particles during a
collision, this article extends the use of WS time delay techniques to
acoustic scattering problems governed by the Helmholtz equation. Expression
for the entries of the WS time delay matrix involving renormalized volume
integrals of energy densities are derived, and shown to hold true independent
of the scatterer’s geometry, boundary condition (sound-soft or sound-hard),
and excitation. Numerical examples show that the eigenmodes of the WS time
delay matrix describe distinct scattering phenomena characterized by well-
defined time delays.

##  I  Introduction

The concept of the Wigner-Smith (WS) time delay matrix $\mathbf{Q}$ was
proposed by Felix Smith to characterize time delays experienced by quantum
particles during a collision [18]. Specifically, Smith showed that the matrix

$\displaystyle\mathbf{Q}=j\mathbf{S}^{\dagger}\frac{\partial\mathbf{S}}{\partial
k}$ (1)

where $\mathbf{S}$ is the scattering matrix of a lossless potential well,
$\partial/\partial k$ denotes the derivative w.r.t. wavenumber $k$, and
$\,{}^{\dagger}$ represents the adjoint operation, fully characterizes the
time particles dwell in a system.

Since its introduction in 1960, the WS time delay matrix has found many
applications in quantum mechanics, photonics, and electromagnetics. In quantum
mechanics, the WS time delay matrix $\mathbf{Q}$ has been used to study
photoionization and photoemission time delays [10, 13], to analyze traversal
times during quantum tunneling [5, 14], and to characterize quantum mechanical
decay mechanisms [7]. In photonics, $\mathbf{Q}$ has been used to study wave
propagation in multimode fibers [6], to shape light flow in disordered media
and complex cavities [4, 12, 3], and to optimize light storage in highly
scattering media [8]. In electromagnetics, WS time delays have been used for
focusing energy in a microwave cavity [2], to characterize group delays of
fields interacting with a two-port waveguide [19], to decompose
electromagnetic fields in terms of modes with well-defined time delays [15],
and to characterize the frequency sensitivity of antenna input impedances
[16].

Applications of WS time delay concepts to acoustic scattering problems have
been few and far in between, however. Exceptions include the use of
$\mathbf{Q}$ to study acoustic resonances in fluid-loaded elastic plates [9],
the study of scattering from two-dimensional fluid slabs in an elastic medium
[17], and the characterization of elastic wave propagation in disordered media
[11].

This work, along with its companion paper “Wigner-Smith Time Delay Matrix for
Acoustic Scattering: Computational Aspects”, extends the authors’ work [15] on
WS methods for Maxwell’s equations (electromagnetics) to the Helmholtz
equation (acoustics). The WS time delay matrix of a lossless acoustic
scatterer possesses several interesting properties that carry over wholesale
from its electromagnetic counterpart, and are summarized next – the reader is
referred to the section on “WS Theory: Systems Perspective” in reference [15]
for a detailed derivation and justification of the statements that follow.

  * •

First, the WS time delay matrix $\mathbf{Q}$ is Hermitian and hence its
diagonal elements are purely real. This property can be derived by using the
symmetry and unitary properties of the scattering matrix, i.e.
$\mathbf{S}^{T}=\mathbf{S}$ and $\mathbf{S}^{\dagger}\mathbf{S}=\mathbf{I}$,
where $\mathbf{I}$ is an identity matrix and $\,{}^{T}$ denotes the transpose
operation.

  * •

Second, the $n$-th diagonal element of $\mathbf{Q}$ represents the average
time delay experienced by an incoming narrowband wave packet that enters the
system via its $n$-th port prior to exiting via any other port.

  * •

Third, $\mathbf{Q}$ and $\mathbf{S}$ can be simultaneously diagonalized as
$\mathbf{Q}=\mathbf{W}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern-0.50003pt$\mathbf{Q}$\kern-0.50003pt}}}\mathbf{W}^{\dagger}$
and $\mathbf{S}=\mathbf{W}^{*}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern-0.50003pt$\mathbf{S}$\kern-0.50003pt}}}\mathbf{W}^{\dagger}$.
$\mathbf{W}$ is a unitary matrix whose columns are eigenvectors of
$\mathbf{Q}$ and are often called “WS modes”.  $\mathbf{Q}$ is a real diagonal
matrix that collects $\mathbf{Q}$’s eigenvalues, often called “WS time delays”
or “WS dwell times”.

  * •

Fourth, the simultaneous diagonalization of $\mathbf{Q}$ and $\mathbf{S}$
implies that WS modes are decoupled and are characterized by well-defined time
delays. In other words, an incoming field composed of narrowband wave packets
that enter the system weighed by the entries of $\mathbf{Q}$’s $n$-th
eigenvector exit the system with a time delay characterized by $\mathbf{Q}$’s
$n$-th eigenvalue.

This paper elucidates the definition and use of the WS time delay matrix for
acoustic scattering problems governed by the Helmholtz equation. Its
contributions are twofold.

  * •

First, it demonstrates that the elements of $\mathbf{Q}$ can be expressed as
renormalized volume integrals of energy density-like quantities involving
fluid potentials. The derived volume integral expressions for elements of
$\mathbf{Q}$ provide an alternative to Eqn. (1), allowing $\mathbf{Q}$’s
computation without knowledge of $\mathbf{S}$ and $\partial\mathbf{S}/\partial
k$. When used in conjunction with Eqn. (1), the volume integral expression for
$\mathbf{Q}$ can be used to compute $\partial\mathbf{S}/\partial k$ provided
$\mathbf{S}$ is known.

  * •

Second, it demonstrates via numerical experimentation that WS modes can be
used to untangle various wave components, including those associated with
corner/edge diffraction, surface waves, ballistic scattering, and resonant
modes, and that each of these is characterized by distinct and well-defined
time delays/dwell times and stored energies.

The two topics above are detailed in Secs. II and Sec. III below. Conclusions
and avenues for future research are provided in Sec. IV. Throughout this
paper, the superscript ′ denotes ${\partial}/{\partial}k$.

##  II WS Theory for Acoustic Scattering

Figure 1: Scattering system excited through free-space port defined on sphere
of radius $R$.

This section shows that the WS time delay matrix for acoustic scattering
problems governed by the Helmholtz equation can be computed directly by
integrating energy-like densities involving the total fluid velocity potential
over $\mathbb{R}^{3}$.

###  II.1 Setup

Consider a closed scatterer $\Omega_{s}$ with surface $d\Omega_{s}$ that
resides in a lossless and homogeneous medium with (real) wavenumber $k$ and is
circumscribed by a sphere of radius $a$ centered at the origin (Fig. 1). Let
$\Omega$ and $d\Omega$ denote the volume and surface of a sphere of radius
$R\gg a$ and $kR\gg 1$, also centered about the origin. The fluid velocity
potential $\phi(\mathbf{x})$ in a source-free medium
$\mathbb{R}^{3}\backslash\Omega_{s}$ satisfies the Helmholtz equation

$\displaystyle\nabla^{2}\phi(\mathbf{x})+k^{2}\phi(\mathbf{x})=0$ (2)

subject to the Dirichlet boundary condition

$\displaystyle\phi(\mathbf{x})=0,\quad\mathbf{x}\in d\Omega_{s}$ (3)

for a sound-soft scatterer or the Neumann boundary condition

$\displaystyle\frac{\partial\phi(\mathbf{x})}{\partial
n(\mathbf{x})}=0,\quad\mathbf{x}\in d\Omega_{s}$ (4)

for a sound-hard scatterer; here $\partial/\partial n(\mathbf{x})$ denotes the
outward normal derivative on $d\Omega_{s}$.

On $d\Omega$, the velocity potential may be expanded in terms of $M$
orthonormal modes ${\cal X}_{p}(\theta,\phi)$. While there exist many
different choices for modes ${\cal X}_{p}(\theta,\phi)$, this paper adopts a
specific realization based on spherical harmonics ${\cal X}_{p}(\theta,\phi)$
with $p=(l,m)$, where $l=0,\ldots,l_{max}$, $m=-l,\ldots,l$, detailed in
Appendix A; $M=(l_{max}+1)^{2}$, $l_{max}=ka+c(ka)^{1/3}$ and typically
$2<c<4$ [20]; these modes are orthonormal, i.e.

$\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}{\cal X}_{p}(\theta,\phi){\cal
X}_{q}^{*}(\theta,\phi)\sin\theta d\theta d\phi=\delta_{p,q}$ (5)

where $\delta_{l,l^{\prime}}$ is a Kronecker delta function.

Assume that the scatterer is excited by a radially-travelling incoming ($i$)
unit-power field $\phi_{p}^{i}(\mathbf{x})$. Using large argument
approximations, this field on $d\Omega$ reads

$\displaystyle\phi_{p,\infty}^{i}(\mathbf{x})={\cal
X}_{p}(\theta,\phi)\frac{e^{jkr}}{r}\,.$ (6)

Next, let $\phi_{p}(\mathbf{x})$ denote the total field throughout $\Omega$
due to the excitation in Eqn. (6). Near $d\Omega$, this field is

$\displaystyle\phi_{p,\infty}(\mathbf{x})=\phi_{p,\infty}^{i}(\mathbf{x})+\phi_{p,\infty}^{o}(\mathbf{x})$
(7)

where $\phi_{p,\infty}^{o}(\mathbf{x})$, the outgoing field near $d\Omega$,
can be expressed as

$\displaystyle\phi_{p,\infty}^{o}(\mathbf{x})$
$\displaystyle=\sum_{m=1}^{M}\mathbf{S}_{mp}{\cal
X}_{m}^{*}(\theta,\phi)\frac{e^{-jkr}}{r}\,.$ (8)

That is, $\phi_{p,\infty}^{o}(\mathbf{x})$ is the superposition of $M$
outgoing modes weighed by wavenumber-dependent scattering coefficients
$\mathbf{S}_{mp}$. The above construct guarantees that $\mathbf{S}$ is
symmetric and unitary, i.e.

$\displaystyle\mathbf{S}^{\dagger}\mathbf{S}$ $\displaystyle=\mathbf{I}$ (9a)
$\displaystyle\mathbf{S}^{T}$ $\displaystyle=\mathbf{S}\,.$ (9b)

In preparation of the derivation of the WS relationship, consider two velocity
potentials, $\phi_{p}(\mathbf{x})$ and $\phi_{q}(\mathbf{x})$. Taking the
derivative w.r.t $k$ of Eqn. (2) with $\phi(\mathbf{x})=\phi_{p}(\mathbf{x})$
yields

$\displaystyle\nabla^{2}\phi_{p}^{\prime}(\mathbf{x})+2k\phi_{p}(\mathbf{x})+k^{2}\phi_{p}^{\prime}(\mathbf{x})$
$\displaystyle=0\,.$ (10)

Multiplying the conjugate of Eqn. (2) with
$\phi(\mathbf{x})=\phi_{q}(\mathbf{x})$ by $\phi_{p}^{\prime}(\mathbf{x})$ and
subtracting from it Eqn. (10) multiplied by $\phi_{q}^{*}(\mathbf{x})$ yields

$\displaystyle\frac{1}{2k}(\phi_{p}^{\prime}(\mathbf{x})\nabla^{2}\phi_{q}^{*}(\mathbf{x})$
$\displaystyle-\phi_{q}^{*}(\mathbf{x})\nabla^{2}\phi_{p}^{\prime}(\mathbf{x}))$
$\displaystyle=\phi_{p}(\mathbf{x})\phi_{q}^{*}(\mathbf{x})\,.$ (11)

Using the identity
$\psi\nabla^{2}\gamma-\gamma\nabla^{2}\psi=\nabla\cdot\left(\psi\nabla\gamma-\gamma\nabla\psi\right)$,
Eqn. (11) simplifies to

$\displaystyle\frac{1}{2k}\nabla\cdot\left(\phi_{p}^{\prime}(\mathbf{x})\nabla\phi_{q}^{*}(\mathbf{x})-\phi_{q}^{*}(\mathbf{x})\nabla\phi_{p}^{\prime}(\mathbf{x})\right)$
$\displaystyle=\phi_{p}(\mathbf{x})\phi_{q}^{*}(\mathbf{x})\,.$ (12)

Next, the RHS of Eqn. (12) is rewritten using Eqn. (2) with
$\phi(\mathbf{x})=\phi_{q}^{*}(\mathbf{x})$ and the vector identity
$\phi\nabla^{2}\gamma=\nabla\cdot(\phi\nabla\gamma)-\nabla\gamma\cdot\nabla\phi$
to obtain

$\displaystyle\frac{1}{2k}\nabla\cdot$
$\displaystyle\left(\phi_{p}^{\prime}(\mathbf{x})\nabla\phi_{q}^{*}(\mathbf{x})-\phi_{q}^{*}(\mathbf{x})\nabla\phi_{p}^{\prime}(\mathbf{x})\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\phi_{p}(\mathbf{x})\phi_{q}^{*}(\mathbf{x})$
$\displaystyle-\frac{1}{2k^{2}}\nabla\cdot(\phi_{p}(\mathbf{x})\nabla\phi_{q}^{*}(\mathbf{x}))$
$\displaystyle+\frac{1}{2k^{2}}\nabla\phi_{q}^{*}(\mathbf{x})\cdot\nabla\phi_{p}(\mathbf{x})\,.$
(13)

Finally, integrating Eqn. (13) over $\Omega$ and using the divergence theorem
yields

$\displaystyle\frac{1}{2k}$
$\displaystyle\int_{d\Omega}\bm{\hat{r}}\cdot\bigg{[}\phi_{p,\infty}^{\prime}(\mathbf{x})\nabla\phi_{q,\infty}^{*}(\mathbf{x})-\phi_{q,\infty}^{*}(\mathbf{x})\nabla\phi_{p,\infty}^{\prime}(\mathbf{x})$
$\displaystyle\quad+\frac{1}{k}\nabla\phi_{q,\infty}^{*}(\mathbf{x})\phi_{p,\infty}(\mathbf{x})\bigg{]}d\mathbf{x}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\phi_{p}(\mathbf{x})\phi_{q}^{*}(\mathbf{x})d\mathbf{x}$
$\displaystyle\quad+\frac{1}{2k^{2}}\int_{\Omega}\nabla\phi_{q}^{*}(\mathbf{x})\cdot\nabla\phi_{p}(\mathbf{x})d\mathbf{x}\,.$
(14)

The above equation assumes a hard or soft-source boundary condition to
eliminate the integration over $d\Omega_{s}$ on the LHS.

###  II.2 WS Relationship

The evaluation of the LHS of Eqn. (14) requires expressions for
$\phi_{p,\infty}^{\prime}(\mathbf{x})$,
$\bm{\hat{r}}\cdot\nabla\phi_{p,\infty}^{\prime}(\mathbf{x})$,
$\phi_{q,\infty}^{*}(\mathbf{x})$, and
$\bm{\hat{r}}\cdot\nabla\phi_{q,\infty}^{*}(\mathbf{x})$. Substituting Eqns.
(6) and (8) into Eqn. (7) and differentiating w.r.t. $k$ yields

$\displaystyle\phi_{p,\infty}^{\prime}(\mathbf{x})$ $\displaystyle=j{\cal
X}_{p}(\theta,\phi)e^{jkr}$
$\displaystyle\quad+\sum_{n=1}^{M}\mathbf{S}_{np}^{\prime}{\cal
X}_{n}^{*}(\theta,\phi)\frac{e^{-jkr}}{r}$
$\displaystyle\quad-j\sum_{n=1}^{M}\mathbf{S}_{np}{\cal
X}_{n}^{*}(\theta,\phi)e^{-jkr}\,.$ (15)

Taking the gradient of Eqn. (15), dotting the result with $\bm{\hat{r}}$, and
using the large argument behavior of $r\rightarrow\infty$ produces

$\displaystyle\bm{\hat{r}}$
$\displaystyle\cdot\nabla\phi_{p,\infty}^{\prime}(\mathbf{x})$
$\displaystyle=-k{\cal X}_{p}(\theta,\phi)e^{jkr}$
$\displaystyle\quad+\sum_{n=1}^{M}\mathbf{S}_{np}^{\prime}{\cal
X}_{n}^{*}(\theta,\phi)\frac{e^{-jkr}}{r}\left(-jk\right)$
$\displaystyle\quad-k\sum_{n=1}^{M}\mathbf{S}_{np}{\cal
X}_{n}^{*}(\theta,\phi)e^{-jkr}.$ (16)

Likewise, substituting Eqns. (6) and (8) into Eqn. (7) while changing $p$ to
$q$, and conjugating the result yields

$\displaystyle\phi_{q,\infty}^{*}(\mathbf{x})$ $\displaystyle={\cal
X}_{q}^{*}(\theta,\phi)\frac{e^{-jkr}}{r}$
$\displaystyle\quad+\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}{\cal
X}_{m}(\theta,\phi)\frac{e^{jkr}}{r}\,.$ (17)

Finally, taking the gradient of Eqn. (17), dotting it with $\bm{\hat{r}}$ and
using the large argument approximation for $r$ results in

$\displaystyle\bm{\hat{r}}\cdot\nabla\phi_{q,\infty}^{*}(\mathbf{x})$
$\displaystyle={\cal
X}_{q}^{*}(\theta,\phi)\frac{e^{-jkr}}{r}\left(-jk\right)$
$\displaystyle+\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}{\cal
X}_{m}(\theta,\phi)\frac{e^{jkr}}{r}\left(jk\right)\,.$ (18)

Next, substituting Eqns. (15), (16), (17), and (18) into the LHS of Eqn. (14)
and evaluating the resulting integral as detailed in Appendix B yields

$\displaystyle\frac{1}{2k}$
$\displaystyle\int_{d\Omega}\bm{\hat{r}}\cdot\bigg{[}\phi_{p,\infty}^{\prime}(\mathbf{x})\nabla\phi_{q,\infty}^{*}(\mathbf{x})-\phi_{q,\infty}^{*}(\mathbf{x})\nabla\phi_{p,\infty}^{\prime}(\mathbf{x})$
$\displaystyle\quad+\frac{1}{k}\nabla\phi_{q,\infty}^{*}(\mathbf{x})\phi_{p,\infty}(\mathbf{x})\bigg{]}d\mathbf{x}$
$\displaystyle=2R\delta_{p,q}+j\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}\mathbf{S}_{mp}^{\prime}.$
(19)

By using Eqn. (19), Eqn. (14) simplifies to

$\displaystyle\widetilde{\mathbf{Q}}_{qp}=2R\delta_{p,q}+j\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}\mathbf{S}_{mp}^{\prime}$
(20)

where $\widetilde{\mathbf{Q}}_{qp}$ reads

$\displaystyle\widetilde{\mathbf{Q}}_{qp}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\phi_{p}(\mathbf{x})\phi_{q}^{*}(\mathbf{x})d\mathbf{x}$
$\displaystyle\quad+\frac{1}{2k^{2}}\int_{\Omega}\nabla\phi_{q}^{*}(\mathbf{x})\cdot\nabla\phi_{p}(\mathbf{x})d\mathbf{x}.$
(21)

To obtain a WS relationship that is independent of $R$, consider the quantity
$\widetilde{\mathbf{Q}}_{qp,\infty}$ obtained by replacing
$(\phi_{p},\nabla\phi_{p})$ and $(\phi_{q},\nabla\phi_{q})$ by
$(\phi_{p,\infty},\nabla\phi_{p,\infty})$ and
$(\phi_{q,\infty},\nabla\phi_{q,\infty})$, i.e.

$\displaystyle\widetilde{\mathbf{Q}}_{qp,\infty}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\phi_{p,\infty}(\mathbf{x})\phi_{q,\infty}^{*}(\mathbf{x})d\mathbf{x}$
$\displaystyle\quad+\frac{1}{2k^{2}}\int_{\Omega}\nabla\phi_{q,\infty}^{*}(\mathbf{x})\cdot\nabla\phi_{p,\infty}(\mathbf{x})d\mathbf{x}.$
(22)

The quantities $\phi_{p,\infty}$, $\phi_{q,\infty}$ in Eqn. (22) are the same
as those in Eqn. (7) though their use is extended to all of
$\mathbf{x}\in\Omega$; a similar interpretation holds true for their
gradients. Evaluating the integral in Eqn. (22) yields

$\displaystyle\widetilde{\mathbf{Q}}_{qp,\infty}$
$\displaystyle=2R\delta_{p,q}\,.$ (23)

Finally, subtracting $\widetilde{\mathbf{Q}}_{qp,\infty}$ from both sides of
Eqn. (20) yields the WS relationship

$\displaystyle\mathbf{Q}_{qp}=j\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}\mathbf{S}_{mp}^{\prime}\,,$
(24)

where the $(q,p)$-th entry of the WS time delay matrix $\mathbf{Q}$ is

$\displaystyle\mathbf{Q}_{qp}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\left[\phi_{p}(\mathbf{x})\phi_{q}^{*}(\mathbf{x})-\phi_{p,\infty}(\mathbf{x})\phi_{q,\infty}^{*}(\mathbf{x})\right]d\mathbf{x}$
$\displaystyle\quad+\frac{1}{2k^{2}}\int_{\Omega}\big{[}\nabla\phi_{q}^{*}(\mathbf{x})\cdot\nabla\phi_{p}(\mathbf{x})$
$\displaystyle\quad\quad\quad-\nabla\phi_{q,\infty}^{*}(\mathbf{x})\cdot\nabla\phi_{p,\infty}(\mathbf{x})\big{]}d\mathbf{x}\,.$
(25)

In matrix form, Eqn. (24) reads

$\displaystyle\mathbf{Q}=j\mathbf{S}^{\dagger}\mathbf{S}^{\prime}\,,$ (26)

showing that the entries of the WS time delay matrix are renormalized volume
integrals of energy-like densities.

###  II.3 Alternative Formulations

The derivation above casts elements of the WS time delay matrix in terms of
integrals involving velocity potentials $\phi_{p}(\mathbf{x})$ and
$\phi_{q}(\mathbf{x})$, and their gradients $\nabla\phi_{p}(\mathbf{x})$ and
$\nabla\phi_{q}(\mathbf{x})$. It is often convenient to compute the entries of
the WS time delay matrix from just the scalar fields or their gradients. The
first alternative formulation, obtained by integrating Eqn. (12), reads

$\displaystyle\mathbf{Q}^{a}$
$\displaystyle=j\mathbf{S}^{\dagger}\mathbf{S}^{\prime}$ (27)

where

$\displaystyle\mathbf{Q}^{a}_{qp}=\widetilde{\mathbf{Q}}_{qp}^{a}+\frac{j}{2k}(-1)^{m}\mathbf{S}_{q,\tilde{p}}-\frac{j}{2k}(-1)^{m}\mathbf{S}_{\tilde{p},q}^{*}$
(28)

and

$\displaystyle\widetilde{\mathbf{Q}}_{qp}^{a}$
$\displaystyle=\int_{\Omega}\left[\phi_{p}(\mathbf{x})\phi_{q}^{*}(\mathbf{x})-\phi_{p,\infty}(\mathbf{x})\phi_{q,\infty}^{*}(\mathbf{x})\right]d\mathbf{x}\,.$
(29)

The derivation of Eqn. (28) requires use of the conjugation property of
spherical harmonics with ${p}=(l,m)$, ${q}=(l^{\prime},m^{\prime})$, and
$\tilde{p}=(l,-m)$ (see Eqn. (35) in Appendix A). The second alternative
formulation, obtained by integrating two times Eqn. (13) and subtracting Eqn.
(12), reads

$\displaystyle\mathbf{Q}^{b}$
$\displaystyle=j\mathbf{S}^{\dagger}\mathbf{S}^{\prime}$ (30)

where

$\displaystyle\mathbf{Q}^{b}_{qp}=\widetilde{\mathbf{Q}}_{qp}^{b}-\frac{j}{2k}(-1)^{m}\mathbf{S}_{q,\tilde{p}}+\frac{j}{2k}(-1)^{m}\mathbf{S}^{*}_{\tilde{p},q}$
(31)

and

$\displaystyle\widetilde{\mathbf{Q}}_{qp}^{b}=$
$\displaystyle\frac{1}{k^{2}}\int_{\Omega}\bigg{[}\nabla\phi_{p}(\mathbf{x})\cdot\nabla\phi_{q}^{*}(\mathbf{x})$
$\displaystyle\quad\quad-\nabla\phi_{p,\infty}(\mathbf{x})\cdot\nabla\phi_{q,\infty}^{*}(\mathbf{x})\bigg{]}d\mathbf{x}\,.$
(32)

It is easily verified that the symmetric formulation in Eqn. (26) is obtained
by adding Eqns. (27) and (30).

##  III Illustrative Examples

This section presents two examples that demonstrate the use of WS methods for
decomposing fields interacting with scatterers into WS modes characterized by
well-defined time delays. The WS modes for sound-soft and sound-hard
scatterers are seen to exhibit different characteristics. Both examples
involve 2D geometries to facilitate visualization of the fields.

###  III.1 Thin Rectangular Strip

First, consider scattering from a 2D rectangular strip of size
$50\mathrm{m}\times 1\mathrm{m}$ centered about $(0,0.25\mathrm{m})$. Note
that the placement of the strip relative to the origin affects the WS modes as
$\mathbf{Q}$ is related to $\mathbf{S}$ and $\mathbf{S}^{\prime}$, both of
which are origin-dependent. The strip is illuminated by $M=111$ incoming
harmonics with $k=1$. Any other incoming field can be expressed as a
superposition of these harmonics. Knowledge of the scattered field associated
with these illuminations is used to construct the strip’s dense $111\times
111$ $\mathbf{Q}$ and $\mathbf{S}$ matrices, and $\mathbf{Q}$ is diagonalized
as $\mathbf{Q}=\mathbf{W}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern-0.50003pt$\mathbf{Q}$\kern-0.50003pt}}}\mathbf{W}^{\dagger}$.
Figs. 2 and 3 show total fields associated with selected WS modes constructed
by weighing incoming harmonics with entries of columns of $\mathbf{W}$ for
sound-soft and sound-hard scatterers, respectively. The corresponding time
delays (diagonal elements of  $\mathbf{Q}$ ) are shown in Figs. 4 and 5. In
what follows, WS modes are ordered by their time-delays (small to large) and
indexed accordingly.

Fig2a.jpeg0.3(a) WS mode #1 Fig2b.jpeg0.3(b) WS mode #3 Fig2c.jpeg0.3(c) WS
mode #5 Fig2d.jpeg0.3(d) WS mode #6 Fig2e.jpeg0.3(e) WS mode #20
Fig2f.jpeg0.3(f) WS mode #50

Figure 2: Selected WS modes of the sound-soft rectangular strip.

Fig3a.jpeg0.3(a) WS mode #1 Fig3b.jpeg0.3(b) WS mode #3 Fig3c.jpeg0.3(c) WS
mode #4 Fig3d.jpeg0.3(d) WS mode #17 Fig3e.jpeg0.3(e) WS mode #108
Fig3f.jpeg0.3(f) WS mode #111

Figure 3: Selected WS modes of the sound-hard rectangular strip. Figure 4: WS
time delays for the sound-soft rectangular strip. Figure 5: WS time delays for
the sound-hard rectangular strip.

Sound-Soft Scatterer: The modes can be classified into three groups:

  1. a.

_Corner Modes_ : WS modes 1-4 illuminate the corners of the strip; total
fields for modes 1 and 3 are shown in Figs. 2a and 2b. Modes 1 and 2
illuminate the two corners on the left (or right) side of the strip with a
common-mode excitation (same magnitude and phase) while modes 3 and 4 excite
the same with a differential mode excitations (same magnitude but opposite
phase). Corner modes experience negative time delays (between $-32\mathrm{s}$
and $-6\mathrm{s}$ in this case), because they spend less time in the system
than they would in the absence of the scatterer.

  2. b.

_Ballistic Modes_ : WS modes $5$-$39$ represent beam-like ballistic (or
geometric optics) modes that specularly reflect off the strip. Distributions
of total fields for modes 5, 6, and 20 are shown in Figs. 2c, 2d, and 2e,
respectively. Note that these modes avoid exciting the corners of the strip.
Ballistic modes experience small negative time delays because these modes
impinge on the strip from the $+\bm{\hat{y}}$ or $-\bm{\hat{y}}$ direction,
reflecting off the strip’s top or bottom surfaces and spending between $0s$ to
$1.5\mathrm{s}$ less in the system depending on the angle of incidence
compared to waves that do not interact with the scatterer (distance from
origin to the top and bottom surface of the strip is $0.75\mathrm{m}$ and
$0.25\mathrm{m}$, respectively).

  3. c.

_Non-propagating Modes_ : WS modes 40-111 do not excite the scatterer. The
total field distribution for one such mode is shown in Fig. 2f. These modes
experience near-zero time delays.

Sound-Hard Scatterer: WS modes for a sound-hard strip likewise can be
categorized into three groups:

  1. a.

_Corner/Surface Wave Modes_ : Modes 1-2 and 106-111 excite both the corners
and the edges of the strip. The distributions of the total fields when the
strip is excited by WS modes 1, 108, and 111 are shown in Figs. 3a, 3e, and
3f, respectively. Modes 1 and 2 only weakly excite the long edges of the strip
as most of their energy rapidly scatters off the corners, resulting in
negative time delays. Modes 106-111, in contrast, strongly excite the long
edges of strip and have large positive time delays (and stored energy) as they
involve surface waves that travel back and forth along the strip. Note that
these surface wave phenomena and the associated positive time delays are
unique to the sound-hard case and absent in the sound-soft case.

  2. b.

_Ballistic Modes_ : WS modes 3-32 represent beam-like ballistic modes that
specularly reflect off the strip. Figs. 3b, 3c, and 3d show the distribution
of the total field when the strip is illuminated by ballistic modes 3, 4, and
17, respectively. The behavior of these modes is nearly identical to that of
ballistic modes for the sound-soft scatterer both in terms of the WS time
delays and the distribution of total field.

  3. c.

_Non-propagating Modes_ : WS modes 33-105 are non-propagating modes with near-
zero time delays. These modes behave similarly to those for the sound-soft
case.

This example demonstrates that WS modes can untangle fields into components
associated with canonical scattering phenomena associated with well-defined
time delays/dwell times and system energies.

###  III.2 Open Cavity

$13.5$$w=3$$13.5$$30$$4$$\bm{\hat{x}}$$\bm{\hat{y}}$$(0,0)$ Figure 6:
Dimensions (in $\mathrm{m}$) of an open cavity.

Consider scattering from the open cavity shown in Fig. 6. The cavity is
illuminated with $M=71$ incoming harmonics with $k=1$. Knowledge of the
scattered field is used to construct the $71\times 71$ $\mathbf{S}$ and
$\mathbf{Q}$ matrices. Next $\mathbf{Q}$ is diagonalized and the total fields
associated with selected WS modes for cavities with sound-soft and sound-hard
boundaries are shown in Figs. 7 and 9, respectively. Their corresponding time
delays are shown in Figs. 10 and 11.

Fig7a.png0.3(a) WS mode #1 Fig7b.png0.3(b) WS mode #5 Fig7c.png0.3(c) WS mode
#8 Fig7d.png0.3(d) WS mode #16 Fig7e.png0.3(e) WS mode #50 Fig7f.png0.3(f) WS
mode #71

Figure 7: Selected WS modes of a sound-soft cavity. Figure 8: WS modes 71 of
the sound-soft cavity with the gap of $w=5~{}\mathrm{m}$.

Fig9a.jpg0.3(a) WS mode #1 Fig9b.jpg0.3(b) WS mode #3 Fig9c.jpg0.3(c) WS mode
#15 Fig9d.jpg0.3(d) WS mode #69 Fig9e.jpg0.3(e) WS mode #70 Fig9f.jpg0.3(f) WS
mode #71

Figure 9: Selected WS modes of the sound-hard cavity. Figure 10: WS time
delays for the sound-soft cavity. Figure 11: WS time delays for the sound-hard
cavity.

Sound-Soft Scatterer: WS modes for the sound-soft cavity can be categorized
into four groups:

  1. a.

_Corner Modes_ : Modes 1-4 excite the corners of the cavity; the field
distribution of WS mode 1 is shown in Fig. 7a. Since these fields quickly
leave the system after scattering off the corners, they have large negative
time delays.

  2. b.

_Ballistic Modes_ : Modes 5-48 specularly reflect off the sides of the cavity
without exciting the gap. Fields distributions for WS modes 5, 8, and 16 are
shown in Figs. 7b, 7c, and 7d, respectively. Note that, unlike for the
rectangular strip, all edges are excited symmetrically. This is attributed to
the cavity being placed symmetrically w.r.t. the origin.

  3. c.

_Non-propagating Modes_ : Modes 49-70 do not excite the scatterer at all; an
example field distribution is shown in Fig. 7e. These modes have near-zero
time delays.

  4. d.

_Cavity Modes_ : Mode 71 excites the interior of the cavity. Fig. 7f shows
that when the gap width is $w=3~{}\mathrm{m}$, the cavity interior is not
strongly excited. Indeed, the cavity’s aperture can be considered as a short
waveguide, which is under cutoff if $w=3~{}\mathrm{m}$. However, increasing
the gap width to $w=5~{}\mathrm{m}$ results in strong fields inside the
cavity, as shown in Fig. 8, because in this case the aperture supports one
propagating mode. Cavity modes have large positive time delays and stored
energies.

Sound-Hard Scatterer: The WS modes can once again be categorized into four
groups:

  1. a.

_Ballistic Modes_ : WS modes 1-36 represent ballistic modes that specularly
reflect off the outer walls of the cavity. Figs. 9a, 9b, and 9c show the total
fields when the sound-hard cavity is illuminated by modes 1, 3, and 15,
respectively. These modes experience negative delays between $-29\mathrm{s}$
and $0\mathrm{s}$ depending on the angle of arrival of the incident field.

  2. b.

_Non-propagating Modes_ : WS modes 37-68 are non-propagating modes that do not
reach the scatterer. They have near-zero time delays.

  3. c.

_Corner/Surface Wave Modes_ : Modes 63-69 excite waves propagating along the
exterior corners and side-walls of the cavity. The total field distribution
for WS mode 69 is shown in Fig. 9d. It can be seen that the fields are
localized near, and launch from, the four corners and side-walls. These modes
have small positive time delays.

  4. d.

_Cavity Modes_ : Modes 70 and 71 are cavity modes. The total field
distribution for these modes is shown in Figs. 9e and 9f. For these WS modes,
the incident field excites the cavity without exciting the corners or side-
walls. These modes represent waves trapped in the cavity, have large stored
energy, and experience large time delays. Note that for the sound-hard
boundary condition, the short aperture waveguide supports at least one mode
even when $w=3~{}\mathrm{m}$, resulting in the strong excitation of the cavity
interior for WS mode 71.

##  IV Conclusion

This paper elucidated a WS theory for acoustic scattering problems governed by
the Helmholtz equation involving sound-soft or sound-hard objects of arbitrary
geometry. The entries of the WS time delay matrix were cast in terms of
renormalized volume integrals involving energy densities. Numerical examples
show that the eigenvectors of the of WS time delay matrix can be used to
untangle scattering phenomena into canonical contributions characterized by
well-defined energies and dwell times.

Current research extending the methodology in this paper is focused on

  1. 1.

The development of fast frequency-sweep methods for computing a scatterer’s
response to broadband excitations;

  2. 2.

The design of acoustic systems and devices resulting in fields that exhibit
desired group delays;

  3. 3.

The phenomenological study of acoustic waves interacting with complicated
structures and metamaterials.

##  Appendix A Modes for Scattering Systems

Incoming fields satisfying the Helmholtz equation in 3D can be expressed as a
superposition of incoming spherical waves

$\displaystyle{\cal I}_{p}^{i}(\mathbf{x})$
$\displaystyle=kj^{l+1}h_{l}^{(1)}(kr){\cal X}_{lm}(\theta,\phi)$ (33)

where $p=(l,m)$ for $l=0,...,L$ and $m=-l,...,l$, and $h_{l}^{(1)}(.)$ is the
spherical Hankel function of type 1 and order $l$. The spherical harmonic
${\cal X}_{p}(\theta,\phi)$ is defined as

$\displaystyle{\cal
X}_{p}(\theta,\phi)=(-1)^{m}\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos\theta)e^{jm\phi}$
(34)

where $P_{l}^{m}(x)$ is the associated Legendre polynomial of degree $l$ and
order $m$ [1]. In addition to the orthogonality property in Eqn. (5), the
spherical harmonics satisfy

$\displaystyle{\cal X}_{p}^{*}(\theta,\phi)$ $\displaystyle=(-1)^{m}{\cal
X}_{\tilde{p}}$ (35)

where $\tilde{p}=(l,-m)$. As $r\rightarrow\infty$, the incoming fields can be
approximated as

$\displaystyle\lim_{r\rightarrow\infty}{\cal
I}^{i}_{lm}(r,\theta,\phi)\cong\frac{e^{jkr}}{r}{\cal X}_{lm}(\theta,\phi)$
(36)

where the large argument approximation of spherical hankel function was used .

##  Appendix B Evaluation of the Surface Integrals

Using the expressions for $\phi_{p,\infty}(\mathbf{x})$,
$\phi_{q,\infty}^{*}(\mathbf{x})$, $\phi_{p,\infty}^{\prime}(\mathbf{x})$,
$\bm{\hat{r}}\cdot\nabla\phi_{q,\infty}^{*}(\mathbf{x})$,
$\bm{\hat{r}}\cdot\nabla\phi_{p,\infty}^{\prime}(\mathbf{x})$ on $d\Omega$,
the surface integral on the LHS of (19) can be evaluated as
$\frac{1}{2k}\left(I_{1}-I_{2}+I_{3}\right)$ where

$\displaystyle I_{1}$
$\displaystyle=\int_{d\Omega}\phi_{p,\infty}^{\prime}(\mathbf{x})\bm{\hat{r}}\cdot\nabla\phi_{q,\infty}^{*}(\mathbf{x})d\mathbf{x}$
(37a)
$\displaystyle=2kR\delta_{pq}-(-1)^{m}\left(j+kR\right)e^{2jkR}\mathbf{S}_{\tilde{p}q}^{*}$
$\displaystyle\quad-
jke^{-2jkR}\mathbf{S}_{qp}^{\prime}+jk\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}\mathbf{S}_{mp}^{\prime}$
$\displaystyle\quad+(-1)^{m}\left(j-kR\right)e^{-2jkR}\mathbf{S}_{q\tilde{p}}$
$\displaystyle I_{2}$
$\displaystyle=\int_{d\Omega}\phi_{q,\infty}^{*}(\mathbf{x})\bm{\hat{r}}\cdot\nabla\phi_{p,\infty}^{\prime}(\mathbf{x})$
(37b) $\displaystyle=-2kR\delta_{qp}-jke^{-2jkR}\mathbf{S}^{\prime}_{qp}$
$\displaystyle\quad-(-1)^{m}kRe^{-2jkR}\mathbf{S}_{q\tilde{p}}-(-1)^{m}kRe^{2jkR}\mathbf{S}_{\tilde{p}q}^{*}$
$\displaystyle\quad-
jk\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}\mathbf{S}_{mp}^{\prime}$ $\displaystyle
I_{3}$
$\displaystyle=\frac{1}{k}\int_{d\Omega}\bm{\hat{r}}\cdot\nabla\phi_{q,\infty}^{*}(\mathbf{x})\phi_{p,\infty}(\mathbf{x})d\mathbf{x}$
(37c)
$\displaystyle=-(-1)^{m}je^{-2jkR}\mathbf{S}_{q\tilde{p}}+j(-1)^{m}e^{2jkR}\mathbf{S}_{\tilde{p}q}^{*}\,.$

Here, indices are defined as $p=(l,m)$, $q=(l^{\prime},m^{\prime})$ and
$\tilde{p}=(l,-m)$. The evaluation of the above integrals was simplified by
using the large argument approximation of Hankel functions and leveraging the
unitarity of $\mathbf{S}$, i.e.
$\sum_{m=1}^{M}\mathbf{S}_{mq}^{*}\mathbf{S}_{mp}=\delta_{q,p}$ .

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