The reflectionless properties of Toeplitz waves and Hankel waves: an analysis via Bessel functions

“One–way waves” are similar to solutions of wave equations but they move in only one direction, and exhibit no propagation in the opposite direction. They are important in the subject of absorbing boundary conditions [4]. Trefethen & Halpern study the well-posedness of one–way wave equations, and they give a nice overview of the history [16]. The literature on reflectionless boundary conditions, and on applications of Bessel function expansions to waves, is extensive. We do not attempt a survey here. An incomplete list of authors includes: Grote, Keller, Givoli, Bayliss & Turkel, Higdon, Ting & Miksis, Lubich & Schadle, Alpert, Greengard & Hagstrom. Particular attention is paid to the issue of whether or not the proposed reflectionless boundary conditions are exact or approximate, and whether or not the conditions are local or global. There are also important connections to the Kirchoff formula and to the Dirichlet–to–Neumann map, and to Born series, that we do not consider here.

We distinguish three settings:

• •

  Fully continuous: both time and space are continuous. The reflectionless boundary condition is the Sommerfeld condition, which in our setting might be applied by imposing

  $$\frac{\mbox{d}u}{\mbox{d}x}=\pm\frac{\mbox{d}u}{\mbox{d}t}$$

  (1)

  on the solution $u(x,t)$ at the left and right boundaries. The choice of the plus or minus sign depends on whether the wave is traveling left or right. Notice this condition (1) is local in time and local in space.

• •

  Fully discrete: Both time and space have been discretized. Engquist & Majda [5, Equation (5.3)] address the issue of the nature of the condition in this setting. Their reflectionless boundary condition is not a simple expression. It is not local in time and it is not local in space.

• •

  Semi-discrete: time remains continuous and space is discretized. Intuitively, we expect the conditions to be local in time, and non-local in space. This situation has been studied by Halpern [7, Section 3].

A well-known model in these settings is a set of masses connected by linear springs, which can be generalised to higher dimensions through an array of springs. The resulting framework is then a linear system of ordinary differential equations.

It is important to note that our purpose in this article is not to study the reflectionless boundary condition per se, nor to study the big field of inverse problems to which it relates; these topics already have an extensive literature. The boundary condition in particular, is an old problem that has been studied by many authors; among the early literature, perhaps the work of Halpern [7] is closest to our own framework. Instead, our purpose is to reveal the properties of the Toeplitz waves and the Hankel waves that we will introduce later. During the course of our analysis, it transpires that these Toeplitz and Hankel waves possess some attractive reflectionless properties.

We revisit the problem with a new perspective, by carefully revealing the algebraic structure of those matrix functions. We find particularly explicit and exact formulas. An attractive outcome of our perspective is to show that the travelling wave can be considered as the sum of two waves that we call the Toeplitz wave and the Hankel wave. We show that these waves can be written as certain linear combinations of even index Bessel functions of the first kind. These expansions, as we will explain, make it clear that the Toeplitz wave and Hankel wave are reflectionless on even, respectively odd, traversals of the domain.

In section 2, we give some background on Toeplitz and Hankel matrix functions. In section 3, we show how to write a travelling wave as the sum of two waves that we call a Toeplitz wave and a Hankel wave. In section 4, we give analytic expressions for the Toeplitz and Hankel waves in terms of certain sums of even index Bessel functions of the first kind. In section 5, we show simulations of these waves using the methods of Nadukandi & Higham [11], and also using methods inspired by our analytic expressions. With our analysis, we are now able to algebraically explain interesting behaviours that have previously been observed in numerical simulations in the literature [9]. We also showcase another benefit of our analysis, namely that it suggests a method of simulation that allows us to choose, in advance, the number of reflections. The paper concludes in section 6 with some observations and discussion for future work.

This section collects together some of the key results from [15] concerning Toeplitz-plus-Hankel matrix functions that we will need later.

The central difference approximation to the Laplacian, which in one space dimension is simply the second derivative, $-\frac{\partial^{2}}{\partial x^{2}}$, is $\bm{K}/h^{2}$, where $\bm{K}$ is the $N\times N$ tridiagonal, symmetric positive definite Toeplitz matrix:

$$\bm{K}=\left(\begin{tabular}[]{ccccc}$2$&-1\\ -1&$2$&-1\\ &$\ddots$&$\ddots$&$\ddots$\\ &&-1&$2$&-1\\ &&&-1&2\end{tabular}\right)$$

(2)

and

$$h=\frac{1}{N+1}.$$

(3)

We can diagonalize the matrix $\bm{K}=\bm{V}\bm{\Lambda}\bm{V}^{\top}=\sum_{k=1}^{N}\lambda_{k}\bm{v}_{k}\bm{v}_{k}^{\top}$ where the eigenvalues appear in the diagonal matrix $\bm{\Lambda}$ and where the eigenvectors form the columns of $\bm{V}$. Given a function $f$ of a scalar argument, we define a matrix function [8] via diagonalization:

$$f(\bm{K})=\bm{V}f(\bm{\Lambda})\bm{V}^{\top}=\sum_{k=1}^{N}f(\lambda_{k})\bm{v}_{k}\bm{v}_{k}^{\top}.$$

(4)

The eigenvalues of the matrix in (2) are:

$$\lambda_{k}=2-2\cos(k\pi h),\qquad k=1,\dots,N,$$

(5)

and the eigenvectors are:

$$\bm{v}_{k}=\sqrt{2h}\Big{(}\sin(k\pi h),\sin(2k\pi h),\dots,\sin(Nk\pi h)\Big{)}^{\top},\qquad k=1,\dots,N.$$

(6)

With these explicit expressions, the $(m,n)$ entry of the matrix function is

$$f(\bm{K})_{m,n}=2h\sum_{k=1}^{N}f(\lambda_{k})\sin(mk\pi h)\sin(nk\pi h).$$

(7)

Entries of the rank one symmetric matrices, denoted $\bm{v}_{k}\bm{v}_{k}^{\top}$ that project onto eigenspaces, are products of sines, of the form $\sin(m\theta)\sin(n\theta)$. Recall the trigonometric identity

$$2\sin\theta_{1}\sin\theta_{2}=\cos(\theta_{1}-\theta_{2})-\cos(\theta_{1}+\theta_{2}).$$

(8)

Thus each entry in $\bm{v}_{k}\bm{v}_{k}^{\top}$ can be written as the sum of a term of the form $\cos((m-n)\theta)$ that leads to a Toeplitz matrix, and another term of the form $\cos((m+n)\theta)$ that leads to a Hankel matrix. Thus, for $k=1,\ldots,N$, the rank one matrix

$$\bm{v}_{k}\bm{v}_{k}^{\top}=\bm{T}_{k}+\bm{H}_{k}$$

(9)

is the sum of a Toeplitz matrix, with $(m,n)$ entries

$$\big{(}\bm{T}_{k}\big{)}_{m,n}=\;\,h\cos\big{(}(m\bm{-}n)k\pi h\big{)}$$

(10)

and a Hankel matrix

$$\big{(}\bm{H}_{k}\big{)}_{m,n}=\bm{-}h\cos\big{(}(m\bm{+}n)k\pi h\big{)},$$

(11)

recalling that $h=\frac{1}{N+1}$.

A sum of Toeplitz matrices is again a Toeplitz matrix, and similarly Hankel matrices are closed under addition. Recalling (4), (7) and (9), we see that as $\bm{K}$ has the strong Toeplitz-plus-Hankel property, all matrix functions of $\bm{K}$ are the sum of a Toeplitz matrix and a Hankel matrix:

$$f(\bm{K})=\sum_{k=1}^{N}f(\lambda_{k})(\bm{T}_{k}+\bm{H}_{k})=\underbrace{\sum_{k=1}^{N}f(\lambda_{k})\bm{T}_{k}}_{\text{Toeplitz}}\;\;+\;\;\underbrace{\sum_{k=1}^{N}f(\lambda_{k})\bm{H}_{k}}_{\text{Hankel}}.$$

(12)

It is helpful to consider the action of the Toeplitz and Hankel parts on a vector ‘input.’ To illustrate this action, consider the so-called shift matrix, which is an example of a Toeplitz matrix. In this particular example, we see the action is a ‘forward shift’ that preserves the orientation of the input:

abc).\left(\begin{tabular}[]{ccccc}&\\ 1&&\\ &1&&\\ &&1&\\ &&&1&\end{tabular}\right)\left(\begin{tabular}[]{c}0\\ a\\ b\\ c\\ 0\end{tabular}\right)=\left(\begin{tabular}[]{c}0\\ 0\\ a\\ b\\ c\\ \end{tabular}\right).

		0	0		(13)

Define $N\times N$ shift matrices $E_{1},\cdots,E_{N-1}$ where $E_{k}$ corresponds to the matrix with 1 on the $k$th upper subdiagonal and 0 elsewhere. We will need these matrices later in (4). Let $E_{0}=I$ be the identity matrix. These matrices and their transposes form a natural basis, $\{E_{0},E_{1},\ldots,E_{N-1},E_{1}^{\top},\ldots E_{N-1}^{\top}\}$, for the space of $N\times N$ Toeplitz matrices. That is, an arbitrary $N\times N$ Toeplitz matrix $T$, with first row $t_{0},t_{1},\ldots t_{N-1}$ and first column $t_{0},t^{1}\ldots,t^{N-1}$, is a linear combination of these shift matrices: $T=t_{0}I+t_{1}E_{1}\ldots t_{n-1}E_{N-1}+t^{1}E_{1}^{\top}+\ldots t^{N-1}E_{N-1}^{\top}$. Thus, we can think of the action of the Toeplitz matrix as a linear combination of these shifts [6]. Likewise, to illustrate the action of a Hankel matrix, consider the ‘Hankel version of the shift matrix.’ In this example, we see a ‘backward shift’ (which is also known as a reversal permutation) that reverses the orientation of the input:

$$\left(\begin{tabular}[]{ccccc}&&&1&\\ &&1&&\\ &1&&&\\ 1&&\\ &\end{tabular}\right)\left(\begin{tabular}[]{c}0\\ a\\ b\\ c\\ 0\end{tabular}\right)=\left(\begin{tabular}[]{c}c\\ b\\ a\\ 0\\ 0\end{tabular}\right).$$

(14)

We define a set of such $N\times N$ matrices, with ones on a backwards diagonal and which we denote by $F_{k}$, and which are depicted later in (39). These matrices are related to the shift matrices by multiplying by the anti-identity matrix $J$, that is $F_{k}=JE_{k}$ for $k=1,\ldots,N-1$, $F_{N}=JE_{0}=JI=J$, and $F_{N+k}=JE_{k}^{\top}$ for $k=1,\ldots,N-1$. The set of matrices $\{F_{k}\}$ form a natural basis for the space of $N\times N$ Hankel matrices. That is, an arbitrary Hankel matrix $H$ can be written as a linear combination $H=h_{1}F_{1}+\ldots h_{2N-1}F_{2N-1}$, where the weights $h_{k}$ in the combination come from the backward diagonals of the Hankel matrix.

This section summarizes observations from [9] that we will need later. It transpires that the Toeplitz part of the solution to the wave equation can be thought of as a type of solution that does not feel the boundaries on even traversals of the domain, whereas the Hankel part of the solution does not feel the boundaries on odd traversals of the domain.

Consider the wave equation

$$\frac{\partial^{2}}{\partial t^{2}}\bm{u}=\frac{\partial^{2}}{\partial x^{2}}\bm{u}$$

(15)

on the domain $-1\leq x\leq 1$ with zero Dirichlet boundary conditions $u(-1,t)=u(1,t)=0$. In the semi-discrete setting, the continuous wave equation (15) is approximated on an equally spaced grid of points $-1,\ldots,-\Delta x,0,\Delta x,\ldots,1$, with spacing $\Delta x=2/(N-1)$, by the linear system of ordinary differential equations

$$\frac{\mbox{d}^{2}}{\mbox{d}t^{2}}\bm{u}=-\frac{\bm{K}}{\Delta x^{2}}\bm{u}$$

(16)

with the matrix from (2). It can be quickly checked by differentiating twice that a solution to our semi-discrete model in (16) is the matrix function

$$\bm{u}(t)=f(\bm{K})\bm{u}(0)=\cos\left(\pm t\frac{\sqrt{\bm{K}}}{\Delta x}\right)\bm{u}(0).$$

(17)

Here we are also using the square root matrix function, and we restrict attention to the special case of a wave equation involving a symmetric graph Laplacian, of which the matrix $\bm{K}$ in (2) is an example. This solution (17) could be compared with the representations of the solutions presented by Nadukandi & Higham [11], who are able to efficiently compute solutions to the wave equation, even in the more general situation where the graph Laplacian matrix in question is not symmetric.

A second order differential equation such as (16) requires two initial conditions. In our solution (17), we are assuming an initial condition $\bm{u}(0)$, and we are also assuming zero initial velocity. If the initial velocity, $\bm{u}^{{}^{\prime}}(0)$, is not zero then the solution involves an extra term:

$$\bm{u}(t)=\cos\left(t\frac{\sqrt{\bm{K}}}{\Delta x}\right)\bm{u}(0)+\Delta x\>\bm{K}^{-\frac{1}{2}}\sin\left(t\frac{\sqrt{\bm{K}}}{\Delta x}\right)\;\bm{u}^{{}^{\prime}}(0).$$

(18)

Our article focuses on the solution in (17).

To connect this solution (17) to the matrix function point of view let us make the particular choice of function

$$f(z)\equiv\cos\left(t\frac{\sqrt{z}}{\Delta x}\right).$$

(19)

Define

$$\bm{T}\equiv\sum_{k=1}^{N}f(\lambda_{k})\bm{T}_{k}$$

(20)

with $\bm{T}_{k}$ as in (10) but now $h=\Delta x=\frac{2}{N-1}$, and

$$\bm{H}\equiv\sum_{k=1}^{N}f(\lambda_{k})\bm{H}_{k},$$

(21)

with $\bm{H}_{k}$ as in (11). The spacing $\Delta x=\frac{2}{N-1}$ is chosen so that the mesh consists of $N$ equally-spaced points, including the endpoints -1 and 1. Now via (12), the solution (17) to the wave equation is the sum of a Toeplitz wave, $\bm{T}\bm{u}(0)$, and a Hankel wave, $\bm{H}\bm{u}(0)$. That is,

$$\bm{u}(t)=\;f(\bm{K})\bm{u}(0)\;=\;\;\;\;\underbrace{\bm{T}\bm{u}(0)}_{\text{Toeplitz wave}}\;\;\;\;+\;\;\;\;\underbrace{\bm{H}\bm{u}(0).}_{\text{Hankel wave}}$$

(22)

Next, we seek to to obtain explicit expressions for the Toeplitz and Hankel waves in terms of certain finite sums of Bessel functions of the first kind.

In this section we derive explicit representations of the Toeplitz and Hankel waves on a symmetric one dimensional domain with a symmetric initial condition. We will show that the nonconstant Toeplitz wave traverses the domain on odd traversals of the domain (the first traversal is a half traversal), while the nonconstant Hankel wave appears on even traversals. These explicit expressions are given as a finite sum of even Bessel functions of the first kind and the number of traversals appears as a parameter on the total number of Bessel functions required. In order to construct the Toeplitz and Hankel waves we need some preliminary lemmas and definitions, some of which we collect from section 2, but here we specialise the expressions to a more convenient form.

We now consider the spatially discretised wave equation based on the central difference approximation on $[-1,1]$ with constant discretisation $\Delta x=\frac{2}{N-1}$, $N$ is odd. Note that in the case $N$ is odd, then 0 is always in the set of mesh points. Hence the discretised spatial operator is $\frac{1}{(\Delta x)^{2}}\>\bm{K}$ and the travelling wave is given by

$$\bm{u}(t)=f\left(\frac{t}{\Delta x}\sqrt{\bm{K}}\right)\>\bm{u}(0),$$

(24)

where $f$ denotes the matrix function $\cos\left(\frac{t}{\Delta x}\sqrt{\bm{K}}\right)$. We note that the eigenvalues of this matrix $\cos\left(\frac{t}{\Delta x}\sqrt{\bm{K}}\right)$ are

$$f(\lambda_{k})=\cos\left(\frac{t}{\Delta x}\sqrt{\lambda_{k}}\right)=\cos\left(2\frac{t}{\Delta x}\sin\frac{k\pi}{2(N+1)}\right),\quad k=1,\cdots,N.$$

(25)

We can now define the Toeplitz and Hankel components of this matrix function. They are

	$\displaystyle\textit{TP}(t)$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{N}\cos\left(\frac{t}{\Delta x}\sqrt{\lambda_{k}}\right)\bm{T}_{k}=\sum_{k=1}^{N}\cos\left(2\frac{t}{\Delta x}\sin\frac{k\pi}{2(N+1)}\right)\bm{T}_{k}$
	$\displaystyle\textit{HK}(t)$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{N}\cos\left(\frac{t}{\Delta x}\sqrt{\lambda_{k}}\right)\bm{H}_{k}=\sum_{k=1}^{N}\cos\left(2\frac{t}{\Delta x}\sin\frac{k\pi}{2(N+1)}\right)\bm{H}_{k}.$

Hence the Toeplitz and Hankel waves are

$$T(t)=\textit{TP}(t)\bm{u}_{0},\quad H(t)=\textit{HK}(t)\bm{u}_{0}$$

(27)

where the full wave is

$$\bm{u}(t)=T(t)+H(t).$$

The choice of notation $T(t)$ and $H(t)$ indicates the close relationship to the matrices appearing in (20), (21), and (22).

In what follows, we will sample time at equally-spaced time points with $t=j\Delta t$, $j$ a positive integer. We are now discretising in time, but note that we are still evaluating the semi-discrete model (where time remains continuous) exactly at these chosen discrete time points. This is as opposed to what is usually termed a ‘fully discrete’ model, where some additional approximation is introduced. We will also assume without loss of generality that $\Delta t=\Delta x$.

Now the Toeplitz and Hankel waves are

	$\displaystyle T(j\Delta t)$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{N}\cos\left(2j\sin\frac{k\pi}{2(N+1)}\right)\bm{T}_{k}\bm{u}_{0}$
	$\displaystyle H(j\Delta t)$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{N}\cos\left(2j\sin\frac{k\pi}{2(N+1)}\right)\bm{H}_{k}\bm{u}_{0}.$

The expressions in (LABEL:eq:eq6) lead us to consider Bessel functions $J_{l}(t)$ of the first kind. In the following, we list some lemmas which yield some important properties of the Bessel functions.

As a consequence of Lemma 4.3,

$$\cos(2j\,\sin\frac{k\pi}{2(N+1)})=J_{0}(2j)+2\sum_{l=1}^{\infty}J_{2l}(2j)\,\cos(l\,\frac{k\pi}{N+1}).$$

(29)

It is clear that we can use (29) in simplifying the expressions for the Toeplitz and Hankel waves in (LABEL:eq:eq6). In particular we can rewrite (LABEL:eq:eq6) as

$$T(j\Delta t)=J_{0}(2j)\left(\sum_{k=1}^{N}\bm{T}_{k}\right)\,\bm{u}_{0}+2\sum_{l=1}^{\infty}J_{2l}(2j)\,\left(\sum_{k=1}^{N}\cos\left(\frac{kl\pi}{N+1}\right)\>\bm{T}_{k}\right)\>\bm{u}_{0}.$$

$$H(j\Delta t)=J_{0}(2j)\left(\sum_{k=1}^{N}\bm{H}_{k}\right)\,\bm{u}_{0}+2\sum_{l=1}^{\infty}J_{2l}(2j)\,\left(\sum_{k=1}^{N}\cos\left(\frac{kl\pi}{N+1}\right)\>\bm{H}_{k}\right)\>\bm{u}_{0}.$$

Using the definition of the Toeplitz and Hankel matrices in (10) and (11) and the trigonometric relation

$$2\cos\theta_{1}\cos\theta_{2}=\cos(\theta_{1}+\theta_{2})+\cos(\theta_{1}-\theta_{2}),$$

then we can write that component $m$, $m=1,\cdots,N$ of the vectors $T(j\Delta t)$ and $H(j\Delta t)$ are

	$\displaystyle(N+1)T_{m}(j\Delta t)$	$\displaystyle=$	$\displaystyle J_{0}(2j)\sum_{n=1}^{N}\sum_{k=1}^{N}\cos\left((m-n)\frac{k\pi}{N+1}\right)\,\bm{u}_{0}$		(30)
		$\displaystyle+$	$\displaystyle\sum_{l=1}^{\infty}J_{2l}(2j)\sum_{n=1}^{N}\sum_{k=1}^{N}\left(\cos\left((l+m-n)\frac{k\pi}{N+1}\right)\right.$
		$\displaystyle+$	$\displaystyle\left.\cos\left((l-(m-n))\frac{k\pi}{N+1}\right)\right)\,\bm{u}_{0}$

	$\displaystyle-(N+1)H_{m}(j\Delta t)$	$\displaystyle=$	$\displaystyle J_{0}(2j)\sum_{n=1}^{N}\sum_{k=1}^{N}\cos\left((m+n)\frac{k\pi}{N+1}\right)\,\bm{u}_{0}$		(31)
		$\displaystyle+$	$\displaystyle\sum_{l=1}^{\infty}J_{2l}(2j)\sum_{n=1}^{N}\sum_{k=1}^{N}\left(\cos\left((l+m+n)\frac{k\pi}{N+1}\right)\right.$
		$\displaystyle+$	$\displaystyle\left.\cos\left((l-(m+n))\frac{k\pi}{N+1}\right)\right)\,\bm{u}_{0}.$

A reminder about notation may be helpful here: $T_{m}(t)$ denotes component $m$ of the time-varying vector that is the Toeplitz wave at time $t$ that was introduced in (LABEL:eq:eq6), whereas $\bm{T}_{k}$ is the matrix defined in (10).

In order to make further progress we use the following lemma, which is known as the Lagrange identity, and a subsequent lemma.

We will use Lemma 4.5 to simplify the Toeplitz wave and the Hankel wave formulation given in (30) and (31). In the case of the Toeplitz wave, $\theta$ can take on the values $(m-n)\frac{\pi}{N+1}$, $(l+m-n)\frac{\pi}{N+1}$, $(l-(m-n))\frac{\pi}{N+1}$ so $p=m-n,\>l+m-n$ or $l-(m-n)$. Therefore we must take care when these values of $p$ are of the form $p=2\rho(N+1)$, $\rho=0,1,2,\cdots$. Similarly, in the case of the Hankel wave, the corresponding $p$ values are $m+n$, $l+m+n$ or $l-(m+n)$.

We note $m-n$ can take on values $k$, say, where $k=-(N-1),\cdots,-1,$ $0,$ $1,\cdots,(N-1)$. Hence for $\rho=0,1,\ldots$ with $p=2\rho(N+1)$ then

$$2l=4(N+1)\rho\mp 2k$$

and the corresponding $L(\theta)=N$ in these cases. We will now simplify the expression $(N+1)T(j\Delta t)$ in vector form.

Introducing the vectors $\bm{\nu}_{k}$, $k=0,\cdots,N-1$, where

	$\displaystyle\bm{\nu}_{0}$	$\displaystyle=$	$\displaystyle\bm{u}_{0}$
	$\displaystyle\bm{\nu}_{k}$	$\displaystyle=$	$\displaystyle(E_{k}+E_{k}^{\top})\,\bm{u}_{0},\quad k=1,\cdots,N-1$

leads to the first part of the Toeplitz wave as

$$F=N\left(\left(J_{0}(2j)+2\sum_{\rho=1}^{R-1}J_{4(N+1)\rho+2k}(2j)+J_{4(N+1)\rho-2k}(2j)\right)\,\bm{\nu}_{k}\right).$$

(32)

We will write the factor at the front as $(N+1\,-1)$. Subtracting off this term and reincorporating into (30) and then using Lemma 6 with $L(\theta)$ either 0 or -1 and finally dividing throughout by $N+1$ leads to

	$\displaystyle T(j\Delta t)$	$\displaystyle=$	$\displaystyle(J_{0}(2j)+2\sum_{\rho=1}^{R-1}J_{4(N+1)\rho}(2j))\,\bm{\nu}_{0}$
		$\displaystyle+$	$\displaystyle\sum_{k=1}^{N-1}\left(J_{2k}(2j)+\sum_{\rho=1}^{R-1}(J_{4(N+1)\rho+2k}(2j)+J_{4(N+1)\rho-2k}(2j))\right)\,\bm{\nu}_{k}$
		$\displaystyle-$	$\displaystyle X,$

where

$$(N+1)\,X=(J_{0}(2j)\,A+2\sum_{l=1}^{\infty}J_{4l}(2j)\,A+2\sum_{l=1}^{\infty}J_{4l-2}(2j)\,B)\,\bm{u}_{0},$$

(34)

where $A$ and $B$ are the $N\times N$ matrices (as an aside, notice that these matrices have the special property that they are simultaneously both Toeplitz and Hankel)

$$A=\left(\begin{array}[]{cccccc}1&0&1&0&1&\cdots\\ 0&1&0&1&0&\cdots\\ 1&0&1&0&1&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \end{array}\right),\quad\quad B=\left(\begin{array}[]{cccccc}0&1&0&1&0&\cdots\\ 1&0&1&0&1&\cdots\\ 0&1&0&1&0&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \end{array}\right).$$

(35)

We can now simplify $X$ by applying Lemma 4.6.

We can now write the characterisation of the Toeplitz wave over the $R$ traversals in two ways: either in terms of the vectors $\bm{\nu}_{k}$ (Theorem 4.1) or increasing values of the Bessel indices (Corollary 4.1.1). A reason to seek results such as these is that they allow us to express the wave as a sum over the number of traversals, which is useful later in computer simulations when we want to control the number of reflections, for example.

We can rewrite (36) in the ascending index of the Bessel function.

We can observe from (37) that the $J_{4(N+1)\rho}$ term is repeated twice and there are always three missing terms when moving to the next traversal - e.g. $J_{2N}$, $J_{2(N+1)}$, $J_{2(N+2)}$ or $J_{6N+4}$, $J_{6N+6}$, $J_{6N+8}$, etc.

We can now construct the Hankel wave based on (31). We must consider the cases where $p=2\rho(N+1),\>\rho=0,\cdots,R-1$ where now $p=m+n,l+m+n$ or $l-(m+n)$. Let $m+n=k$; then $k$ can take values $2,3,\cdots,2N$. Hence

$$l\pm k=2\rho(N+1)$$

so

$$2l=4(N+1)\rho\mp 2k,\quad k=2,\cdots,2N,\quad\rho=0,\cdots,R-1.$$

(38)

For these values $L(\theta)=N$.