Pulse reflection in a random waveguide with a turning point

# Pulse reflection in a random waveguide with a turning point

Liliana Borcea111Department of Mathematics, University of Michigan, Ann Arbor, MI 48109. borcea@umich.edu    Josselin Garnier222Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau Cedex, France. josselin.garnier@polytechnique.edu
###### Abstract

We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and monotone change of the width of the waveguide and small and rapid fluctuations of the boundary, on the scale of the wavelength. These fluctuations are modeled as random. The turning point is many wavelengths away from the source, which emits a pulse that propagates toward the turning point, where it is reflected. To focus attention on this reflection, we assume that the waveguide supports a single propagating mode from the source to the turning point, beyond which all the waves are evanescent. We consider a scaling regime where scattering at the random boundary has a significant effect on the reflected pulse. In this regime scattering from the random boundary away from the turning point is negligible, while scattering from the random boundary around the turning point results in a strong, deterministic pulse deformation. The reflected pulse shape is not the same as the emitted one. It is damped, due to scattering at the boundary, and is deformed by dispersion in the waveguide. The reflected pulse also carries a random phase.

Key words. Turning waves, random waveguide, pulse stabilization

## 1 Introduction

Guided waves arise in a wide range of applications in electromagnetics [10], optics and communications [25], underwater acoustics [19], and so on. The classical theory of guided waves relies on the separability of the wave equation in ideal waveguides with straight walls and filled with homogeneous media [29]. It decomposes the wave field in independent waveguide modes, which are special solutions of the wave equation. The modes are either propagating waves along the axis of the waveguide or evanescent waves. They do not interact with each other and have constant amplitudes determined by the source excitation.

We study sound waves in two-dimensional waveguides with varying cross section and slowly bending axis, where the waveguide effect is due to reflecting boundaries, modeled for simplicity as sound soft. The three-dimensional case and other boundary conditions can be treated similarly, and do not involve conceptual differences. We refer to [18, 26] for examples of numerical studies of waves in slowly varying waveguides, and to [25] for local mode decompositions of the wave field, where the modes are coupled, and their amplitudes vary along the waveguide axis. An analysis of such a decomposition is given in [2, 31], and the transition of propagating modes to evanescent ones at turning points in slowly changing waveguides is studied in [5]. Here we analyze this wave transition at a turning point in a random waveguide with small and rapid random fluctuations of the boundary on the scale of the wavelength, in addition to the slow variations.

The wave field is generated by a source which emits a pulse with central frequency and bandwidth . It is the superposition of a countable set of modes, of which only finitely many propagate. To focus attention on the turning point, we consider a central frequency such that there is a single propagating mode between the source and the turning point. We also assume that the slow variation of the waveguide width is monotone, so that no propagation occurs beyond the turning point. Due to energy conservation, the propagating mode is reflected at the turning point and returns to the source location. The goal of the paper is to analyze the pulse shape carried by this reflected wave.

Sound wave propagation in random waveguides is analyzed in [19, 11, 14, 13, 16] for the case of waveguides filled with a random medium, and in [4, 7, 17] for the case of waveguides with random perturbations of straight boundaries. We also refer to [25, 3] for the analysis of electromagnetic waves in random waveguides. A main difficulty arising in the extension of these results to random waveguides with slowly varying cross section is due to the turning points, studied in this paper.

An analysis of random multiple scattering of turning waves is given in [20, 21], in the context of wave propagation in randomly layered media. These results are relevant to our study, specially the stochastic averaging theorem in [21]. In this paper we derive from first principles a stochastic equation for the reflection coefficient of the propagating mode in the random waveguide, and study in detail its statistics, using the limit theorem in [21]. To characterize the reflected pulse, we carry out a multi-frequency analysis of the reflection coefficient whose phase has a non-trivial random frequency dependence. We quantify the standard deviation of the random fluctuations of the boundary that trigger strong modifications of the amplitude and shape of the reflected pulse. We show that such random fluctuations have negligible effect on the pulse away from the turning point, but near the turning point the effect is strong and leads to a deterministic pulse deformation and damping. This pulse stabilization result is similar, but different from the ones obtained in layered media in [27, 9, 23, 22], in locally layered media in [30], in time-dependent layered media in [8], and in three-dimensional random media in [15]. In these references the medium is random, not the boundary, there is no turning point, and pulse deformation is observed when the standard deviation of the random fluctuations is larger than the one considered here. In this paper we explain why the random fluctuations have a stronger effect close to the turning point than away from it.

The paper is organized as follows: We begin in section LABEL:sect:formulation with the formulation of the problem and state the pulse stabilization result in section LABEL:sect:result. The proof of this result is in section LABEL:sect:proof. We end with a summary in section LABEL:sect:sum.

## 2 Formulation of the problem

We describe in section LABEL:sect:form1 the setup of the problem, and define in section LABEL:sect:form1p the scaling regime. Then we give in section LABEL:sect:form2 the mode decomposition of the wave field, and derive the stochastic differential equation satisfied by the propagating mode. The remainder of the paper is concerned with the analysis of this equation.

### 2.1 Setup

Consider a two-dimensional waveguide occupying the semi-infinite domain , with sound soft boundary consisting of the union of two curves, as illustrated in Figure LABEL:fig:setup. We refer to as the bottom boundary and to as the top boundary. The waveguide has a slowly bending axis parametrized by the arc length . Ideally, and would be symmetric with respect to this axis, but the top boundary is perturbed by small fluctuations. The waveguide is filled with a homogeneous medium with wave speed , and the excitation is due to a point source at location , that emits the pulse

 f(t)=cos(ωot)F(Bt). \hb@xt@.01(2.1)

This pulse is modeled by a periodic carrier signal at frequency , and a real-valued, smooth envelope function of dimensionless argument. Its Fourier transform is supported in the interval , so the Fourier transform of (LABEL:eq:fm1),

 \hb@xt@.01(2.2)

is supported in the frequency interval centered at , with bandwidth , and its negative image . Since is real valued,

 \hb@xt@.01(2.3)

where the bar denotes complex conjugate. We take , so that approximates the wavelength at all frequencies in the support of , and suppose that is small with respect to the arc length distance of order from the source to the turning point.

The wave field is modeled by the acoustic pressure , the solution of the wave equation

 \hb@xt@.01(2.4)

with homogeneous Dirichlet boundary conditions

 p(t,\itbfx)=0,\itbfx∈∂Ω,  t∈R. \hb@xt@.01(2.5)

Prior to the excitation the medium is quiescent,

 p(t,\itbfx)≡0,t≪0. \hb@xt@.01(2.6)

It is convenient to write equations (LABEL:eq:waveeq)–(LABEL:eq:waveeq2) in the orthogonal curvilinear coordinate system with axes along and , the unit tangent and normal vectors to the axis of the waveguide, at arc length . These vectors change slowly in , on the length scale , according to the Frenet-Serret formulas

 ∂zτ(zL)=1Lκ(zL)\itbfn(zL),∂z\itbfn(zL)=−1Lκ(zL)τ(zL), \hb@xt@.01(2.7)

where is the curvature. We parametrize the points by , using

 \itbfx=\itbfx∥(z)+r\itbfn(zL), \hb@xt@.01(2.8)

where is on the waveguide axis, at arc length , and is the coordinate in the direction of the normal at . This coordinate lies in the interval , with at the bottom boundary

 r−(z)=−D(z/L)2, \hb@xt@.01(2.9)

and at the randomly perturbed top boundary

 r+(z)=D(z/L)2[1+1(−∞,0)(z)σν(zℓ)]. \hb@xt@.01(2.10)

Here is the width of the unperturbed waveguide, a smooth (at least three times continuously differentiable) and monotonically increasing function that varies slowly in , on the scale . The top boundary has small and rapid random fluctuations on the left of the source, and is the indicator function of the negative axis , smoothed near the origin. The fluctuations are modeled by the zero-mean stationary process of dimensionless argument, with autocorrelation function

 R(ζ)=E[ν(ζ)ν(0)]. \hb@xt@.01(2.11)

This process is mixing, with rapidly decaying mixing rate, as defined for example in [28, section 2], and it is bounded, with bounded first two derivatives, almost surely. We normalize so that

 R(0)=1,∫∞−∞dζR(ζ)=1 [or O(1)], \hb@xt@.01(2.12)

and control the amplitude of the fluctuations in (LABEL:eq:fm6) by the standard deviation , and their spatial scale by the correlation length . The Fourier transform of

 ˆR(k)=∫∞−∞dζeikζR(ζ)=∫∞−∞dζcos(kζ)R(ζ)

is the power spectral density of the stationary process . It is an even and nonnegative function.

In the curvilinear coordinate system the source is located at , and the wave equation (LABEL:eq:waveeq) becomes

 ⎡⎢ ⎢ ⎢⎣∂2r−1Lκ(zL)∂r1−rLκ(zL)+∂2z[1−rLκ(zL)]2+rL2κ′(zL)∂z[1−rLκ(zL)]3−1c2∂2t⎤⎥ ⎥ ⎥⎦p(t,r,z) =∣∣∣1−r⋆Lκ(0)∣∣∣−1f(t)δ(z)δ(r−r⋆), \hb@xt@.01(2.13)

for , and , with boundary conditions (LABEL:eq:waveeq2) given by

 p(t,r−(z),z)=p(t,r+(z),z)=0,∀t∈R, z∈R. \hb@xt@.01(2.14)

Here denotes the derivative of the curvature and we used the parametrization (LABEL:eq:fm4) of the points in the waveguide, with

 ∂r\itbfx=\itbfn(zL),∂z\itbfx=[1−rLκ(zL)]τ(zL),

and Lamé coefficients

to write the Laplacian

 Δ =1hrhz[∂r(hrhzh2r∂r)+∂z(hrhzh2z∂z)],

and the Dirac delta at ,

 δ(\itbfx−\itbfx⋆)=1hrhzδ(z)δ(r−r⋆).

The problem is to analyze the wave field at time , where is the duration of the emitted pulse . This models the reflected wave in the random section of the waveguide, which contains the turning point.

### 2.2 The scaling regime

We define here a scaling regime where the random boundary fluctuations have a significant effect on the reflected wave. The regime is defined by the standard deviation of the random fluctuations, and the relation between the important length scales in the problem: the central wavelength , the correlation length of the random fluctuations, the scale of the slow variations of the waveguide, and the width of the cross section.

The length scales are ordered as

 L≫D∼λo∼ℓ, \hb@xt@.01(2.15)

where denotes “of the same order as”. In this scaling regime the central wavelength is of the same order as the correlation length of the medium, and is much smaller than the typical propagation distance, so that the waves interact efficiently with the boundary fluctuations. We model (LABEL:eq:sc0) using the small, dimensionless parameter

 ε=ℓL≪1, \hb@xt@.01(2.16)

and use asymptotic analysis in the limit to characterize the reflected wave.

The relation between the waveguide width and the central wavelength determines the number

 N(z)=⌊2D(z/L)/λo⌋

of propagating modes in the local mode decomposition of the wave , at given , where denotes the integer part. To simplify the analysis we assume that the central frequency of the pulse is such that for , where is the arc length at the turning point, satisfying

 λo=2D(zT(ωo)/L). \hb@xt@.01(2.17)

The turning point is assumed simple, meaning that , and by the monotonicity of we have for . Consistent with the slow variations of the waveguide on the scale , we suppose that .

We know from the study [4] of waveguides with randomly perturbed straight boundaries that the interaction of the waves with the boundary fluctuations gives an order one net scattering effect over the distance scaled as in (LABEL:eq:sc1), when the standard deviation of the fluctuations is of the order . Thus, we take

 σ=√εσε, \hb@xt@.01(2.18)

with at most of order one with respect to . It will be adjusted later, so that the effect of the random fluctuations of the waveguide boundary on the reflected pulse is of order one as .

The duration of is inverse proportional to the bandwidth , and must be much smaller than the travel time from the source to the turning point and back, otherwise would not be a pulse. This implies the scaling relation

 1ωo≪1B≪Lc∼1εωo, \hb@xt@.01(2.19)

where we used (LABEL:eq:sc0) and . We show in section LABEL:sect:proof that the characterization of the probability distribution of the reflected pulse involves the joint distribution of the reflection coefficients at frequencies spaced by . We choose

 Bωo∼√ε, \hb@xt@.01(2.20)

so that (LABEL:eq:sc3) is satisfied and the phases of the frequency-dependent reflection coefficients have statistically dependent and independent components. This gives the pulse stabilization result after Fourier synthesis.

#### 2.2.1 The scaled variables

We scale the arc length by , and the waveguide width and cross-range coordinate by ,

 ~z=z/L,  ~r=r/ℓ,  ~D(~z)=D(z/L)/ℓ. \hb@xt@.01(2.21)

The scaled frequency is

 ~ω=ωℓc. \hb@xt@.01(2.22)

The central frequency is . By (LABEL:eq:sc4) the bandwidth is such that so we introduce the scaled bandwidth defined by

 ~B=Bℓc1√ε. \hb@xt@.01(2.23)

The scaled wavenumber is

 ~k(~ω)=k(ω)ℓ. \hb@xt@.01(2.24)

All scaled quantities are of order one in the scaling regime described just above.

#### 2.2.2 The scaled equation

We assume henceforth that the variables are scaled, and simplify the notation by dropping the tilde. We take the Fourier transform of (LABEL:eq:fm7) with respect to time, and denote by the wave field in the scaled variables. After multiplying the resulting equation by we obtain

 [∂2z+(1−εrκ(z))2ε2(k2(ω)+∂2r)−κ(z)(1−εrκ(z))ε∂r+εrκ′(z)(1−εrκ(z))∂z]ˆp(ω,r,z) =ˆfε(ω)εδ(r−r⋆)δ(z), \hb@xt@.01(2.25)

with

 ˆfε(ω)=(1−εr⋆κ(0))2√εB[ˆF(ω−ωo√εB)+¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ˆF(−ω−ωo√εB)], \hb@xt@.01(2.26)

and homogeneous Dirichlet boundary conditions

 ˆp(ω,r±(z),z)=0,r−(z)=−D(z)2,r+(z)=D(z)2[1+√εσεν(zε)], \hb@xt@.01(2.27)

for all in the support of and . The wave is outgoing at , because there are no random fluctuations there, and decays exponentially (is evanescent) at .

### 2.3 Mode decomposition

To define the mode decomposition, we change coordinates to map the random boundary fluctuations to the coefficients of the wave equation (LABEL:eq:md1). This way we obtain a linear differential operator that has an asymptotic expansion in , and acts on functions that vanish at the unperturbed boundary for all . The modes are defined using the spectral decomposition of the leading part of , and they have random amplitudes satisfying stochastic differential equations driven by the process , with excitation given by jump conditions at , where the source lies.

#### 2.3.1 The random change of coordinates

We use the following change of coordinates that maps the random boundary fluctuations to the wave operator

 r=ρ+(2ρ+D(z))4√εσεν(zε),∀z<0, \hb@xt@.01(2.28)

where is in the unperturbed domain . There are no random fluctuations for , so there. Substituting in (LABEL:eq:md1) and using the chain rule, we obtain after straightforward calculations that

 ˆpε(ω,ρ,z)=ˆp(ω,ρ+(2ρ+D(z))4√εσεν(zε),z) \hb@xt@.01(2.29)

satisfies the equation

 Lεˆpε(ω,ρ,z)=ˆfε(ω)εδ(ρ−r⋆)δ(z),∀ρ∈(−D(z)2,D(z)2),  z∈R, \hb@xt@.01(2.30)

and the boundary conditions (LABEL:eq:md3) become

 ˆpε(ω,±D(z)/2,z)=0. \hb@xt@.01(2.31)

The operator is given by

 +O(√ε)]∂2ρz+O(ε)∂z+O(ε−1/2)∂2ρ+O(ε−1/2)∂ρ, \hb@xt@.01(2.32)

where are differential operators with respect to , with coefficients that depend on . These operators depend on only through and the argument of , , and . The leading operator is

 L0 =k2(ω)+∂2ρ, \hb@xt@.01(2.33)

its first perturbation depends linearly on the random process ,

 L1=−σεν(zε)∂2ρ−σεν′′(zε)(2ρ+D(z))4∂ρ, \hb@xt@.01(2.34)

the second perturbation is quadratic in ,

 L2=3σ2ε4ν2(zε)∂2ρ+σ2εν′2(zε)[(2ρ+D(z))216∂2ρ+(2ρ+D(z))4∂ρ] +σ2εν(zε)ν′′(zε)(2ρ+D(z))8∂ρ−κ(z)[2ρ(k2(ω)+∂2ρ)+∂ρ]. \hb@xt@.01(2.35)

#### 2.3.2 The waveguide modes

The self-adjoint operator , acting on functions that vanish at for any fixed , has the eigenfunctions

 yj(ρ,z)=[2D(z)]1/2sin[(2ρ+D(z))2μj(z)],μj(z)=πjD(z),  j=1,2,…, \hb@xt@.01(2.36)

and eigenvalues , for The eigenfunctions form an orthonormal basis in , so we can decompose the wave field at any as

 ˆpε(ω,ρ,z)=∞∑j=1ˆuεj(ω,z)yj(ρ,z), \hb@xt@.01(2.37)

where are waves in one dimension, called the waveguide modes. Substituting (LABEL:eq:decomp) in (LABEL:eq:perteq2), using the orthogonality of the eigenfunctions and the identities given in appendix LABEL:ap:identities, we obtain

 [∂2z+k2(ω)−μ2j(z)ε2]ˆuεj(ω,z)+σεε3/2[μ2j(z)ν(zε)+14ν′′(zε)]ˆuεj(ω,z) +18ν(zε)ν′′(zε)}ˆuεj(ω,z)−σ2ε4ν(zε)ν′(zε)∂zˆuεj(ω,z)=Cεj(ω,z,{ˆuεq}q≠j), \hb@xt@.01(2.38)

for . Here we neglected the remainder of order , and denoted by the coupling terms that depend on the modes , for . The curvature of the axis of the waveguide appears only in these terms. The equations for are simpler, because there are no random fluctuations in the right-hand side. They, are obtained from (LABEL:eq:decomp1) by setting to zero all the terms that depend on the process .

The first term in the wave equations (LABEL:eq:decomp1) shows that the th mode is a propagating wave when , and it is evanescent when the opposite inequality holds. By our scaling assumptions we have a single propagating mode for , the one indexed by . This interacts with the evanescent modes via the coupling term . We refer to [4, section 3.3] for the analysis of such an interaction. It shows that the evanescent modes can be expressed in terms of , so that we can close the wave equation for this propagating mode. We do not give here this calculation, because it is basically the same as in [4]. The result is that the contribution of the evanescent modes consists of an additional term in the equation for , that is similar to the quadratic one in the fluctuations, written in the curly bracket in (LABEL:eq:decomp1). We will see in section LABEL:sect:proof that this term is negligible when is scaled so that the reflected pulse retains a deterministic shape. For the sake of brevity, we do not include the contribution of the evanescent modes which play no role in the end.

#### 2.3.3 The equation for the propagating mode

We can simplify the equation for the propagating mode using integrating factors, by redefining the unknown

 ˆuε(ω,z) =ˆuε1(ω,z)exp[ε1/2σε4ν(zε)−εσ2ε16ν2(zε)]=ˆuε1(ω,z)[1+O(ε1/2)]. \hb@xt@.01(2.39)

Substituting in equation (LABEL:eq:decomp1) for , we obtain

 ∂2zˆuε(ω,z)+[k2(ω)−μ2(z)ε2+σεμ2(z)ε3/2ν(zε)+σ2εεgε(ω,z)]ˆuε(ω,z)=0 \hb@xt@.01(2.40)

for , with the simplified notation

 μ(z)=μ1(z)=πD(z),gε(ω,z)=−34μ2(z)ν2(zε)−π212ν′2(zε), \hb@xt@.01(2.41)

where the contribution of the evanescent waves is not written as it vanishes in our scaling regime. The excitation comes from the jump conditions at the source, with defined in (LABEL:eq:md2),

 ˆuε(ω,0+)−ˆuε(ω,0−) =0, \hb@xt@.01(2.42) ∂zˆuε(ω,0+)−∂zˆuε(ω,0−) =ε−1ˆfε(ω)y1(r⋆,0). \hb@xt@.01(2.43)

The remainder of the paper is concerned with the analysis of the solution of (LABEL:eq:decomp4), with initial condition defined by (LABEL:eq:jump1)–(LABEL:eq:jump2), outgoing condition at and exponential decay beyond the turning point, where the mode is evanescent.

## 3 The reflection coefficient and statement of results

We begin in section LABEL:sect:res1 with the decomposition of in forward and backward going waves. This allows us to define the reflection coefficient in section LABEL:sect:res2, and then state the pulse stabilization result in section LABEL:sect:res3. This result is derived in section LABEL:sect:proof under the assumption that the turning point of the mode is simple, for any in the support of . The frequency-dependent turning point is defined by

 k(ω)=μ(zT(ω))=πD(zT(ω)), \hb@xt@.01(3.1)

and it is unique due to the monotonicity of .

### 3.1 The forward and backward going waves

Let us write equation (LABEL:eq:decomp4) as a first-order system of stochastic differential equations

 ∂z(ˆuε(ω,z)ˆvε(ω,z)) =iε(01k2(ω)−μ2(z)0)(ˆuε(ω,z)ˆvε(ω,z)) +[iσε√εμ2(z)ν(zε)+iσ2εgε(ω,z)](0010)(ˆuε(ω,z)ˆvε(ω,z)), \hb@xt@.01(3.2)

for the vector with components and Let also be a flow of smooth and invertible matrices, and define the vector

 (ˆaε(ω,z)ˆbε(ω,z))=\itbfMε,−1(ω,z)(ˆuε(ω,z)ˆvε(ω,z)), \hb@xt@.01(3.3)

which satisfies equations

 ∂z(ˆaε(ω,z)ˆbε(ω,z))=\itbfMε,−1(ω,z){iε(01k2(ω)−μ2(z)0)\itbfMε(ω,z)−∂z\itbfMε(ω,z) +[iσε√εμ2(z)ν(zε)+iσ2εgε(ω,z)](0010)\itbfMε(ω,z)}(ˆaε(ω,z)ˆbε(ω,z)), \hb@xt@.01(3.4)

derived from (LABEL:eq:res3), where denotes the inverse of . The purpose of the decomposition (LABEL:eq:res4) is to remove the leading deterministic coupling term in (LABEL:eq:res5) by a proper choice of , so that we can analyze the effect of the random fluctuations. Then, we can associate the random fields and to the amplitudes of the forward and backward going waves for the mode , at .

#### 3.1.1 The propagator

The leading coupling term in (LABEL:eq:res5) vanishes when , the exact propagator matrix in the unperturbed, slowly changing waveguide. This is the solution of the flow problem

 ∂z\itbfMε⋆(ω,z)=iε(01k2(ω)−μ2(z)0)\itbfMε⋆(ω,z),z<0,

with chosen so that we have the usual wave decomposition at , as in a waveguide with straight boundaries. We work with an approximate propagator, which does not make the first line in the right-hand side of (LABEL:eq:res5) exactly zero, but it ensures that its contribution to (LABEL:eq:res5) converges to zero in the limit , uniformly in , and its expression is explicit.

As in [24], is the WKB approximation of . It is a matrix with structure

 \hb@xt@.01(3.5)

where we recall that the bar denotes complex conjugate. The structure in (LABEL:eq:res6) is like in waveguides with straight boundaries, and ensures energy conservation, as follows later in the section. The entries in (LABEL:eq:res6) are defined in terms of the function

 ϕω(z)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∫zzT(ω)dz′√k2(ω)−μ2(z′), zT(ω)≤z≤0,−∫zT(ω)zdz′√μ2(z′)−k2(ω),z

which in turn defines

 ηεω(z)={ε−2/3[3ϕω(z)/2]2/3,zT(ω)≤z≤0,−ε−2/3[−3ϕω(z)/2]2/3,z

and

 Qω(z)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩[3ϕω(z)/2]1/6[k2(ω)−μ2(z)]1/4,zT(ω)≤z≤0,[−3ϕω(z)/2]1/6[μ2(z)−k2(ω)]1/4,z

Note that is positive, at least twice continuously differentiable, and at the turning point it satisfies

 Qω(zT(ω))=γ−1/6ω,∂zQω(zT(ω))=θω5γ7/6ω,∂2zQω(zT(ω))=3ρω7γ7/6ω+9θ2ω35γ13/6ω, \hb@xt@.01(3.9)

where

 γω =−∂z[μ2(z)]∣∣z=zT(ω)=2k3(ω)πD′(zT(ω))>0, \hb@xt@.01(3.10) θω =12∂2z[μ2(z)]∣∣z=zT(ω),    ρω=16∂3z[μ2(z)]∣∣z=zT(ω). \hb@xt@.01(3.11)

The function vanishes at the turning point, and its derivative is given by

 ∂zηεω(z)=ε−2/3Q−2ω(z),∀z<0. \hb@xt@.01(3.12)

The entries of the propagator matrix (LABEL:eq:res6) are defined by

 Mε11(ω,z)=ε−1/6√πQω(z)e−iϕω(0)/ε+iπ/4[Ai(−ηεω(z))−iBi(−ηεω(z))], \hb@xt@.01(3.13)

and

 Mε21(ω,z)= −iε∂zMε11(ω,z) = −ε1/6√πQω(z)e−iϕω(0)/ε−iπ/4[A′i(−ηεω(z))−iB′i(−ηεω(z))] +ε5/6√πQ′ω(z)e−iϕω(0)/ε−iπ/4[Ai(−ηεω(z))−iBi(−ηεω(z))], \hb@xt@.01(3.14)

in terms of the Airy functions [1, chapter 10] denoted by and .

The next lemma, proved in appendix LABEL:ap:propagator, shows that approximates the exact propagator, and that it is an invertible matrix with constant determinant.

###### Lemma 3.1

The matrix-valued process (LABEL:eq:res6), with entries defined by equations (LABEL:eq:res15)–(LABEL:eq:res16), satisfies

 \hb@xt@.01(3.15)

and

 det\itbfMε(ω,z)=2, \hb@xt@.01(3.16)

for all .

The next lemma, proved in appendix LABEL:ap:propagator, describes the propagator as approaches , where the source lies.

###### Lemma 3.2

When and the entries (LABEL:eq:res15)–(LABEL:eq:res16) of have the following asymptotic expansions in ,

 Mε11(ω,z)=[k2(ω)−μ2(0)]−1/4{exp[iε(ϕω(z)−ϕω(0))]+O(ε)},

and

 Mε21(ω,z)=[k2(ω)−μ2(0)]1/4{exp[iε(ϕω(z)−ϕω(0))]+O(ε)}.

The leading terms in these expansions are the entries of the propagator in waveguides with straight boundaries and width .

Using this result in (LABEL:eq:res4) and (LABEL:eq:res6), we obtain a wave decomposition like in waveguides with straight walls [12, chapter 20]. The wave field is, for ,

 ˆuε(ω,z)≈[k2(ω)−μ2(0)]−1/4[ˆaε(ω,z)exp(iε∫z0dz′√k2(ω)−μ2(z′)) −ˆbε(ω,z)exp(−iε∫z0dz′√k2(ω)−μ2(z′))], \hb@xt@.01(3.17)

and its derivative is

 ∂zˆuε(ω,z)≈iε[k2(ω)−μ2(0)]1/4[ˆaε(ω,z)exp(iε∫z0dz′√k2(ω)−μ2(z′)) +ˆbε(ω,z)exp(−iε∫z0dz′√k2(ω)−μ2(z′))], \hb@xt@.01(3.18)

with relative error of order .

In the vicinity of the turning point, for , the Airy functions and their derivatives are bounded, as are and its derivatives, described in (LABEL:eq:res10). We obtain that the entries in the first row of are large, of order , and the entries in the second row are small, of order .

Beyond the turning point, at , the entries of