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Research Article

Mathematical

Methods in the

Applied Sciences

Received XXXX

(www.interscience.wiley.com) DOI: 10.1002/sim.0000

MOS subject classification: 35L05, 35C20, 78M35

On mechanical waves and Doppler shifts

from moving boundaries

Ivan C. Christova†∗ and C. I. Christovb‡

We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity

and arriving at a stationary observer. In the classical Doppler effect, Xs(t) = vt is the location of the source with constant

velocity v . In the present work, however, we consider a source co-located with a moving boundary x = Xs(t), where Xs(t)

can have an arbitrary functional form. For “slowly moving” boundaries (i.e., ones for which the timescale set by the

mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple-scale

asymptotic analysis of the moving-boundary problem for the linear wave equation. We obtain a closed-form leading-order

(with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only

in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extending the

applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (a) Xs for non-

uniform boundary motion can be inserted in place of the constant velocity v in the classical Doppler formula and (b) that

the non-uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating

and oscillatory boundary motion are worked out and illustrated. Copyright c© 2016 John Wiley & Sons, Ltd.

Keywords: Doppler effect; accelerating source; multiple-scales expansion; wave equation; moving boundary

1. Introduction

The classical (or non-relativistic) Doppler effect [1] is concerned with the change in observed frequency of a mechanical wave

when its emitter is in relative motion with respect to the observer [2, 3]. The shifted frequency ωD, measured by a stationary

observer downstream (along the direction of propagation of the wave) from the emitter, goes (see, e.g., [2]) as

ωD =ω

1− v/c, (1)

where ω is the frequency of the emitted waves, v is the velocity of the source, and c is the phase speed of infinitesimal waves

in the particular medium under consideration (e.g., the speed of sound for an acoustic wave). The quantity ∆ω := ωD − ω is

aTheoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USAbDepartment of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504, USA∗Correspondence to: School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA. E-mail: christov@purdue.edu; URL:

http://christov.tmnt-lab.org.

Contract/grant sponsor: Los Alamos National Laboratory (LANL) is operated by Los Alamos National Security, L.L.C. for the National Nuclear

Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396.†Present address: School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA.‡Prof. C. I. Christov passed away prior to submission of the manuscript. This paper is dedicated to his memory.

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termed the Doppler shift. For a source moving towards the observer (v > 0), the measured frequency ωD is larger than the

emitted frequency ω, whereas for a source moving away from the observer (v < 0), ωD < ω. The Doppler effect is a staple of

classical wave physics, and the applications of (1) in telecommunications, meteorology, medicine, etc. are so numerous that we

do not attempt to list them here (see, e.g., [2, 3, 4, 5]). However, more than 150 years after Christian Doppler’s proposal, novel

aspects of the Doppler effect continue to be uncovered [6].

The case of translation of the source along the line x = Xs(t) ≡ vt at constant velocity v has been exhaustively treated and

is well understood. For non-uniform source velocities, i.e., Xs(t) 6= const., the problem has been analyzed more recently [7, 8]

due to its relevance for acoustics in moving and inhomogeneous media [9, 10, 11, 12]. Simple acoustic laboratory experiments

[13, 14] have been performed showing the effects of acceleration of the emitter. Specifically, higher harmonics appear in the

spectrum of the received signal, and the spectrum itself becomes markedly asymmetric. In the analysis of the data, however, it is

common to formally replace v with Xs(t) in (1) [13, 14]. This manipulation is justified by the solutions, given in [8, Chap. 5], of

the three-dimensional (3D0 initial-boundary-value problem (IBVP) for the wave equation with the acoustic source modeled as a

singular term on the right hand side of the linear wave equation [7, 8, 15]. Here, we would like to pose a different variant of this

problem: if the source were attached to a moving boundary of a one-dimensional (1D) domain, then can we still replace v with

Xs(t) in (1), or would corrections arise from a formal mathematical analysis? Posing the problem in this manner also provides a

natural generalization of some fundamental IBVPs, which we review in context below, studied in the mathematics literature.

From the mathematical point of view, some basic sketches of the theory of such mechanical wave motions have been presented

in the context of asymptotic and perturbation methods [16, 17, 18, 19]. However, to the best of our knowledge, an analysis of

wave propagation from an emitter co-located with a boundary of non-uniform velocity cannot be found in the literature. Thus, in

this paper, we provide a formal perturbative solution based on the method of multiple scales for the case in which the timescale

set by the mechanical motion of the emitter is large compared to the inverse frequency of the emitted wave. We show that for

general boundary velocity given by Xs(t), the expression for the shifted frequency (1) can be immediately modified as

ωD(t) =ω

1− Xs(t)/c, (2)

within the assumed order of approximation. This result is, of course, exactly in agreement with the corresponding Doppler shift

found from the general 3D solution for a moving source in a homogeneous stationary medium [8, eq. (5.23)]. An additional

physical effect obtained by our analysis is that the amplitude of the wave is also affected by the non-uniform boundary motion,

which can also be inferred from the the general 3D solution for a moving source in a homogeneous stationary medium [8, §5.1].

To this end, in §2, we reformulate the moving-boundary problem for the wave equation into an equivalent problem on a fixed

domain for a dispersive, variable-coefficient wave equation. In §3 we give the leading-order solution by the method of multiple

scales. Then, in §4, the solution is illustrated for a decelerating boundary and for an oscillatory boundary motion. Finally, in §5,

conclusions are stated and a broader context for the present results and their applicability is proposed.

2. Position of the problem

A plethora of mechanical wave phenomena are governed by the (1 + 1) dimensional linear wave partial differential equation

(PDE)∂2U

∂t2− c2

∂2U

∂x2= 0, Xs(t) < x <∞, 0 < t <∞, (3)

where U = U(x, t) can be, e.g., the acoustic potential [4], the elastic displacement [20] or even the temperature field (under

certain nonclassical theories of thermoelasticity) [21, §2.3], to name a few, and c is the phase speed of infinitesimal waves in

the material medium. In the present work, (3) is subject to the boundary condition

U(

Xs(t), t)

= eiωt , Xs(t) := vt + aα(Ωt), t > 0, (4)

which represents an accelerating moving source emitting monochromatic harmonic waves with frequency ω, where v , a and Ω

are some positive constants and α is a dimensionless function (nonlinear in its argument). Here, a has the dimension of length

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Mathematical

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and Ω has the dimension of inverse time. We have chosen this particular functional form for Xs(t) so that the classical Doppler

effect is easily recovered in the final results by setting α = 0 and/or taking the limit a→ 0+. We take α such that α(0) = 0,

without loss of generality, so that Xs(0) = 0. By convention, we work with complex exponentials since (3) is a linear equation,

and the real part of U is taken at the end of the calculation.

In addition, we must supplement eqs. (3) and (4) with the radiation condition

limx→∞

(

∂U

∂x+ iκU

)

= 0, (5)

where κ is the spatial wave number, and the “+” is chosen so that only waves that are outgoing at x =∞ are allowed [22, §28].

Without loss of generality, homogeneous initial conditions, U(x, 0) = ∂U∂t (x, 0) = 0, can be imposed because, for the present

purposes, we are only interested in the influence of the boundary condition, meaning that ∀x <∞ ∃t = x/c such that any

non-zero initial condition has propagated past this location, and only the effects due to the boundary condition are “felt” there.†

George Carrier’s “spaghetti problem” [23] regarding the normal modes of a string being shortened due to its accelerated

withdrawal into an orifice motivated some early analytical work by Balazs [24] and Greenspan [25] on the Dirichlet IBVP for

(3) on a finite domain with moving and/or accelerating boundaries. A pernicious feature of these problems is reflections from

the boundaries, leading to analytical solutions in the form of trigonometric series [24, 25, 26]. The Dirichlet problem can also

be solved for general boundary motions using nonlinear transformations of the independent variables [27, 28] and multiple-scale

asymptotics [29]. Integral representations for the solution to the half-space problem with a moving boundary have also recently

been proposed [30] on the basis of advanced transform techniques [31, 32, 33, 34]. In contrast, we study the physical (rather

than abstract) half-space problem for harmonic mechanical waves in order to discern any frequency and/or amplitude shifts due

to the non-uniform motion of the boundary.

We choose to convert eqs. (3)–(5) into a boundary-value problem on [0,∞) by introducing the moving frame coordinate

ξ = x − Xs(t) ≡ x − vt − aα(Ωt), (6)

while keeping the time coordinate the same. Then, the temporal and spatial partial derivatives transform as

∂2

∂t2=

∂2

∂t2− 2(v + aΩα′)

∂2

∂ξ∂t− aΩ2α′′

∂

∂ξ+ (v + aΩα′)2

∂2

∂ξ2,

∂2

∂x2=

∂2

∂ξ2, (7)

where a prime indicates differentiation with respect to the argument of α. Letting U(x, t) = U(ξ, t) and introducing (7) into

(3), we obtain

∂2U

∂t2− 2(v + aΩα′)

∂2U

∂ξ∂t− aΩ2α′′

∂U

∂ξ−

[

c2 − (v + aΩα′)2]∂2U

∂ξ2= 0, 0 < ξ <∞, 0 < t <∞. (8)

Equation (8) is now a dispersive wave equation with variable coefficients. The general theory of dispersive waves under such

equations is described by Whitham [5, Chap. 11]. Some remarks on the theory of such PDEs, including analysis of the Lie

symmetries, were given by Bluman [35]. Analytical solutions for special choices of the coefficient have been provided, e.g., for

v = a = 0 and c = c(x) [36, 37, 38] or v = a = 0 and c = c(x, t) [39], with further generalization given in [40]. Others have

considered the case of discontinuous c(x) [41]. More recently, the asymptotic properties of localized solutions for v = a = 0 but

c = c(x) have been examined in detail [42, 43]. Some special constant coefficient cases of (8) arise in the study of low-frequency

modulation of acoustic radiation forces [44]. Unfortunately, all these results are too specialized to be of immediate use in our

analysis of (8).

Before proceeding further, we must enforce some limitations on v , a and Ω. Obviously, at any time t, we must have

|Xs(t)| ≡ |v + aΩα′(Ωt)| < c, (9)

†This type of argument could be generalized to “arbitrary” initial conditions as long as they are constrained to produce waves satisfying the radiation

condition (5).

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i.e., the instantaneous velocity of the boundary must be less that the phase speed of waves, otherwise (8) changes type from

hyperbolic to elliptic, and the problem becomes ill-posed (and unphysical). In other words, the boundary motion is subsonic.‡ We

consider the case when aΩ/c (a type of acceleration-based Strouhal number) is O(1), i.e., the time scale set by the acceleration

of the source and the time scale on which its acceleration varies are comparable. Since we introduced the parameter a, we are free

to normalize α so that maxt≥0 |α′(t)| = 1. Consequently, a necessary condition for the inequality (9) to hold is aΩ/c < 1− v/c .

The most important assumption we make, however, is that ω ≫ Ω, i.e., the frequency of the emitted wave ω is much larger

than the frequency of the mechanical motion Ω associated with the acceleration of the source.§ (Equivalently, the timescale Ω−1

set by the mechanical motion is large in comparison to the time scale ω−1 of wave propagation; i.e., a “slowly moving” source.)

This assumption defines the small parameter for the upcoming asymptotic expansion. Therefore, we introduce the following

dimensionless independent variables and dimensionless parameters:

τ = ωt, η = ξω/c, β := v/c, δ := aΩ/c, ǫ := Ω/ω ≪ 1. (10)

Note that the non-dimensionalization of the dependent variable is arbitrary since (8) is a homogeneous linear equation, hence it

is invariant under re-scaling of the dependent variable.

Now, letting U(ξ, t) = U(η, τ) and making use of these dimensionless variables from (10), (8) becomes

∂2U

∂τ 2− 2

[

β + δα′(ǫτ)] ∂2U

∂η∂τ− ǫδα′′(ǫτ)

∂U

∂η−

1−[

β + δα′(ǫτ)]2 ∂2U

∂η2= 0, 0 < η <∞, 0 < τ <∞. (11)

Recall that, here, primes stand for differentiation of a function with respect to its argument (in this case, ǫτ).

Finally, the boundary and radiation condition from eqs. (4) and (5) become

U(0, τ) = eiτ , limη→∞

(

∂U

∂η+ ikU

)

= 0, (12)

where k = κc/ω is the dimensionless wave number.

Hence, we have transformed our original movin- boundary problem into a variable-coefficient problem on the half-line.

Unfortunately, our problem does not appear to yield itself to a closed-form solution. However, the variable coefficients in (11)

are slowly varying, i.e., they depend only on ǫτ . Thus, we proceed by perturbation methods as in [29, 44, 45].

3. Solution by a multiple-scales expansion

Equation (11) is a linear wave equation with slowly varying coefficients, which makes it an ideal candidate for a multiple-scales

asymptotic expansion [16, 17, 18, 19, 46, 47], the generalization of Cole’s two-variable expansion procedure [48, Chap. 3]. To

this end, we introduce the “fast” time t0 = τ , the “slow” time t1 = ǫτ , the “short” spatial coordinate y0 = η and the “long”

spatial coordinate y1 = ǫη. For convenience, we first rewrite (11) as

∂2U

∂τ 2− 2β(t1)

∂2U

∂η∂τ−

[

1− β2(t1)]∂2U

∂η2− ǫδα′′(t1)

∂U

∂η= 0, (13)

‡Note that, in the case of the IBVP for wave propagation in a homogeneous medium at rest, in which the acoustic source modeled as a singular

term on the right hand side of (3) (see, e.g., [8, Chap. 5]), it possible to also consider supersonic sources.§Note that, in the case of the IBVP for wave propagation in a homogeneous medium at rest, in which the acoustic source modeled as a singular

term on the right hand side of (3) (see, e.g., [8, Chap. 5]), it possible to find an analytical solution without requiring that ω ≫ Ω.

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where β(t1) := β + δα′(t1) is a function of the slow time alone. Then, we let U(η, τ) = U(y0, y1, t0, t1) with its partial derivatives

transforming as∂U

∂τ=∂U

∂t0+ ǫ

∂U

∂t1,

∂2U

∂τ 2=∂2U

∂t20+ 2ǫ

∂2U

∂t0∂t1+ ǫ2

∂2U

∂t21,

∂U

∂η=∂U

∂y0+ ǫ

∂U

∂y1,

∂2U

∂η2=∂2U

∂y 20+ 2ǫ

∂2U

∂y0∂y1+ ǫ2

∂2U

∂y 21,

∂2U

∂η∂τ=

∂2U

∂y0∂t0+ ǫ

∂2U

∂y0∂t1+ ǫ

∂2U

∂y1∂t0+ ǫ2

∂2U

∂y1∂t1.

(14)

Upon introducing (14) into (13) and keeping only leading-order terms and terms proportional to ǫ, we obtain

∂2U

∂t20− 2β(t1)

∂2U

∂y0∂t0−

[

1− β2(t1)]∂2U

∂y 20+ 2ǫ

∂2U

∂t0∂t1− 2ǫβ(t1)

∂2U

∂y0∂t1

− 2ǫβ(t1)∂2U

∂y1∂t0− 2ǫ

[

1− β2(t1)] ∂2U

∂y0∂y1− ǫδα′′(t1)

∂U

∂y0+O(ǫ2) = 0. (15)

We proceed in the usual manner by a regular expansion of the dependent variable:

U(y0, y1, t0, t1) = U0(y0, y1, t0, t1) + ǫU1(y0, y1, t0, t1) + · · · . (16)

In turn, the (first) boundary condition from (12) becomes

U0(0, 0, t0, t1) = eit0 , Uj (0, 0, t0, t1) = 0 (j > 0). (17)

Then, at the leading order, (15) becomes

L0[U0] = 0, L0 :=∂2

∂t20− 2β

∂2

∂y0∂t0− (1− β2)

∂2

∂y 20. (18)

Here, it is important to recall that β does not depend on the fast time t0. Therefore, according to the multiple-scales expansion

procedure, it is considered a constant at this order. This is equivalent to the assumption that the wavenumber k depends on the

slow time t1, which is sometimes referred to as the “generalized” method of multiple scales [16, §6.4].

Clearly, a solution of eqs. (18) and (17) of the form¶

U0(y0, y1, t0, t1) = A0(y1, t1)eit0−iky0+iψ0(y1,t1) (19)

exists provided that

− 1− 2βk +(

1− β2)

k2 = 0 =⇒ k =β ± 1

1− β2=±1

1∓ β. (20)

Here, we must pick the upper sign in the expression for k (⇒ k = 1/(1− β) > 0) to satisfy the radiation condition (12) (i.e., to

have waves that are propagating away from the source and outgoing at η =∞) and set A0(0, t1) = 1, ψ0(0, t1) = 0 to satisfy

the boundary condition (17). Notice that the earlier assumptions that aΩ/c < 1− v/c ⇔ δ < 1− β and that maxt≥0 |α′(t)| = 1

guarantees that (1− β) > 0⇒ 1/(1− β) > 0 in the accelerating case (i.e., the case of δα′ > 0). In general, the sign of β

depends on whether the boundary is moving towards or away from the observer, the latter being situated somewhere on the

positive abscissa. For β > 0, we have k > 1 (the wave is “shorter”), i.e., the observed pitch is higher for a source moving towards

the observer. Conversely, for β < 0, k < 1 (the wave is “longer”), which means the observed pitch is lower for a receding source.

Continuing to O(ǫ), we must now solve

L0[U1] = −2∂2U0∂t0∂t1

+ 2β∂2U0∂y0∂t1

+ 2β∂2U0∂y1∂t0

+ 2(

1− β2) ∂2U0∂y0∂y1

+ δα′′∂U0∂y0

. (21)

¶Here, we use the structure of the boundary condition to infer the form of the solution. More generally, for arbitrary excitations, one can introduce

characteristic coordinates and proceed along the lines of the Appendix.

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Denoting the right-hand side above as F, it can be evaluated based on the solution for U0 from (19):

F(y0, y1, t0, t1) =

− 2

(

i∂A0∂t1−A0

∂ψ0

∂t1

)

+ 2β

(

−ik∂A0∂t1+ kA0

∂ψ0

∂t1

)

+ 2β

(

i∂A0∂y1

−A0∂ψ0

∂y1

)

+ 2(

1− β2)

(

−ik∂A0∂y1

+ kA0∂ψ0

∂y1

)

− iδα′′kA0

eit0−iky0+iψ0 . (22)

This right-hand side of (21) will produce secular terms because eit0−iky0+iψ0 is in the nullspace of L0. Therefore, we must choose

A0 and ψ0 so that F ≡ 0. Separating the real and imaginary parts of (22) and assuming a nontrivial solution A0 6= 0, we obtain

(1 + βk)∂ψ0

∂t1−

[

β −(

1− β2)

k]∂ψ0

∂y1= 0, (23a)

(1 + βk)∂A0∂t1−

[

β −(

1− β2)

k]∂A0∂y1

= − 12δα′′kA0. (23b)

We proceed by the method of characteristics [49, Chap. II], which transforms (23) into a set of ordinary differential equations

(ODEs):

dt1ds= 1 + βk, (24a)

dy1ds= −

[

β −(

1− β2)

k]

, (24b)

dψ0ds= 0, (24c)

dA0ds= − 12δα

′′kA0, (24d)

where we have introduced the notation ψ0(y1, t1) = ψ0(s) and A0(y1, t1) = A0(s). To find the characteristics, we need to

integrate the first two ODEs in (24), keeping in mind that β = β(t1), subject to the “initial” condition that s = 0 when t1 = 0.

Therefore, upon using the expression for k from (20), we have

ds

dt1= 1− β(t1) =⇒ s(y1, t1) =

∫ t1

0

1− β(t) dt+ C1(y1), (25a)

ds

dy1= 1 =⇒ s(y1, t1) = y1 + C2(t1). (25b)

Solving for the arbitrary functions C1,2 between the two equations and recalling that β(t1) ≡ β + δα′(t1), we obtain

s(y1, t1) =

∫ t1

0

1− β(t) dt+ y1 = (1− β)t1 − δα(t1) + y1. (26)

From (24d), we obtain

ln∣

∣A0(

s(y1, t1))∣

∣− ln∣

∣A0(

s(0, t1))∣

∣ = −1

2

∫ s(y1,t1)

s(0,t1)

δα′′(t1(ς))

1− β(t1(ς))dς, (27)

where the limits of integration were determined by the fact that our boundary condition is given at y0 = y1 = 0. Making a change

of variables in the integral by using (25a) and recalling that A0(

s(0, t1))

= 1 from the boundary condition (17), we have

ln |A0(y1, t1)| = −1

2

∫ s(y1 ,t1)

s(0,t1)

δα′′(t) dt =⇒ A0(y1, t1) = exp

−δ

2

[

α′(

s(y1, t1))

− α′(

s(0, t1))]

. (28)

If α′′(t) is continuous, then, by the mean value theorem for integrals, we can write a more compact expression: A0(y1, t1) =

exp

− δ2α′′(ς)y1

, where ς ∈(

s(0, t1), s(y1, t1))

is to be determined based on the functional form of α′′.

The ODE for the phase (24c) gives ψ0(s) = ψ0(

s(0, t1))

= 0 ∀t1 after applying the boundary condition (17). Substituting the

positive solution from (20) for k, (28) for A0 and ψ0 = 0 into (19) and then eliminating s using (26) completes the leading-order

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solution:

U(η, τ) ∼ exp

−δ

2

[

α′(

(1− β)ǫτ − δα(ǫτ) + ǫη)

− α′(

(1− β)ǫτ − δα(ǫτ))

]

exp

i

[

τ −η

1− β(ǫτ)

]

. (29)

Upon returning to the original (dimensional) variables in the stationary frame, we have

U(x, t) ∼ exp

−δ

2

[

α′(

(1− β)Ωt + xΩ/c − βΩt − 2δα(Ωt))

− α′(

(1− β)Ωt − δα(Ωt))

]

× exp

iω

1− β − δα′(Ωt)

[

t − x/c − δα′(Ωt)t + δΩ−1α(Ωt)]

. (30)

This asymptotic result is valid for arbitrary δ, provided that the speed of the emitter never exceeds the phase speed of waves

in the medium [recall the discussion before (10)], which corresponds to the requirement that the wave equation in the moving

frame remains hyperbolic, namely δ < 1− β. Furthermore, we have to keep in mind that (30) is valid only for x < ct due to the

finite speed of propagation of waves under (3). For x > ct, U = 0 since we stipulated homogeneous initial conditions.

Finally, we would like to make a brief comparison between (30) and the corresponding solution [8, Eq. (5.15)] to the the

IBVP for wave propagation in a homogeneous medium at rest, in which the acoustic source is modeled as a singular term on

the right-hand side of (3). Although (30) and [8, Eq. (5.15)] look quite different, they have some conceptual similarities. The

Doppler frequency including Xs appears in the harmonic exponential term of both. Likewise, both (30) and [8, Eq. (5.15)] feature

amplitude modulation but the functional form of the amplitude differs because (30) is for a moving-boundary IBVP, while [8,

Eq. (5.15)] is for a moving source in a homogeneous medium at rest. Finally, while [8, Eq. (5.15)] includes a summation over

all solutions to the retarded time equation [8, Eq. (5.7)], (30) does not because it is posed on a moving domain.

4. Discussion

The prefactor of the bracketed expression inside the second exponential in the asymptotic solution (30) gives the frequency of

the observed wave at a distance x from the emitter at time t. Thus, we learn that the emitted frequency ω is shifted to the

observed frequency ω/[1− β − δα′(Ωt)], within O(ǫ), due to the acceleration of the boundary. The shifted frequency has the

same functional form as in the non-accelerating case (1), except that v/c ≡ β is replaced by Xs(t)/c ≡ β + δα′(Ωt) [recall the

second equation in (4)], consistent with (2). Consequently, the wave experiences frequency modulation‖ (rather than a simple

shift) since α′ depends upon t. In addition, within the same asymptotic order of approximation, there is an amplitude modulation

of the waveform as embodied by first exponential in (30). Note, however, that all time dependences in (30) are upon Ωt ≡ ǫωt

[using (10)], which is the slow time variable defined previously. Taking the limit δ → 0 (no acceleration, Xs(t)/c → β = const.),

(29) becomes

U(x, t) ∼ exp

iω

1− β

[

t − x/c]

, (31)

which is a harmonic wave with constant amplitude and constant frequency, the latter given by the ordinary Doppler formula

(1), as required. Furthermore, note that for certain choices of α, the first exponential in eqs. (29) and (30) could lead to the

increase of the amplitude of the wave.

To summarize: the effects on the observed waveform due to the acceleration of the boundary are (i) amplitude modulation,

as made explicit by the first exponential in eqs. (29) and (30); (ii) frequency modulation of the same functional form as for the

case of an emitter with constant velocity; (iii) a time-dependent phase shift, namely δα′(Ωt)t − δΩ−1α(Ωt).

Next, based on (30), we can define two quantities that characterize the waveform: the frequency modulation (FM) factor

FM(t) :=1

1− β − δα′(Ωt)(32)

‖Frequency modulation in the Doppler spectrum of underwater acoustic waves due to source or receiver motion has been measured [50].

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and the amplitude modulation (AM) factor

AM(x, t) := exp

−δ

2

[

α′(

(1− β)Ωt + xΩ/c − βΩt − 2δα(Ωt))

− α′(

(1− β)Ωt − δα(Ωt))

]

. (33)

Although our treatment applies to mechanical waves, we borrow this terminology from the radio wave transmission literature,

wherein frequency modulation of the emitted wave at the source is used for ultra-high frequency (UHF) communications, while

amplitude modulation of the emitted wave at the source is preferred for low frequency (LF) communications [51]. Additionally,

as was the case with (2), the FM factor (32) is consistent with the FM factor found from the general three-dimensional solution

for a moving source in a homogeneous stationary medium [8, eq. (5.23)].

FM and AM can be rewritten as functions of the three dimensionless parameters (β, δ and ǫ), the dimensionless time τ = ωt

and the dimensionless distance ωx/c :

FM(t) =1

1− β − δα′(ǫωt), (34a)

AM(x, t) = exp

−δ

2

[

α′(

ǫ(1− β)ωt + ǫωx/c − ǫβωt − 2δα(ǫωt))

− α′(

(1− β)ǫωt − δα(ǫωt))

]

. (34b)

Note that while the frequency modulation is only a function of time, the amplitude modulation is both a function of time and

space. Consequently, the wave amplitude measured by an observer depends on the observer’s instantaneous distance from the

emitter.

Next, we consider two illustrative examples.

4.1. A decelerating boundary

First, consider the case of a continuously decelerating boundary We take

α(t) = 1− e−t (35)

so that α′(t) = e−t, maxt≥0 |α′(t)| = 1 and α(0) = 0. Then, evaluating (34) using (35) yields

FM(t) =1

1− β − δe−ǫωt, (36a)

AM(x, t) = exp

δ

2eδ−2δe

−ǫωt[

eδe−ǫωt−ǫ(1−β)ωt − eδ−ǫωt−ǫωx/c+2ǫβωt

]

. (36b)

The FM factor is not singular thanks to our earlier restriction of δ < 1− β.

We can compute the following asymptotic limits of (36):

FM→

1

1− β − δ, t → 0;

1

1− β, t →∞;

(37a)

AM→

exp

δ

2(1− e−ǫωx/c)

, t → 0;

1, t →∞.

(37b)

As t →∞, we observe that FM and AM reduce to the classical Doppler relations. This is expected as an exponentially-decaying

acceleration quickly becomes negligible.

Figure 1 shows plots of FM and AM for some representative values of the dimensionless parameters. Meanwhile, fig. 2 shows

the waveform Re[AM(x, t)eiFM(t)ω(t−x/c)] at fixed ωx/c . To simplify the discussion, the time-dependent phase shift has been

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Figure 1. Frequency (left) and amplitude (right, at ωx/c = 10) modulation factors for ǫ = 0.1 and β = 0; δ = −0.2 (solid), δ = −0.1 (dashed) and δ = −0.05

(dotted).

Figure 2. Waveform Re[AM(x, t)eiFM(t)ω(t−x/c)] observed (solid curves) by two receivers in the fixed frame, as a function of dimensionless time; ǫ = 0.1, β = 0,

δ = −0.2; ωx/c = 0.1 (left) and ωx/c = 10 (right). Dashed curves represent the envelope AM from (36b), while the dotted curves represent the harmonic

wave Re[eiω(t−x/c)], as it would propagate away from a stationary source.

Figure 3. Frequency (left) and amplitude (right, at ωx/c = 10) modulation factors for ǫ = 0.1 and β = 0; δ = 0.2 (solid), δ = 0.1 (dashed) and δ = 0.05

(dotted).

neglected and, without loss of generality, β = 0 is used in these figures. The plots show that the effects due to acceleration

disappear as t →∞ (i.e., as the boundary velocity becomes uniform or, in this case of β = 0, the boundary becomes stationary)

as evidenced by the increasing overlap between the solid curves, Re[AM(x, t)eiFM(t)ω(t−x/c)], and dotted curves, Re[eiω(t−x/c)],

in fig. 2 for large ωt. This observation is supported by the long-time asymptotics given by (37). As can be seen in fig. 2, the

frequency shift due to the acceleration of the boundary is more pronounced for ωx/c ≪ 1 (close to the source), while the

amplitude shift is more pronounced for ωx/c ≫ 1 (farther downstream).

4.2. Periodic oscillations of the boundary

Second, consider an oscillating boundary motion. In this case, we take

α(t) = sin(t) (38)

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Figure 4. Waveform Re[AM(x, t)eiFM(t)ω(t−x/c)] observed (solid curves) by two receivers in the fixed frame, as a function of dimensionless time; ǫ = 0.1, β = 0,

δ = 0.2; ωx/c = 0.1 (left) and ωx/c = 10 (right). Dashed curves represent the envelope AM from (39b), while the dotted curves represent the harmonic wave

Re[eiω(t−x/c)], as it would propagate away from a stationary source.

so that α′(t) = cos(t), maxt≥0 |α′(t)| = 1 and α(0) = 0. Evaluating (34) using (38) yields

FM(t) =1

1− β − δ cos(ǫωt), (39a)

AM(x, t) = exp

−δ

2

[

cos(

ǫ(1− β)ωt + ǫωx/c − ǫβωt − 2δ sin(ǫωt))

− cos(

ǫ(1− β)ωt − δ sin(ǫωt))

]

. (39b)

Once again, δ < 1− β guarantees that the latter expressions are free of singularities. The expression (39a) for FM agrees with

the heuristic expression in [14, eq. (6)]. These time-periodic frequency and amplitude modulations can also be loosely interpreted

as combined vibrato and tremolo produced by, e.g., a Leslie loudspeaker [52, pp. 521–523].

The standard Doppler effect, as quantified by β, is to change the mean of the FM and AM factors, so we take β = 0 for the

remainder of this subsection without loss of generality. This assumption corresponds to an oscillating boundary with zero net

displacement. As in §4.1, figures 3 and 4 illustrate, respectively, the frequency and amplitude modulation factors, i.e., FM and

AM, and the waveform Re[AM(x, t)eiFM(t)ω(t−x/c)] at fixed ωx/c , neglecting the time dependent phase shift and with β = 0.

Once again, fig. 4 highlights the fact that the amplitude shift due to the acceleration of the boundary is more pronounced for

ωx/c ≫ 1 (far downstream from the source). In this example, however, the frequency modulation remains a pernicious feature

for all t because FM(t)9 1/(1 − β) = 1 (for β = 0) as t →∞.

5. Conclusion

In this work, we investigated the effect of the acceleration of a moving boundary of a 1D domain on the emitted mechanical plane

waves from a co-located source in a homogeneous resting medium, which represents a new variant of George Carrier’s “spaghetti

problem” [23]. The hyperbolic partial differential equation describing the wave motion was posed as a boundary-value problem

with a moving “inlet” boundary condition at the emitter’s position (i.e., at the moving boundary) and a radiation condition at

infinity. This problem was transformed to a dispersive hyperbolic PDE with variable coefficients in the moving frame.

The small parameter ǫ := Ω/ω, which represents the ratio of the characteristic time scale of the emitted wave to the

characteristic time scale on which the mechanical oscillations of the boundary take place, was used in a multiple-scales asymptotic

expansion of the solution of the boundary-value problem in the moving frame. Specifically, we derived the solution given in (30),

which consists of an envelope propagating over the Doppler shifted carrier wave. This solution is a formal new result for moving

IBVPs. Futhermore, since the governing PDE in the moving frame (8) has variable coefficients, it also has, in particular, a

variable phase speed; thus, the results in §3 generalize the problem and solution from the appendix of [53].

It is interesting to note that, in the case of a decelerating boundary motion considered in §4.1, the amplitude modulation is

pure attenuation. A similar a effect is experienced by a harmonic wave traveling through a porous medium under the nonlinear

theory of acoustics [54]. In the latter case the source is stationary, but the medium through which the wave travels induces signal

loss, and the signal itself experiences nonlinear effects due to compressibility. Of course, in the present work, the attenuation is

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purely due to the motion of the boundary (it arises in the absence of dissipative or nonlinear effects, although it would also be

of interest to consider the case of complex dispersive moving media [15]).

More generally, our discussion of the Doppler effect in the context of moving IBVPs has important implications for a number

of other wave phenomena. Inhomogeneities of the carrier medium can cause diffraction of acoustic and/or elastic waves, which

can be interrogated using techniques similar to the present mathematical framework [45]. In plasmas, the variable refractive

index of the medium can also cause frequency modulations that are not explainable by the classical Doppler effect [55].

We note that, in our illustrated examples in §4, we neglected the effect of the time-dependent phase shift caused by the

acceleration of the boundary. It is conceivable that this feature could explain the experimental observations [13, 14] of generation

of higher harmonics in the spectrum of the received signal. Future work also includes studying the evolution of a Gaussian wave

packet under (8) in the spirit of [56].

Finally, we emphasize that our work does not treat the relativistic case, i.e., the field of moving electromagnetic charges,

which has been examined in detail in textbooks [57, Chap. 8].

Acknowledgements

I.C.C. was partially supported by the LANL/LDRD Program through a Feynman Distinguished Fellowship. I.C.C. thanks an

anonymous reviewer for incisive comments, which significantly improved the presentation, and for providing key references.

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Appendix. Formulation using characteristic coordinates

Consider again the governing PDE (11):

∂2U

∂τ 2− 2β(τ1)

∂2U

∂η∂τ−

[

1− β2(τ1)]∂2U

∂η2− ǫδα′′(τ1)

∂U

∂η= 0, 0 < η <∞, 0 < τ <∞. (A.1)

In general, we can impose “initial” conditions of the form

U(0, τ) = U(τ),∂U

∂η(0, τ) = 0, (A.2)

where U(·) is some given excitation, and η plays the role of the “time-like” variable.

Without the simplifying assumption of a harmonic excitation, implementing the multiple-scale expansion proceeds by

introducing characteristic coordinates for the slow scales only (see, e.g., [18, §3.9] and also [58] for illuminating applications of

this technique):

θ1 = η + (β + 1)τ, θ2 = η + (β − 1)τ, τ1 = ǫτ, η1 = ǫη. (A.3)

As before, replacing derivates using the chain rule and letting U(η, τ) = U(θ1, θ2, η1, τ1) transforms (A.1) is into

− 4∂2U

∂θ1∂θ2= −2ǫ

∂

∂τ1

(

∂U

∂θ1−∂U

∂θ2

)

+ 2ǫ∂

∂η1

[

(1 + β)∂U

∂θ1+ (1− β)

∂U

∂θ2

]

+ ǫδα′′(

∂U

∂θ1+∂U

∂θ2

)

+O(ǫ2). (A.4)

The boundary condition is specified at η = 0, which corresponds to θ1 = (β + 1)τ , θ2 = (β − 1)τ and η1 = 0. Therefore, (A.2)

becomes

U(θ1, θ2, 0, τ1) = U

(

θ1

β + 1

)

= U

(

θ2

β − 1

)

,

(

∂

∂θ1+

∂

∂θ2+ ǫ

∂

∂η1+ · · ·

)

U(θ1, θ2, 0, τ1) = 0. (A.5)

We proceed by a regular expansion of the independent variable

U(θ1, θ2, η1, τ1) = U0(θ1, θ2, η1, τ1) + ǫU1(θ1, θ2, η1, τ1) + · · · . (A.6)

At the leading order, we obtain the following “initial-value” problem:

∂2U0∂θ1∂θ2

= 0, (A.7a)

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U0(θ1, θ2, 0, τ1) = U

(

θ1

β + 1

)

,

(

∂

∂θ1+

∂

∂θ2

)

U0(θ1, θ2, 0, τ1) = 0. (A.7b)

The general d’Alembert-type solution to (A.7) is

U0 = G0,1(θ1, η1, τ1) + G0,2(θ2, η1, τ1), (A.8)

where G0,1 and G0,2 are to be determined.

The O(ǫ) problem is

− 4∂2U1∂θ1∂θ2

= −2∂

∂τ1

(

∂U0∂θ1−∂U0∂θ2

)

+ 2∂

∂η1

[

(1 + β)∂U0∂θ1+ (1− β)

∂U0∂θ2

]

+ δα′′(

∂U0∂θ1+∂U0∂θ2

)

. (A.9)

Substituting the O(1) solution from (A.8) into (A.9) and solving, we find that

U1 = G1,1(θ1, η1, τ1) + G1,2(θ2, η1, τ1)

−1

4θ2

(

−2∂

∂τ1+ 2(1 + β)

∂

∂η1+ δα′′

)

G0,1 −1

4θ1

(

2∂

∂τ1+ 2(1− β)

∂

∂η1+ δα′′

)

G0,2, (A.10)

where G1,1,2(θ1,2, η1, τ1) satisfy the homogeneous PDE for U1. To suppress secular terms, we must require that

−∂G0,1∂η1

+1

1 + β(τ1)

∂G0,1∂τ1

= −δα′′(τ1)

2[1 + β(τ1)]G0,1, (A.11a)

∂G0,2∂η1

+1

1− β(τ1)

∂G0,2∂τ1

= −δα′′(τ1)

2[1− β(τ1)]G0,2. (A.11b)

This is a pair of uncoupled scalar hyperbolic PDEs. We proceed by the method of characteristics:

dη1ds1,2

= ∓1,dτ1ds1,2

=1

1± β(τ1),

dG0,1,2ds1,2

= −δα′′(τ1)

2[1± β(τ1)]G0,1,2. (A.12)

Recalling that β(t1) ≡ β + δα′(t1), the first two equations in (A.12) give

s1,2(η1, τ1) =

∫ τ1

0

1± β(t) dt∓ η1 = (1± β)τ1 ± δα(τ1)∓ η1. (A.13)

Integrating the third ODE in (A.12), we obtain

∫ s1,2(η1,τ1)

s1,2(0,τ1)

dG0,1,2G0,1,2

= −

∫ s1,2(η1,τ1)

s1,2(0,τ1)

δα′′(τ1(ς))

2[1± β(τ1(ς))]dς = −

δ

2

∫ s1,2(η1,τ1)

s1,2(0,τ1)

α′′(τ1) dτ1, (A.14)

where we used the second equation in (A.12) to change variables in the integral and the lower limit was chosen because the

initial condition (A.5) is specified at η1 = 0. Finally, performing the integration in (A.14) and eliminating s1,2 using (A.13), we

arrive at

G0,1,2(θ1,2, η1, τ1) = G0,1,2(

θ1,2, τ1)

exp

−δ

2

[

α′(

(1± β)τ1 ± δα(τ1)∓ η1)

− α′(

(1± β)τ1 ± δα(τ1))]

, (A.15)

where G0,1,2(

θ1,2, τ1)

are uniquely determined by the “initial” condition at η1 = 0, namely (A.7b). The amplitude modulation is

identical to that in (29), while the frequency modulation (and classical Doppler shift) are already “built into” the characteristic

variables θ1,2.

14 Copyright c© 2016 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2016, 00 1–14

Prepared using mmaauth.cls