Plane waves

As suggested by (3.2.9), when looking for solutions of (3.1.1) on an infinite domain, it is appropriate to seek travelling waves of the form

w(x, t) = A expi(kx− ωt), (3.2.20) rather than seeking a discrete set of normal modes. The real constants |A|, k and ω represent the amplitude, wave-number and frequency, respectively, of the wave (3.2.20). The wave-number is related to the wave-length λ by k = 2π/λ, so large values of k correspond to short waves and vice versa. We can alternatively write (3.2.20) in the form

w(x, t) = A expik(x− ct), where c = ω

k; (3.2.21)

the quantity c is known as the phase velocity. From (3.2.21) we can easily see that c represents the speed at which wave crests propagate. We can also see that the minus sign included by convention in (3.2.20) ensures that the wave propagates in the positive x-direction when ω and k are both positive. We can generalise this approach to the two-dimensional wave equation (3.1.2) by looking for a plane wave solution of the form

w(x, y, t) = A expi(k1x + k2y− ωt) 

= A exp 

ik· x − ωt, (3.2.22) where we define the wave-vector k = (k1, k2)T. By writing (3.2.22) in the form

w = A exp 

i|k|X − ωt, where X = k· x

|k| , (3.2.23)

we observe that it represents a harmonic wave travelling in the direction of k at speed ω/|k|. The phase velocity, at which the wave crests propagate, is thus given by

c = ωk

|k|2. (3.2.24)

When we substitute (3.2.20) into (3.1.1), we find that the amplitude can be nonzero only if

ω2 = k2T /. (3.2.25a)

This dispersion relation tells us how the frequency of any given wave depends on its wave-length. We can also view (3.2.25a) as an equation for the phase velocity, namely

3.2 Normal modes and plane waves 111

which tells us that all waves move at the same speed, irrespective of their wave-length. Waves with this property are called non-dispersive, to distin- guish them from dispersive waves in which c varies with k.

Similarly, substitution of (3.2.22) into the two-dimensional wave equation (3.1.2) leads to the dispersion relation

ω2=|k|2T /ς. (3.2.26)

The phase speed

|c| = c =T /ς (3.2.27)

is still independent of the wave-vector k, so these two-dimensional waves are also non-dispersive.

As in (3.2.9), the general solution of the wave equation on an infinite do- main may be written as a linear superposition of harmonic waves, travelling in both directions, in the form of a Fourier integral. In other words, we can write w(x, y, t) =  _∞ −∞  _∞ −∞A(k1, k2) exp  i (k1x + k2y− ω(k1, k2)t) + B(k1, k2) exp  i (k1x + k2y + ω(k1, k2)t) dk1dk2, (3.2.28a) or w(x, t) =  R2 A(k) exp  i (k· x − ω(k)t) + B(k) exp  i (k· x + ω(k)t) dk, (3.2.28b) where ω(k) is given by the dispersion relation (3.2.26) and the amplitude functions A(k) and B(k) can be found from the Fourier transform of the initial conditions. By writing the displacement in the form (3.2.28b), we indicate that this and all the other results in this section apply also to the wave equation in three space dimensions.

3.2.3 Scattering

Scattering refers generally to the problem of irradiating a target with a known incoming wave-field and trying to determine the resultant scattered field that is the result of the presence of the target. This idea is important, for example, in tomography, seismology and ultrasonic testing, where we try to infer, non-invasively, the properties of some inhomogeneities in a bulk elastic medium by measuring the scattered wave-fields that they produce.

The basic idea is illustrated by the problem of an elastic string, modelled by (3.1.1), stretched along the x-axis with a point mass m attached at the origin. The equation of motion for the mass leads to the boundary conditions

w(0−, t) = w(0+, t), ∂w ∂x(0+, t)− ∂w ∂x(0−, t) = m T ∂2w ∂t2 (3.2.29) at x = 0, where T is the tension in the string. Now suppose we send in a known incident wave of the form w = ei(kx−ωt) from x = −∞, where we assume that ω > 0 and k > 0 so that the wave is travelling in the positive x-direction. We can work in the frequency domain to write the resulting displacement field in the form w(x, t) = A(x)e−iωt, where

A(x) = 

eikx+ cRe−ikx x < 0,

cTeikx x > 0. (3.2.30)

Thus, apart from the prescribed incoming wave, the scattered wave-field is outgoing from the target, with reflection and transmission coefficients cRand cT that can be determined from the boundary conditions (3.2.29), as shown in Exercise 3.1. As a very simple example of a tomography problem we could, for instance, work out the size of the mass by measuring the amplitude|cR| of the reflected waves.

The extension to higher-dimensional problems present us with a serious mathematical challenge. Indeed, even the solution of Helmholtz’ equation (3.2.11) to model plane wave incidence at a scatterer on which A = 0, say, is beyond the scope of this book. The two principal difficulties are the following.

(i) Assuming an incident wave eikx, as in (3.2.30), we have to solve the Helmholtz equation for the scattered wave ˜A = A − eikx, subject to

˜

A = eikx on the scatterer. This means that there is no simple solution by separation of variables, even when the scatterer is a circle.

(ii) In the far field, it is not good enough simply to say that ˜A → 0 as we did in many of the elastostatic problems of Chapter 2. We now need to capture the physical requirement that ˜A be outgoing from the scatterer, again as in (3.2.30). It can be shown (see Exercise 3.3 and Bleistein, 1984, Section 6.4) that, as the radial coordinate r tends to infinity, all possible solutions of (3.2.11) take the form ˜A→ ˜A_±(θ)r−1/2e±ikrand, since ˜A must be multiplied by e−iωt to find the scattered wave, only the positive sign is appropriate to describe outward-travelling waves. This is equivalent to saying that ˜A satisfies the Sommerfeld radiation condition

r1/2  ∂ ˜A ∂r − ik ˜A  → 0 as r→ ∞. (3.2.31)

3.2 Normal modes and plane waves 113

Then the canonical tomography problem for (3.2.11) is to reconstruct the shape of the scatterer given the directivity function ˜A+(θ).

We note that, in the absence of any incident field, a circular boundary can radiate waves in which, from (3.2.16), A depends only on r and is a linear combination of J0(kr) and Y0(kr). Using the fact that

J0(kr)± i Y0(kr)→ )

2 πkre

±i(kr−(π/4)) _{as r → ∞, (3.2.32)} we see that, to describe outward-propagating waves, A must tend to a mul- tiple of the Hankel function H₀(1)(kr) = J0(kr) + i Y0(kr) as r→ ∞.