Exercise: Use the method of images to construct i) the Dirichlet, and ii) the

Neumann, Green function for the region „¦, consisting of everything to the

right of the screen. Use your Green functions to write the solution to the

di¬raction problem in this region a) in terms of the values of ψ on the aperture

surface AB, b) in terms of the values of (n · )ψ on the aperture surface.

In each case, assume that the boundary data are identically zero on the

dark side of the screen. Your expressions should coincide with the Rayleigh-

Sommerfeld di¬raction integrals of the ¬rst and second kind, respectively2 .

Explore the di¬erences between the predictions of these two formul¦ and

that of Kirchho¬ for case of the di¬raction of a plane wave incident on the

aperture from the left.

2

M. Born and E. Wolf Principles of Optics 7th (expanded) edition, section 8.11.

Chapter 7

The Mathematics of Real

Waves

Waves are found everywhere in the physical world, but we often need more

than the simple wave equation to understand them. The principal compli-

cations are non-linearity and dispersion. In this chapter we will digress a

little from our monotonous catalogue of linear problems, and describe the

mathematics lying behind some commonly observed, but still fascinating,

phenomena.

7.1 Dispersive waves

In this section we will investigate the e¬ects of dispersion, the dependence

of the speed of propagation on the frequency of the wave. We will see that

dispersion has a profound e¬ect on the behaviour of a wave-packet.

7.1.1 Ocean Waves

The most commonly seen dispersive waves are those on the surface of water.

Although often used to illustrate wave motion in class demonstrations, these

waves are not as simple as they seem.

In chapter one we derived the equations governing the motion of water

with a free surface. Now we will solve these equations. Recall that we

described the ¬‚ow by introducing a velocity potential φ such that, v = φ,

and a variable h(x, t) which is the depth of the water at abscissa x.

183

184 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

y P

0

g

h(x,t)

ρ0

x

Water with a free surface.

Again looking back to chapter one, we see that the ¬‚uid motion is determined

by imposing

2

φ=0 (7.1)

everywhere in the bulk of the ¬‚uid, together with boundary conditions

‚φ

= 0, on y = 0, (7.2)

‚y

‚φ 1

+ ( φ)2 + gy = 0, on the free surface y = h, (7.3)

‚t 2

‚h ‚φ ‚h ‚φ

’ + = 0, on the free surface y = h. (7.4)

‚t ‚y ‚x ‚x

Recall the physical interpretation of these equations: The vanishing of the

Laplacian of the velocity potential simply means that the bulk ¬‚ow is incom-

pressible

· v ≡ 2 φ = 0. (7.5)

The ¬rst two of the boundary conditions are also easy to interpret: The ¬rst

says that no water escapes through the lower boundary at y = 0. The second,

a form of Bernoulli™s equation, asserts that the free surface is everywhere at

constant (atmospheric) pressure. The remaining boundary condition is more

obscure. It states that a ¬‚uid particle initially on the surface stays on the

surface. Remember that we set f (x, y, t) = h(x, t) ’ y, so the water surface

is given by f (x, y, t) = 0. If the surface particles are carried with the ¬‚ow

then the convective derivative of f ,

df ‚f

≡ + (v · )f, (7.6)

dt ‚t

7.1. DISPERSIVE WAVES 185

should vanish on the free surface. Using v = φ and the de¬nition of f , this

reduces to

‚h ‚φ ‚h ‚φ

’

+ = 0, (7.7)

‚t ‚x ‚x ‚y

which is indeed the last boundary condition.

Using our knowledge of solutions of Laplace™s equation, we can immedi-

ately write down a wave-like solution satisfying the boundary condition at

y=0

φ(x, y, t) = a cosh(ky) cos(kx ’ ωt). (7.8)

The tricky part is satisfying the remaining two boundary conditions. The

di¬culty is that they are non-linear, and so couple modes with di¬erent

wave-numbers. We will get around the di¬culty by restricting ourselves to

small amplitude waves, for which the boundary conditions can be linearized.

Suppressing all terms that contain a product of two or more small quantities,

we are left with

‚φ

+ gh = 0, (7.9)

‚t

‚h ‚φ

’ = 0. (7.10)

‚t ‚y

Because of the linearization, these equations should be applied at y = h0 ,

the equilibrium surface of the ¬‚uid. It is convenient to eliminate h to get

‚2φ ‚φ

+g = 0, on y = h0 . (7.11)

‚t2 ‚y

Enforcing this condition on φ leads to the dispersion equation

ω 2 = gk tanh kh0 , (7.12)

relating the frequency to the wave-number.

Two limiting cases are of interest:

i) Long waves on shallow water: Here kh0 1, and, in this limit,

ω = k gh0 .

√

ii) Waves on deep water: Here, kh0 1, leading to ω = gk.

186 CHAPTER 7. THE MATHEMATICS OF REAL WAVES

For deep water, the velocity potential becomes

φ(x, y, t) = aek(y’h0 ) cos(kx ’ ωt). (7.13)

We see that the disturbance due to the surface wave dies away exponentially,

and becomes very small only a few wavelengths below the surface.

Remember that the velocity of the fuid is v = φ. To follow the motion

of individual particles of ¬‚uid we must solve the equations

dx

= vx = ’akek(y’h0 ) sin(kx ’ ωt),

dt

dy

= vy = akek(y’h0 ) cos(kx ’ ωt). (7.14)

dt