Stone M., Goldbart P. Mathematics for Physics: A Guided Tour for Graduate Students

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6.6 Further exercises and problems 225

(b) Solve the same problem by using a Fourier integral expansion in terms of sin kx on

the half-line 0 < x < ∞and obtaining the time evolution of the Fourier coefficients.

Invert the transform and show that your answer reduces to that of part (a). (Hint:

replace the initial condition by θ(x,0) = θ

−x

, so that the Fourier transform

converges, and then take the limit  → 0 at the end of your calculation.)

Exercise 6.15: Seasonal heat waves. Suppose that the measured temperature of the air

above the arctic permafrost at time t is expressed as a Fourier series

θ(t) = θ

∞



n=1

cos nωt,

where the period T = 2π/ω is one year. Solve the heat equation for the soil temperature,

∂θ

∂t

= κ

∂

∂z

,0< z < ∞

with this boundary condition, and find the temperature θ(z, t) at a depth z below the

surface as a function of time. Observe that the subsurface temperature fluctuates with

the same period as that of the air, but with a phase lag that depends on the depth. Also

observe that the longest-period temperature fluctuations penetrate the deepest into the

ground. (Hint: for each Fourier component, write θ as Re[A

(z) exp inωt], where A

a complex function of z.)

The next problem is an illustration of a Dirichlet principle.

Exercise 6.16: Helmholtz–Hodge decomposition. Given a three-dimensional region 

with smooth boundary ∂, introduce the real Hilbert space L

vec

() of finite-norm vector

fields, with inner product

u, v=





u · v d

Consider the spaces L ={v : v =∇φ} and T ={v : v = curl w} consisting of

vector fields in L

vec

() that can be written as gradients and curls, respectively. (Strictly

speaking, we should consider the completions of these spaces.)

(a) Show that if we demand that either (or both) of φ and the tangential component of

w vanish on ∂, then the two spaces L and T are mutually orthogonal with respect

to the L

vec

() inner product.

Let u ∈ L

vec

(). We will try to express u as the sum of a gradient and a curl by

seeking to make the distance functional

[φ, w]=u −∇φ − curl w

def





|u −∇φ −curl w|

equal to zero.

226 6 Partial differential equations

(b) Show that if we find a w and φ that minimize F

[φ, w], then the residual vector field

def

= u −∇φ − curl w

obeys curl h = 0 and div h = 0, together with boundary conditions determined by

the constraints imposed on φ and w:

(i) If φ is unconstrained on ∂, but the tangential boundary component of w is

required to vanish, then the component of h normal to the boundary must be zero.

(ii) If φ = 0on∂, but the tangential boundary component of w is unconstrained,

then the tangential boundary component of h must be zero.

(iii) If φ = 0on∂ and also the tangential boundary component of w is required

to vanish, then h need satisfy no boundary condition.

of the three boundary conditions of the previous part, we have a Helmholtz–Hodge

decomposition

u =∇φ + curl w + h

into unique parts that are mutually L

vec

() orthogonal. Observe that the residual

vector field h is harmonic – i.e. it satisfies the equation ∇

h = 0, where

∇

def

=∇(div h) − curl (curl h)

is the vector Laplacian acting on h.

If u is sufficiently smooth, there will exist φ and w that minimize the distance

u −∇φ − curl w and satisfy the boundary conditions. Whether or not h is needed in

the decomposition is another matter. It depends both on how we constrain φ and w, and

on the topology of . At issue is whether or not the boundary conditions imposed on h

are sufficient to force it to be zero. If  is the interior of a torus, for example, then h can

be non-zero whenever its tangential component is unconstrained.

The Helmholtz–Hodge decomposition is closely related to the vector-field eigenvalue

problems commonly met with in electromagnetism or elasticity. The next few exercises

lead up to this connection.

Exercise 6.17: Self-adjointness and the vector Laplacian. Consider the vector Laplacian

(defined in the previous problem) as a linear operator on the Hilbert space L

vec

() .

(a) Show that







u · (∇

v) − v · (∇





∂

{

(n · u) div v −(n · v) div u

−u · (n × curl v) + v ·(n × curl u)

}

6.6 Further exercises and problems 227

(b) Deduce from the identity in part (a) that the domain of ∇

coincides with the domain

of (∇

)

†

, and hence the vector Laplacian defines a truly self-adjoint operator with

a complete set of mutually orthogonal eigenfunctions, when we take as boundary

conditions one of the following:

(i) Dirichlet–Dirichlet: n · u = 0 and n × u = 0on∂;

(ii) Dirichlet–Neumann: n · u = 0 and n × curl u = 0on∂;

(iii) Neumann–Dirichlet: div u = 0 and n × u = 0on∂;.

(iv) Neumann–Neumann: div u = 0 and n × curl u = 0on∂:

α(n · u) + β div u = 0,

λ(n × u) + µ(n × curl u) = 0,

where αβ, µνcan be position dependent, also give rise to a truly self-adjoint

operator.

Problem 6.18: Cavity electrodynamics and the Hodge–Weyl decomposition. Each of the

self-adjoint boundary conditions in the previous problem gives rise to a complete set of

mutually orthogonal vector eigenfunctions obeying

−∇

= k

For these eigenfunctions to describe the normal modes of the electric field E and the

magnetic field B (which we identify with H as we will use units in which µ

= 

1) within a cavity bounded by a perfect conductor, we need to additionally impose

the Maxwell equations div B = div E = 0 everywhere within , and to satisfy the

perfect-conductor boundary conditions n × E = n · B = 0.

(a) For each eigenfunction u

corresponding to a non-zero eigenvalue k

, define

curl (curl u

), w

=−

∇(div u

so that u

= v

+ w

. Show that v

and w

are, if non-zero, each eigenfunctions

of −∇

with eigenvalue k

. The vector eigenfunctions that are not in the null-

space of ∇

can therefore be decomposed into their transverse (the v

, which obey

div v

= 0) and longitudinal (the w

, which obey curl w

= 0) parts. However, it is

not immediately clear what boundary conditions the v

and w

separately obey.

(b) The boundary-value problems of relevance to electromagnetism are:



−∇

= k

, within ,

n · h

= 0, n × curl h

= 0, on ∂;

228 6 Partial differential equations

ii)



−∇

= k

, within ,

div e

= 0, n × e

= 0, on ∂;

iii)



−∇

= k

, within ,

div b

= 0, n × curl b

= 0, on ∂.

These problems involve, respectively, the Dirichlet–Neumann, Neumann–Dirichlet

and Neumann–Neumann boundary conditions from the previous problem.

Show that the divergence-free transverse eigenfunctions

def

curl (curl h

), E

def

curl (curl e

), B

def

curl (curl b

)

obey n · H

= n × E

= n × curl B

= 0 on the boundary, and that from these and

the eigenvalue equations we can deduce that n ×curl H

= n ·B

= n ·curl E

= 0

on the boundary. The perfect-conductor boundary conditions are therefore satisfied.

Also show that the corresponding longitudinal eigenfunctions

def

∇(div h

), 

def

∇(div e

), β

def

∇(div b

)

obey the boundary conditions n · η

= n × 

= n × β

= 0.

the Dirichlet–Dirichlet boundary condition is not compatible with a longitudi-

nal+transverse decomposition. (A purely transverse wave incident on such a

boundary will, on reflection, acquire a longitudinal component.)

(d) Show that

0 =





· H

x =







· E

x =





· B

but that the v

and w

obtained from the Dirichlet–Dirichlet boundary condition u

’s

are not in general orthogonal to each other. Use the continuity of the L

vec

() inner

product

→ x ⇒x

, y→x, y

to show that this individual-eigenfunction orthogonality is retained by limits of sums

of the eigenfunctions. Deduce that, for each of the boundary conditions (i)–(iii) (but

not for the Dirichlet–Dirichlet case), we have the Hodge–Weyl decomposition of

vec

() as the orthogonal direct sum

vec

() = L ⊕ T ⊕ N ,

6.6 Further exercises and problems 229

where L, T are respectively the spaces of functions representable as infinite sums

of the longitudinal and transverse eigenfunctions, and N is the finite-dimensional

space of harmonic (null-space) eigenfunctions.

Complete sets of vector eigenfunctions for the interior of a rectangular box, and for each

of the four sets of boundary conditions we have considered, can be found in Morse and

Feshbach §13.1.

Problem 6.19: Hodge–Weyl and Helmholtz–Hodge. In this exercise we consider the

problem of what classes of vector-valued functions can be expanded in terms of the

various families of eigenfunctions of the previous problem. It is tempting (but wrong)

to think that we are restricted to expanding functions that obey the same boundary

conditions as the eigenfunctions themselves. Thus, we might erroniously expect that

the E

are good only for expanding functions whose divergence vanishes and have

vanishing tangential boundary components, or that the η

can expand out only curl-free

vector fields with vanishing normal boundary component. That this supposition can be

false was exposed in section 2.2.3, where we showed that functions that are zero at the

endpoints of an interval can be used to expand out functions that are not zero there.

The key point is that each of our four families of u

constitute a complete orthonormal

set in L

vec

(), and can therefore be used expand any vector field. As a consequence,

the infinite sum

∈ T can, for example, represent any vector-valued function

u ∈ L

vec

() provided only that u possesses no component lying either in the subspace

L of the longitudinal eigenfunctions 

, or in the nullspace N .

(a) Let T =E

 be space of functions representable as infinite sums of the E

. Show

that

E



⊥

={u : curl u = 0 within , n × u = 0on∂}.

Similarly show that





⊥

={u : div u = 0 within , no condition on ∂},

H



⊥

={u : curl u = 0 within , no condition on ∂},

η



⊥

={u : div u = 0 within , n · u = 0on∂},

B



⊥

={u : curl u = 0 within , no condition on ∂}.

β



⊥

={u : div u = 0 within , no condition on ∂},

(b) Use the results of part (a) and the Helmholtz–Hodge decomposition to show that

E

={u ∈ L

vec

() : u = curl w, no condition on w on ∂}.



={u ∈ L

vec

() : u =∇φ, where φ = 0on∂},

H

={u ∈ L

vec

() : u = curl w, where n × w = 0on∂},

η

={u ∈ L

vec

() : u =∇φ, no condition on φ on ∂},

230 6 Partial differential equations

B

={u ∈ L

vec

() : u = curl w, where n × w = 0on∂}.

β

={u ∈ L

vec

() : u =∇φ, where φ = 0on∂}.

vec

() = L ⊕ T ⊕ N

for each of the three vector Laplacian boundary condition families (i), (ii) and (iii)

coincides the Helmholtz–Hodge decompositions under the conditions (i), (ii) and

(iii) in problem 6. 6.16

(d) As an illustration of the practical distinctions between the decompositions in part

(c), take  to be the unit cube in R

, and u = (1, 0, 0) a constant vector field. Show

that with conditions (i) we have u ∈ L, but for (ii) we have u ∈ T , and for (iii) we

have u ∈ N .

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 complications are nonlinearity

and dispersion. In this chapter we will describe the mathematics lying behind some

commonly observed, but still fascinating, phenomena.

7.1 Dispersive waves

In this section we will investigate the effects of dispersion, the dependence of the speed

of propagation on the frequency of the wave. We will see that dispersion has a profound

effect on the behaviour of a wavepacket.

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 1 we derived the equations governing the motion of water with a free

surface. Now we will solve these equations. Recall that we described the flow by intro-

ducing a velocity potential φ such that v =∇φ, and a variable h(x, t) which is the depth

of the water at abscissa x (see Figure 7.1).

h(x,t)



Figure 7.1 Water with a free surface.

231

232 7 The mathematics of real waves

Again looking back to Chapter 1, we see that the fluid motion is determined by

imposing

∇

φ = 0 (7.1)

everywhere in the bulk of the fluid, together with boundary conditions

∂φ

∂y

= 0, on y = 0, (7.2)

∂φ

∂t

(∇φ)

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

∂h

∂t

−

∂φ

∂y

∂h

∂x

∂φ

∂x

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

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

the velocity potential simply means that the bulk flow is incompressible

div v =∇

φ = 0. (7.5)

The first two of the boundary conditions are also easy to interpret: the first 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 fluid 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 flow

then the convective derivative of f ,

def

∂f

∂t

+ (v ·∇)f , (7.6)

should vanish on the free surface. Using v =∇φ and the definition of f , this reduces to

∂h

∂t

∂φ

∂x

∂h

∂x

−

∂φ

∂y

= 0, (7.7)

which is indeed the last boundary condition.

Using our knowledge of solutions of Laplace’s equation, we can immediately 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 difficulty is that

they are nonlinear, and so couple modes with different wavenumbers. We will circumvent

the difficulty by restricting ourselves to small-amplitude waves, for which the boundary

7.1 Dispersive waves 233

conditions can be linearized. Suppressing all terms that contain a product of two or more

small quantities, we are left with

∂φ

∂t

+ gh = 0, (7.9)

∂h

∂t

−

∂φ

∂y

= 0. (7.10)

Because φ is already a small quantity, and the wave amplitude is a small quantity,

linearization requires that these equations should be imposed at the equilibrium surface

of the fluid y = h

. It is convenient to eliminate h to get

∂

∂t

+ g

∂φ

∂y

= 0, on y = h

. (7.11)

Inserting (7.8) into this boundary condition leads to the dispersion equation

= gk tanh kh

, (7.12)

relating the frequency to the wavenumber.

Two limiting cases are of particular interest:

(i) Long waves on shallow water: Here kh

1, and, in this limit,

ω = k



(ii) Waves on deep water: Here, kh

 1, leading to ω =



gk.

For deep water, the velocity potential becomes

φ(x, y, t) = ae

k(y−h

)

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 fluid is v =∇φ. To follow the motion of individual

particles of fluid we must solve the equations

= v

=−ake

k(y−h

)

sin(kx − ωt),

= v

= ake

k(y−h

)

cos(kx − ωt). (7.14)

This is a system of coupled nonlinear differential equations, but to find the small-

amplitude motion of particles at the surface we may, to a first approximation, set

234 7 The mathematics of real waves

Figure 7.2 Circular orbits in deep water surface waves.

x = x

, y = h

on the right-hand side. The orbits of the surface particles are therefore

approximately

x(t) = x

−

cos(kx

− ωt),

y (t) = y

−

sin(kx

− ωt). (7.15)

For right-moving waves, the particle orbits are clockwise circles. At the wave crest the

particles move in the direction of the wave propagation; in the troughs they move in the

opposite direction. Figure 7.2 shows that this motion results in an up-down-asymmetric

cycloidal wave profile.

When the effect of the bottom becomes significant, the circular orbits deform into

ellipses. For shallow water waves, the motion is principally back-and-forth with motion

in the y-direction almost negligible.

7.1.2 Group velocity

The most important effect of dispersion is that the group velocity of the waves – the

speed at which a wavepacket travels – differs from the phase velocity – the speed at

which individual wave crests move. The group velocity is also the speed at which the

energy associated with the waves travels.

Suppose that we have waves with dispersion equation ω = ω(k). A right-going

wavepacket of finite extent (Figure 7.3), and with initial profile ϕ(x), can be Fourier

analysed to give

ϕ(x) =



∞

−∞

2π

A(k)e

ikx

. (7.16)

At later times this will evolve to

ϕ(x, t) =



∞

−∞

2π

A(k)e

ikx−iω(k)t

. (7.17)