Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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356 9 Boundary-Value Problems
9.6.4 Retarded Green Function
If one performs a Fourier analysis of both the time and spatial dependencies for
Ρ
r,t
3
k
2Π
3
Ω
2Π
k,
rΩt
˜
Ρ
Ω
k (9.213)
Ψ
r,t
3
k
2Π
3
Ω
2Π
k,
rΩt
˜
Ψ
Ω
k (9.214)
the inhomogeneous wave equation takes the form
k
2
Ω
c
2
˜
Ψ
Ω
k
˜
Ρ
Ω
k
˜
Ψ
Ω
k
˜
Ρ
Ω
k
k
2
Ω
c
2
(9.215)
where, in this approach, k is not restricted to Ω/c. The spatial wave function is obtained
by inverting the Fourier transform
Ψ
r,tc
2
3
k
2Π
3
Ω
2Π
k,
rΩt
˜
Ρ
Ω
k
Ω
2
k
2
c
2
(9.216)
but care must be exercised in handling the singularity. It is simplest to consider the Green
function for a point source in both space and time, which satisfies an equation of the form
1
c
2
(
2
(t
2
+
2
G
r,tb
r
'
,t
'
4Π∆
r
r
'
∆
t t
'
(9.217)
with boundary conditions to be specified later. Thus, using
Ρ
r,t4Π∆
r
r
'
∆t t
'
˜
Ρ
Ω
k4Π Exp
k ,
r
'
Ωt
'
(9.218)
we find that
G
r,tb
r
'
,t
'
4Πc
2
3
k
2Π
3
Ω
2Π
Exp
k ,
r
r
'
Ωt t
'
Ω
2
k
2
c
2
(9.219)
represents a wave produced by a point source at position
r
'
at time t
'
.
The angular part of the integral can be performed easily using
Exp
k ,
R4Π
j
kRY
ˆ
k,Y
ˆ
R (9.220)
where
R
r
r
'
and R
R, such that
G
r,tb
r
'
,t
'
2c
2
Π
0
kk
2
j
0
kR
Ω
2Π
ExpΩt t
'
Ω
2
k
2
c
2
(9.221)
is limited by spherical symmetry to 0. The integrand exhibits poles on the real axis
of the complex frequency plane at Ωkc. The integral does not have a unique value
9.6 Inhomogeneous Wave Equation 357
Re Z
Im Z
kc kc
tt
Re Z
Im Z
kc kc
t!t
Figure 9.6. Contours for Retarded Green Function.
because the poles lie on the integration path, but we can appeal to causality to determine
boundary conditions that specify the appropriate contour. Thus, we require G to vanish for
t<t
'
such that causes precede effects. This is accomplished by employing a contour that
is slightly above the real axis, such that Ω!ΩΕwhere Ε is a positive infinitesimal, and
closing in the contour in the upper half-plane for t<t
'
or the lower half-plane for t>t
'
,
as shown in Fig. 9.6. Alternatively, one could imagine shifting the poles slightly below the
real axis, such that Ω
kc Εand taking the limit Ε!0
after evaluating the integral.
Either way we obtain
Ω
2Π
ExpΩt t
'
Ω Ε
2
k
2
c
2
Sinkct t
'
kc
<t t
'
(9.222)
where < is the unit step function.
The retarded Green function then takes the form
G
r,tb
r
'
,t
'
2c
Π
<t t
'
R
0
k SinkR Sinkct t
'
c
Π
<t t
'
R
0
k
Cos
k
R ct t
'
Cos
k
R ct t
'
c
2Π
<t t
'
R
k
ExpkR ct t
'
.
Exp
k
R ct t
'
(9.223)
Therefore, we finally obtain a retarded Green function
G
r,tb
r
'
,t
'
c
∆
r
r
'
ct t
'
<t t
'
r
r
'
(9.224)
describing a spherical delta-function shell propagating outwards with velocity c. The con-
tribution that propagates inward for earlier times is suppressed by the step function. How-
ever, for some problems it might be appropriate to consider either the advanced Green
function, G
, or a standing wave obtained using different boundary conditions.
The wave function for a variable source can now be computed using
Ψ
r,tc
∆
r
r
'
ct t
'
<t t
'
r
r
'
Ρ
r
'
,t
'
V
'
t
'
(9.225)
358 9 Boundary-Value Problems
such that
Ψ
r,t
Ρ
r
'
,t
r
r
'
c
r
r
'
V
'
(9.226)
resembles the solution to the static Poisson equation except that the potential at position
r
contributed by a source at
r
'
is determined by an earlier, or retarded, time t
r
r
'
/cas
expected for a finite propagation velocity.
9.6.5 Lippmann–Schwinger Equation
The time-independent Schrödinger equation
2
2m
+
2
V
r
Ψ
rEΨ
r (9.227)
for positive energies E
2
k
2
/ 2m can be expressed in a form
+
2
k
2
Ψ
rΝ
rΨ
r (9.228)
that strongly resembles the inhomogeneous Helmholtz equation except that the source term
is proportional to the wave function itself. Here we define Ν2mV /
2
for convenience.
Thus, a formal solution can be expressed in the form of the Lippmann–Schwinger equation
Ψ
rΦ
r
G
r,
r
'
Ν
r
'
Ψ
r
'
3
r
'
(9.229)
where Φ is a solution to the homogeneous equation
+
2
k
2
Φ
r0 (9.230)
and where the Green function satisfies
+
2
k
2
G
r,
r
'
∆
r
r
'
(9.231)
and with outgoing boundary conditions becomes
G
k
r,
r
'
1
4Π
Expk
r
r
'
r
r
'
(9.232)
Note that one must always be aware that the normalization of the Green function depends
upon the form chosen for its point source and tends to vary with the application.
With the unknown wavefunction Ψ appearing on both sides of this integral equation,
one may question whether anything has been accomplished. Nevertheless, the Lippmann–
Schwinger equation has proven to be very valuable in developing approximations and
analyzing the general properties of scattering solutions. For example, if the potential is
sufficiently weak, we can assume that its effect is relatively small and use
Ψ*ΦΨ
r*Φ
r
G
r,
r
'
V
r
'
Φ
r
'
3
r
'
(9.233)
9.6 Inhomogeneous Wave Equation 359
to obtain the Born approximation. This approximation can be improved by substituting the
first-order approximation back into the Lippmann–Schwinger equation to obtain
Ψ
r*Φ
r
3
r
'
G
r,
r
'
V
r
'
Φ
r
'
3
r
'
3
r
''
G
r,
r
'
V
r
'
G
r
'
,
r
''
V
r
''
Φ
r
''
(9.234)
Although it is often difficult to prove that the series converges, iteration of this successive-
approximation procedure provides a multiple-scattering series that is formally exact. The
notation for this series can be simplified considerably by using an operator representation
of the Lippmann–Schwinger equation
ΨΦGΝΨ (9.235)
Next we define the transition operator T by the operator equation
TΦΝΨ (9.236)
and multiply both sides of the Lippmann–Schwinger equation by Ν, such that
ΝΨ ΝΦ ΝGΝΨ TΦΝΝGT Ψ (9.237)
and conclude that the transition operator satisfies
T ΝΝGT T ΝΝGΝΝGΝGΝ (9.238)
For elastic scattering by a short-ranged potential, one naturally chooses a plane wave for Φ
and uses the outgoing Green function such that
Ψ
rExp
k ,
r
m
2Π
2
Expk
r
r
'
r
r
'
V
r
'
Ψ
r
'
3
r
'
(9.239)
For distances much larger than the size of the source, we can approximate the phase of the
Green function using
r D r
'
r
r
'
*r1
ˆr ,
r
'
r
Expk
r
r
'
r
r
'
*
Expkr
r
Exp
k
'
,
r
'
(9.240)
where
k
'
kˆr is the wave vector for the scattered wave. Note that we must treat the phase
carefully because it varies rapidly over the size of the source, but need not be overly con-
cerned with the denominator because the magnitude of the Green function varies slowly
when r D r
'
. Thus, the asymptotic wavefunction takes the form
r D r
'
Ψ
r Exp
k ,
r f
k
'
,
k
Expkr
r
(9.241)
where the scattering amplitude is identified as
f
k
'
,
k
m
2Π
2
Exp
k
'
,
r
'
V
r
'
Ψ
r
'
3
r
'
(9.242)
360 9 Boundary-Value Problems
Although this result is exact, it is not especially useful without a suitable approximation
for Ψ. Provided that the potential is short-ranged and sufficiently weak, we can use the
Born approximation to replace Ψ with Φ and obtain
f
q*
m
2Π
2
Exp
q ,
r
'
V
r
'
3
r
'
(9.243)
where
q
k
k
'
is the momentum transfer to the target.
Problems for Chapter 9
1. Between the sheets
Two infinite parallel conducting planes at z 0,Lare held at Ψ0.
a) Using cylindrical coordinates,
r Ξ, Φ,z, demonstrate that the Dirichlet Green func-
tion takes the form
G
r,
r
'
4
L
m
3
4
4
4
4
4
5
ExpmΦ Φ
'
n1
Sink
n
z Sink
n
z
'
I
m
k
n
Ξ
<
K
m
k
n
Ξ
>
6
7
7
7
7
7
8
(9.244)
where k
n
nΠ
L
and where I
m
and K
m
are modified Bessel functions of the first and second
kinds, respectively.
b) Show that this Green function can also be expressed in the form
G
r,
r
'
2
m
ExpmΦ Φ
'
0
kJ
m
kΞJ
m
kΞ
'
Sinhkz
<
SinhkL z
>
SinhkL
(9.245)
c) Use either of these representations to evaluate the electrostatic potential produced by a
point charge on the z axis at height h between grounded planes.
d) Find expressions for the surface charge densities, Σ
0
and Σ
L
, on the conducting planes
and compute the total charge on each.
2. Polar caps
Suppose that a portion of a sphere of radius R contained within Θ < Θ
0
, described as a north
polar cap, is maintained at potential V
0
and that the corresponding south polar cap with
Θ > ΠΘ
0
is held at potential V
0
while the remainder of the sphere is grounded.
a) Develop an expansion for the electrostatic potential within the sphere. The coefficients
may involve Legendre polynomials with fixed argument. Which terms contribute?
b) Write the corresponding expansion for r>R. Describe the asymptotic behavior of the
potential for r D R and find an explicit expression for the effective dipole moment.
Problems for Chapter 9 361
3. Dirichlet Green function for two-dimensional semicircle
Consider a two-dimensional semicircular region defined by (0 Ξa, 0 ΦΠ). Use
the eigenfunctions of
+
2
Λ
2
n,m
Ψ
r0, Ψa, Φ ΨΞ, 0 ΨΞ, Π 0 (9.246)
to expand the two-dimensional Dirichlet Green function for the Poisson equation
+
2
G
r,
r
'
4Π∆
r
r
'
(9.247)
Although numerical values for the eigenvalues are not required, you must provide the
equations that determine them.
4. Piece of pie
Suppose that an electrode is shaped like a piece of pie – a wedge of radius a and opening
angle Α. Evaluate the two-dimensional Dirichlet Green function using separation of vari-
ables in polar coordinates, (Ξ, Φ).
5. Cylinder with grounded endcaps
An empty cylinder of radius a with its axis along the ˆz axis has grounded endcaps (Ψ0)
at z 0 and z L while its curved surface is held at potential Ψa, Φ,zV Φ,z.
a) Develop an expansion for the electrostatic potential ΨΞ, Φ,z within the cylinder and
express the coefficients in terms of the appropriate integral over VΦ,z.
b) Determine the coefficients for the simple case V Φ,zV
0
1 2<ΦΠwhere < is
the unit step function.
6. Cylinder with opposite potentials on its endcaps
The curved surface of a cylinder of radius a is grounded while the endcaps at z L/2
are maintained at opposite potentials ΨΞ, Φ, L/2V Ξ, Φ.
a) Develop an expansion for the electrostatic potential ΨΞ, Φ,z within the cylinder and
express the coefficients in terms of the appropriate integral over VΞ, Φ.
b) Determine the coefficients for the simple case VΞ, ΦV
0
where V
0
is constant.
7. Cylinder with diametrically opposed electrodes
Suppose that a long conducting cylinder of radius a has edge electrodes at potentials ΨV
for Θ Α and ΨV for Θ Π Α. The remaining portions of the cylinder are held at
Ψ0. Evaluate the electrostatic potential in both interior and exterior regions.
8. Split-sphere acoustic antenna
Sound waves produced by a split-sphere antenna with radius R satisfy
1
c
2
(
2
Ψ
(t
2
+
2
Ψ0, ΨR, Θ, Φ,t
L
M
M
N
M
M
O
A
Ωt
0 < Θ <
Π
2
A
Ωt
Π
2
< Θ < Π
(9.248)
Evaluate Ψ
r,t for r>Rwith outgoing boundary conditions.
362 9 Boundary-Value Problems
9. Acoustical waves from vibrating cylinder
The surface of an infinite cylinder of radius R vibrates harmonically such that the pressure
at the surface is given by
ΨR, Θ,t f Θ
Ωt
(9.249)
where the angular function f Θ is prescribed. The pressure for r>Rsatisfies the wave
equation
1
c
2
(
2
(t
2
+
2
Ψ0 (9.250)
with outgoing boundary conditions. Construct a formal solution for the field.
10. Acoustical wave guides
Density displacements Ψ
r,t for a gas in a confined volume satisfy a wave equation of the
form
1
c
2
(
2
Ψ
(t
2
+
2
Ψ0,
(Ψ
(n
0 (9.251)
where c is the sound velocity and (Ψ/ (n ˆn ,
+Ψ is the normal derivative at a boundary
surface.
a) Evaluate the normal modes for a long rectangular wave guide with cross-sectional
dimensions a b. Show that for most modes there is a minimum frequency for trans-
mission. Compare the phase and group velocities.
b) Evaluate the normal modes for a long cylindrical wave guide with cross-sectional radi-
us R. Show that for most modes there is a minimum frequency for transmission. Com-
pare the phase and group velocities.
11. Acoustic modes
Sound waves in a confined volume satisfy a wave equation of the form
1
v
2
(
2
Ψ
(t
2
+
2
Ψ0,
(Ψ
(n
0 (9.252)
where c is the sound velocity and (Ψ/ (n ˆn ,
+Ψ is the normal derivative at a boundary
surface. The normal modes take the form
Ψ
i
r,tΨ
i
r
Ω
i
t
(9.253)
where the eigenfrequencies Ω
i
may be degenerate, requiring additional indices to distin-
guish between degenerate modes.
a) Evaluate the normal modes and eigenfrequencies in a box with dimensions a b c.
b) Evaluate the normal modes and eigenfrequencies for a sphere of radius R.
Problems for Chapter 9 363
c) Evaluate the normal modes and eigenfrequencies for a cylinder of radius R and length L.
12. Organ pipe
The pressure fluctuation Ψ in an organ pipe of length L satisfies the wave equation
1
c
2
(
2
Ψx, t
(t
2
(
2
Ψx, t
(x
2
0 (9.254)
where c is the speed of sound. The end at x 0 is open while the end at x L is partly
closed such that the boundary conditions are
Ψ0,t0 ,
ΨΑ
(Ψ
(x
xL
0 (9.255)
where Α is a constant.
a) Construct the normal modes and derive the equation that determines the characteristic
frequencies. Describe a graphical solution to this equation.
b) Demonstrate that the normal modes are orthogonal.
c) Suppose that the pressure in the pipe is in equilibrium for t<0 and that constant
pressure Ψ
0
is applied to the open end for t 2 0. Find a series representation for Ψx, t.
13. Vibrations of a membrane wedge
Suppose that a taut membrane is stretched across the area described by a Ξb,
Θ
1
ΘΘ
2
. Develop the vibrational eigenfunctions and the condition satisfied by the
eigenfrequencies. Demonstrate that the eigenfunctions are orthogonal and provide a for-
mal expansion for the initial-value problem (specified initial displacement \Ξ, Θ,t 0).
It is not necessary to obtain an explicit formula for the radial normalization integral.
14. Critical mass
Suppose that the neutron density Ψ in a fissionable material satisfies an inhomogeneous
diffusion equation of the form
1
Κ
(Ψ
(t
+
2
ΨΛΨ (9.256)
where Κ is the diffusion constant and Λ depends upon the fission probability per neutron.
For simplicity assume that Ψ0 on the surface of the material (neutrons escape rapidly at
the surface or are absorbed by the surrounding medium).
a) Suppose that the material forms a sphere of radius R. Determine the critical radius R
c
,
beyond which the neutron density is unstable (exponentially increasing) and produces
an explosion.
b) Determine the critical radius R
1
, for a hemisphere of the same material. If two hemi-
spheres that are barely stable are brought together as a sphere, the final configuration
will be unstable. Express the explosive time constant Τ, in terms of Κ and Λ such that
ΨG
t/Τ
.
364 9 Boundary-Value Problems
15. Heating simple solids
Suppose that a simple solid (brick, sphere, cylinder, etc.) with uniform initial temperature
is immersed at time t 0 in a heat bath. The temperature Ψ
r,t within the material
satisfies
1
Κ
(Ψ
(t
+
2
Ψ, Ψ
r, 0Ψ
i
, Ψ
R, t 2 0Ψ
f
(9.257)
where
R is on the surface.
a) Determine the temperature distribution for positive times within a brick with 0 x a,
0 y b, and 0 z c.
b) Determine the temperature distribution for positive times within a sphere with r R.
What is the asymptotic time dependence of the central temperature?
c) Determine the temperature distribution for positive times within a cylinder with ΞR
and 0 z L.
16. Cooling sphere
A solid sphere of radius R has a spherically symmetric initial temperature distribution
Ψ
0
r and cools at its surface according to Newton’s law of cooling
r R ΨΑR
(Ψ
(r
0 (9.258)
If one can neglect the radial variation of the diffusion constant, this problem provides a
simple model for the cooling of a planet.
a) Develop a series representation for Ψr,t and express the coefficients in terms of Ψ
0
.
b) Evaluate the special case where Ψ
0
is constant and Α is negligible. Plot the temperature
dependence for several radii.
17. Diffusion in a cylinder with constant surface temperature
A long solid cylinder of radius a has initial temperature distribution Ψ
0
Ξ, Φ and constant
surface temperature Ψa, Φ0. The temperature satisfies the diffusion equation
1
Κ
(Ψ
(t
+
2
Ψ (9.259)
where the variation with height is ignored. Develop a series expansion for ΨΞ, Φ,t and
express the coefficients in terms of Ψ
0
.
18. Surface waves on ideal fluid
The velocity field
v
+Ψ for the flow of an ideal fluid can be expressed in terms of
a velocity potential Ψ that satisfies Laplace’s equation, +
2
Ψ0, subject to appropriate
boundary conditions. Suppose that a fluid with equilibrium depth h is essentially infinite
in two dimensions, x and y, but has a flat floor. The boundary conditions are then
(Ψ
(z
1
g
(
2
Ψ
(t
2
z0
0,
(Ψ
(z
zh
0 (9.260)
Problems for Chapter 9 365
a) Find a general expression Ψ
Ω
x, y, z,t for a plane wave with frequency Ω that propagates
in the x direction.
b) Determine and sketch the phase and group velocities as functions of kh, where k is the
wave number.
19. Standing waves on ideal fluid in rectangular tank
The velocity field
v
+Ψ for the flow of an ideal fluid can be expressed in terms of
a velocity potential Ψ that satisfies Laplace’s equation, +
2
Ψ0, subject to appropriate
boundary conditions. Suppose that a fluid with equilibrium depth h is in a rectangular tank
with dimensions a b and a flat floor. The boundary conditions are then
(Ψ
(z
1
g
(
2
Ψ
(t
2
z0
0,
(Ψ
(z
zh
0,
(Ψ
(y
x0,a
0,
(Ψ
(x
y0,b
0 (9.261)
a) Construct standing-wave solutions Ψ
n,m
x, y, z,t and determine their frequencies.
b) Construct general solutions that satisfy the initial condition
v
z
x, y, 0, 0v
0
x, y (9.262)
20. Seasonal variation of ground temperature
In the flat-earth approximation, the temperature Ψ at depth z can be described by a one-
dimensional diffusion equation of the form
(
2
Ψ
(z
2
1
Κ
(Ψ
(t
(9.263)
where Κ is the thermal diffusion constant. Suppose that the external temperature can be
approximated by a sinusoidal variation of the form
Ψ0,tΨ
0
Ψ
1
SinΩt (9.264)
where Ω2Π/T for period T.
a) Solve for Ψz, t and determine the penetration depth d and the phase delay for propa-
gation of thermal waves into the ground.
b) The penetration depth for annual variations is approximately 3 m. Determine the phase
delay for annual variations. Also determine the corresponding penetration depth and
phase delay for daily variations. Discuss the separability of these periods.
c) Obtain Ψz, t for combined annual and diurnal variations.
21. Current distribution in wire
One can show that the current density
J satisfies an equation of the form
+
2
ΕΜ
c
2
(
2
(t
2
4ΠΣΜ
c
2
(
(t
J 0 (9.265)
when the permittivity Ε, permeability Μ, and conductivity Σ are constant. The radial distri-
bution of current flowing with sinusoidal time dependence along a long straight wire with
circular cross-section of radius a can be represented as
J
z
Ξ,tRe
ΨΞ ExpΩt
(9.266)