Exercises

21.6 EXERCISES
Using plane polar coordinates, the radial (i.e. normal) derivative of this function is given
by
∂G(r, r0 )
r
=
· ∇G(r, r0 )
∂ρ
|r|
q(r − r1 )
r
r − r0
.
+
=
·
2π|r|
|r − r0 |2
|r − r1 |2
Using (21.107), on the perimeter of the circle ρ = a the radial derivative takes the form
2
∂G(r, r0 ) |r| − r · r0
1
q|r|2 − q(a2 /|r0 |2 )r · r0
=
+
2
2
2
2
∂ρ
2π|r|
|r − r0 |
(a /|r0 | )|r − r0 |
ρ=a
2
1
1
2
=
|r| + q|r0 | − (1 + q)r · r0 ,
2πa |r − r0 |2
where we have set |r|2 = a2 in the second term on the RHS, but not in the first. If we take
q = 1, the radial derivative simplifies to
1
∂G(r, r0 ) ,
=
∂ρ ρ=a
2πa
or 1/L, where L is the circumference, and so (21.108) with q = 1 is the required Neumann
Green’s function.
Since ρ(r) = 0, the solution to our boundary-value problem is now given by (21.106) as
u(r0 ) = u(r)C −
G(r, r0 )f(r) dl(r),
C
where the integral is around the circumference of the circle C. In plane polar coordinates
r = ρ cos φ i + ρ sin φ j and r0 = ρ0 cos φ0 i + ρ0 sin φ0 j, and again using (21.107) we find
that on C the Green’s function is given by
1
a
ln |r − r0 | + ln
G(r, r0 )|ρ=a =
|r − r0 | + c
2π
|r0 |
a
1
2
ln |r − r0 | + ln
+c
=
2π
|r0 |
2
a
1
ln a + ρ20 − 2aρ0 cos(φ − φ0 ) + ln
+c .
(21.109)
=
2π
ρ0
Since dl = a dφ on C, the solution to the problem is given by
2π
a
u(ρ0 , φ0 ) = uC −
f(φ) ln[a2 + ρ20 − 2aρ0 cos(φ − φ0 )] dφ.
2π 0
The contributions of the final two terms terms in the Green’s function (21.109) vanish
2π
because 0 f(φ) dφ = 0. The average value of u around the circumference, uC , is a freely
specifiable constant as we would expect for a Neumann problem. This result should be
compared with the result (21.104) for the corresponding Dirichlet problem, but it should
be remembered that in the one case f(φ) is a potential, and in the other the gradient of a
potential. 21.6 Exercises
21.1
Solve the following first-order partial differential equations by separating the
variables:
∂u
∂u
∂u
∂u
(a)
−x
= 0;
(b) x
− 2y
= 0.
∂x
∂y
∂x
∂y
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PDES: SEPARATION OF VARIABLES AND OTHER METHODS
21.2
21.3
A cube, made of material whose conductivity is k, has as its six faces the planes
x = ±a, y = ±a and z = ±a, and contains no internal heat sources. Verify that
the temperature distribution
πx
2κπ 2 t
πz
u(x, y, z, t) = A cos
sin
exp − 2
a
a
a
obeys the appropriate diffusion equation. Across which faces is there heat flow?
What is the direction and rate of heat flow at the point (3a/4, a/4, a) at time
t = a2 /(κπ 2 )?
The wave equation describing the transverse vibrations of a stretched membrane
under tension T and having a uniform surface density ρ is
2
∂2 u
∂2 u
∂ u
=ρ 2.
+
T
∂x2
∂y 2
∂t
Find a separable solution appropriate to a membrane stretched on a frame of
length a and width b, showing that the natural angular frequencies of such a
membrane are given by
π 2 T n2
m2
ω2 =
+
,
ρ
a2
b2
21.4
where n and m are any positive integers.
Schrödinger’s equation for a non-relativistic particle in a constant potential region
can be taken as
∂u
2 ∂ 2 u
∂2 u
∂2 u
= i .
+
+
−
2m ∂x2
∂y 2
∂z 2
∂t
(a) Find a solution, separable in the four independent variables, that can be
written in the form of a plane wave,
ψ(x, y, z, t) = A exp[i(k · r − ωt)].
Using the relationships associated with de Broglie (p = k) and Einstein
(E = ω), show that the separation constants must be such that
p2x + p2y + p2z = 2mE.
(b) Obtain a different separable solution describing a particle confined to a box
of side a (ψ must vanish at the walls of the box). Show that the energy of
the particle can only take the quantised values
2 π 2 2
(n + n2y + n2z ),
2ma2 x
where nx , ny and nz are integers.
E=
21.5
Denoting the three terms of ∇2 in spherical polars by ∇2r , ∇2θ , ∇2φ in an obvious
way, evaluate ∇2r u, etc. for the two functions given below and verify that, in each
case, although the individual terms are not necessarily zero their sum ∇2 u is zero.
Identify the corresponding values of and m.
B 3 cos2 θ − 1
(a) u(r, θ, φ) = Ar2 + 3
.
r
2
B
(b) u(r, θ, φ) = Ar + 2 sin θ exp iφ.
r
21.6
Prove that the expression given in equation (21.47) for the associated Legendre
function Pm (µ) satisfies the appropriate equation, (21.45), as follows.
768
21.6 EXERCISES
(a) Evaluate dPm (µ)/dµ and d2 Pm (µ)/dµ2 , using the forms given in (21.47), and
substitute them into (21.45).
(b) Differentiate Legendre’s equation m times using Leibnitz’ theorem.
(c) Show that the equations obtained in (a) and (b) are multiples of each other,
and hence that the validity of (b) implies that of (a).
21.7
Continue the analysis of exercise 10.20, concerned with the flow of a very viscous
fluid past a sphere, to find the full expression for the stream function ψ(r, θ). At
the surface of the sphere r = a, the velocity field u = 0, whilst far from the sphere
ψ (Ur2 sin2 θ)/2.
Show that f(r) can be expressed as a superposition of powers of r, and
determine which powers give acceptable solutions. Hence show that
U
a3
ψ(r, θ) =
2r2 − 3ar +
sin2 θ.
4
r
21.8
The motion of a very viscous fluid in the two-dimensional (wedge) region −α <
φ < α can be described, in (ρ, φ) coordinates, by the (biharmonic) equation
∇2 ∇2 ψ ≡ ∇4 ψ = 0,
together with the boundary conditions ∂ψ/∂φ = 0 at φ = ±α, which represent
the fact that there is no radial fluid velocity close to either of the bounding walls
because of the viscosity, and ∂ψ/∂ρ = ±ρ at φ = ±α, which impose the condition
that azimuthal flow increases linearly with r along any radial line. Assuming a
solution in separated-variable form, show that the full expression for ψ is
ψ(ρ, φ) =
21.9
21.10
ρ2 sin 2φ − 2φ cos 2α
.
2 sin 2α − 2α cos 2α
A circular disc of radius a is heated in such a way that its perimeter ρ = a has
a steady temperature distribution A + B cos2 φ, where ρ and φ are plane polar
coordinates and A and B are constants. Find the temperature T (ρ, φ) everywhere
in the region ρ < a.
Consider possible solutions of Laplace’s equation inside a circular domain as
follows.
(a) Find the solution in plane polar coordinates ρ, φ, that takes the value +1
for 0 < φ < π and the value −1 for −π < φ < 0, when ρ = a.
(b) For a point (x, y) on or inside the circle x2 + y 2 = a2 , identify the angles α
and β defined by
y
y
and
β = tan−1
.
α = tan−1
a+x
a−x
Show that u(x, y) = (2/π)(α + β) is a solution of Laplace’s equation that
satisfies the boundary conditions given in (a).
(c) Deduce a Fourier series expansion for the function
tan−1
21.11
sin φ
sin φ
+ tan−1
.
1 + cos φ
1 − cos φ
The free transverse vibrations of a thick rod satisfy the equation
a4
∂2 u
∂4 u
+ 2 = 0.
∂x4
∂t
Obtain a solution in separated-variable form and, for a rod clamped at one end,
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PDES: SEPARATION OF VARIABLES AND OTHER METHODS
x = 0, and free at the other, x = L, show that the angular frequency of vibration
ω satisfies
1/2 1/2 ω L
ω L
= − sec
.
cosh
a
a
21.12
21.13
[ At a clamped end both u and ∂u/∂x vanish, whilst at a free end, where there is
no bending moment, ∂2 u/∂x2 and ∂3 u/∂x3 are both zero. ]
A membrane is stretched between two concentric rings of radii a and b (b > a).
If the smaller ring is transversely distorted from the planar configuration by an
amount c|φ|, −π ≤ φ ≤ π, show that the membrane then has a shape given by
2m
cπ ln(b/ρ) 4c am
b
m
cos mφ.
u(ρ, φ) =
−
ρ
−
2 ln(b/a)
π m odd m2 (b2m − a2m ) ρm
A string of length L, fixed at its two ends, is plucked at its mid-point by an
amount A and then released. Prove that the subsequent displacement is given by
∞
(2n + 1)πct
8A (−1)n
(2n + 1)πx
u(x, t) =
cos
,
sin
π 2 (2n + 1)2
L
L
n=0
where, in the usual notation, c2 = T /ρ.
Find the total kinetic energy of the string when it passes through its unplucked
position, by calculating it in each mode (each n) and summing, using the result
∞
0
21.14
21.15
21.16
21.17
π2
1
=
.
(2n + 1)2
8
Confirm that the total energy is equal to the work done in plucking the string
initially.
Prove that the potential for ρ < a associated with a vertical split cylinder of
radius a, the two halves of which (cos φ > 0 and cos φ < 0) are maintained at
equal and opposite potentials ±V , is given by
∞
4V (−1)n ρ 2n+1
u(ρ, φ) =
cos(2n + 1)φ.
π n=0 2n + 1 a
A conducting spherical shell of radius a is cut round its equator and the two
halves connected to voltages of +V and −V . Show that an expression for the
potential at the point (r, θ, φ) anywhere inside the two hemispheres is
∞
(−1)n (2n)!(4n + 3) r 2n+1
u(r, θ, φ) = V
P2n+1 (cos θ).
22n+1 n!(n + 1)!
a
n=0
[ This is the spherical polar analogue of the previous question. ]
A slice of biological material of thickness L is placed into a solution of a
radioactive isotope of constant concentration C0 at time t = 0. For a later time t
find the concentration of radioactive ions at a depth x inside one of its surfaces
if the diffusion constant is κ.
Two identical copper bars are each of length a. Initially, one is at 0 ◦ C and the
other is at 100 ◦ C; they are then joined together end to end and thermally isolated.
Obtain in the form of a Fourier series an expression u(x, t) for the temperature
at any point a distance x from the join at a later time t. Bear in mind the heat
flow conditions at the free ends of the bars.
Taking a = 0.5 m estimate the time it takes for one of the free ends to
attain a temperature of 55 ◦ C. The thermal conductivity of copper is 3.8 ×
102 J m−1 K−1 s−1 , and its specific heat capacity is 3.4 × 106 J m−3 K−1 .
770
21.6 EXERCISES
21.18
A sphere of radius a and thermal conductivity k1 is surrounded by an infinite
medium of conductivity k2 in which far away the temperature tends to T∞ .
A distribution of heat sources q(θ) embedded in the sphere’s surface establish
steady temperature fields T1 (r, θ) inside the sphere and T2 (r, θ) outside it. It can
be shown, by considering the heat flow through a small volume that includes
part of the sphere’s surface, that
k1
∂T1
∂T2
− k2
= q(θ) on
∂r
∂r
r = a.
Given that
q(θ) =
21.19
21.20
∞
1
qn Pn (cos θ),
a n=0
find complete expressions for T1 (r, θ) and T2 (r, θ). What is the temperature at
the centre of the sphere?
Using result (21.74) from the worked example in the text, find the general
expression for the temperature u(x, t) in the bar, given that the temperature
distribution at time t = 0 is u(x, 0) = exp(−x2 /a2 ).
Working in spherical polar coordinates r = (r, θ, φ), but for a system that has
azimuthal symmetry around the polar axis, consider the following gravitational
problem.
(a) Show that the gravitational potential due to a uniform disc of radius a and
mass M, centred at the origin, is given for r < a by
r
1 r 2
1 r 4
2GM
1 − P1 (cos θ) +
P2 (cos θ) −
P4 (cos θ) + · · · ,
a
a
2 a
8 a
and for r > a by
1 a 2
1 a 4
GM
1−
P2 (cos θ) +
P4 (cos θ) − · · · ,
r
4 r
8 r
where the polar axis is normal to the plane of the disc.
(b) Reconcile the presence of a term P1 (cos θ), which is odd under θ → π − θ,
with the symmetry with respect to the plane of the disc of the physical
system.
(c) Deduce that the gravitational field near an infinite sheet of matter of constant
density ρ per unit area is 2πGρ.
21.21
In the region −∞ < x, y < ∞ and −t ≤ z ≤ t, a charge-density wave ρ(r) =
A cos qx, in the x-direction, is represented by
∞
eiqx
ρ̃(α)eiαz dα.
ρ(r) = √
2π −∞
The resulting potential is represented by
∞
eiqx
V (r) = √
Ṽ (α)eiαz dα.
2π −∞
Determine the relationship between Ṽ (α) and ρ̃(α), and hence show that the
potential at the point (0, 0, 0) is
∞
A
sin kt
dk.
π0 −∞ k(k 2 + q 2 )
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PDES: SEPARATION OF VARIABLES AND OTHER METHODS
21.22
21.23
21.24
Point charges q and −qa/b (with a < b) are placed, respectively, at a point P , a
distance b from the origin O, and a point Q between O and P , a distance a2 /b
from O. Show, by considering similar triangles QOS and SOP , where S is any
point on the surface of the sphere centred at O and of radius a, that the net
potential anywhere on the sphere due to the two charges is zero.
Use this result (backed up by the uniqueness theorem) to find the force with
which a point charge q placed a distance b from the centre of a spherical
conductor of radius a (< b) is attracted to the sphere (i) if the sphere is earthed,
and (ii) if the sphere is uncharged and insulated.
Find the Green’s function G(r, r0 ) in the half-space z > 0 for the solution of
∇2 Φ = 0 with Φ specified in cylindrical polar coordinates (ρ, φ, z) on the plane
z = 0 by
#
1
for ρ ≤ 1,
Φ(ρ, φ, z) =
1/ρ for ρ > 1.
Determine the variation of Φ(0, 0, z) along the z-axis.
Electrostatic charge is distributed in a sphere of radius R centred on the origin.
Determine the form of the resultant potential φ(r) at distances much greater than
R, as follows.
(a) Express in the form of an integral over all space the solution of
∇2 φ = −
ρ(r)
.
0
(b) Show that, for r r ,
|r − r | = r −
r · r
+O
r
1
.
r
(c) Use results (a) and (b) to show that φ(r) has the form
M
d·r
1
φ(r) =
+ 3 +O 3 .
r
r
r
Find expressions for M and d, and identify them physically.
21.25
21.26
Find, in the form of an infinite series, the Green’s function of the ∇2 operator for
the Dirichlet problem in the region −∞ < x < ∞, −∞ < y < ∞, −c ≤ z ≤ c.
Find the Green’s function for the three-dimensional Neumann problem
∇2 φ = 0
for z > 0
and
Determine φ(x, y, z) if
#
f(x, y) =
21.27
∂φ
= f(x, y)
∂z
on z = 0.
δ(y) for |x| < a,
0
for |x| ≥ a.
Determine the Green’s function for the Klein–Gordon equation in a half-space
as follows.
(a) By applying the divergence theorem to the volume integral
φ(∇2 − m2 )ψ − ψ(∇2 − m2 )φ dV ,
V
obtain a Green’s function expression, as the sum of a volume integral and a
surface integral, for the function φ(r ) that satisfies
∇2 φ − m2 φ = ρ
772