Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.6 EXERCISES

(a) EvaluatedPm(μ)/dμandd^2 Pm(μ)/dμ^2 , using the forms given in (21.47), and

substitute them into (21.45).

(b) Differentiate Legendre’s equationmtimes 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 exercise10.20, concerned with the flow of a very viscous
fluid past a sphere, to find the full expression for the stream functionψ(r, θ). At
the surface of the spherer=a, the velocity fieldu= 0 , whilst far from the sphere
ψ(Ur^2 sin^2 θ)/ 2.
Show thatf(r) can be expressed as a superposition of powers ofr,and
determine which powers give acceptable solutions. Hence show that

ψ(r, θ)=

U

4

(

2 r^2 − 3 ar+

a^3

r

)

sin^2 θ.

21.8 The motion of a very viscous fluid in the two-dimensional (wedge) region−α<
φ<αcan be described, in (ρ, φ) coordinates, by the (biharmonic) equation

∇^2 ∇^2 ψ≡∇^4 ψ=0,

together with the boundary conditions∂ψ/∂φ=0atφ=±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=±ρatφ=±α, which impose the condition

that azimuthal flow increases linearly withralong any radial line. Assuming a

solution in separated-variable form, show that the full expression forψis

ψ(ρ, φ)=

ρ^2

2

sin 2φ− 2 φcos 2α

sin 2α− 2 αcos 2α

.

21.9 A circular disc of radiusais heated in such a way that its perimeterρ=ahas
a steady temperature distributionA+Bcos^2 φ,whereρandφare plane polar
coordinates andAandBare constants. Find the temperatureT(ρ, φ)everywhere
in the regionρ<a.
21.10 Consider possible solutions of Laplace’s equation inside a circular domain as
follows.

(a) Find the solution in plane polar coordinatesρ, φ, that takes the value +1

for 0<φ<πand the value−1for−π<φ<0, whenρ=a.

(b) For a point (x, y) on or inside the circlex^2 +y^2 =a^2 , identify the anglesα

andβdefined by

α=tan−^1

y

a+x

and β=tan−^1

y

a−x

.

Show thatu(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

sinφ

1+cosφ

+tan−^1

sinφ

1 −cosφ

.

21.11 The free transverse vibrations of a thick rod satisfy the equation

a^4

∂^4 u

∂x^4

+

∂^2 u

∂t^2

=0.

Obtain a solution in separated-variable form and, for a rod clamped at one end,