Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

x= 0, and free at the other,x=L, show that the angular frequency of vibration

ωsatisfies

cosh

(

ω^1 /^2 L

a

)

=−sec

(

ω^1 /^2 L

a

)

.

[ At a clamped end bothuand∂u/∂xvanish, whilst at a free end, where there is
no bending moment,∂^2 u/∂x^2 and∂^3 u/∂x^3 are both zero. ]
21.12 A membrane is stretched between two concentric rings of radiiaandb(b>a).
If the smaller ring is transversely distorted from the planar configuration by an
amountc|φ|,−π≤φ≤π, show that the membrane then has a shape given by

u(ρ, φ)=

cπ

2

ln(b/ρ)

ln(b/a)

−

4 c

π

∑

modd

am

m^2 (b^2 m−a^2 m)

(

b^2 m

ρm

−ρm

)

cosmφ.

21.13 A string of lengthL, fixed at its two ends, is plucked at its mid-point by an
amountAand then released. Prove that the subsequent displacement is given by

u(x, t)=

∑∞

n=0

8 A(−1)n

π^2 (2n+1)^2

sin

[

(2n+1)πx

L

]

cos

[

(2n+1)πct

L

]

,

where, in the usual notation,c^2 =T/ρ.

Find the total kinetic energy of the string when it passes through its unplucked

position, by calculating it in each mode (eachn) and summing, using the result

∑∞

0

1

(2n+1)^2

=

π^2

8

.

Confirm that the total energy is equal to the work done in plucking the string
initially.
21.14 Prove that the potential forρ<aassociated with a vertical split cylinder of
radiusa, the two halves of which (cosφ>0andcosφ<0) are maintained at
equal and opposite potentials±V,isgivenby

u(ρ, φ)=

4 V

π

∑∞

n=0

(−1)n

2 n+1

(ρ

a

) 2 n+1

cos(2n+1)φ.

21.15 A conducting spherical shell of radiusais cut round its equator and the two
halves connected to voltages of +Vand−V. Show that an expression for the
potential at the point (r, θ, φ) anywhere inside the two hemispheres is

u(r, θ, φ)=V

∑∞

n=0

(−1)n(2n)!(4n+3)

22 n+1n!(n+1)!

(r

a

) 2 n+1

P 2 n+1(cosθ).

[ This is the spherical polar analogue of the previous question. ]
21.16 A slice of biological material of thicknessLis placed into a solution of a
radioactive isotope of constant concentrationC 0 at timet=0.Foralatertimet
find the concentration of radioactive ions at a depthxinside one of its surfaces
if the diffusion constant isκ.
21.17 Two identical copper bars are each of lengtha. Initially, one is at 0◦Candthe
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 expressionu(x, t) for the temperature
at any point a distancexfrom the join at a later timet. Bear in mind the heat
flow conditions at the free ends of the bars.
Takinga=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 Jm−^1 K−^1 s−^1 , and its specific heat capacity is 3. 4 × 106 Jm−^3 K−^1.