Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.9 EXERCISES

(b) CalculateF(s) on either side of the branch cut, evaluate the integral and

hence determinef(t).

(c) Confirm that the derivative with respect tosof the Laplace transform

integral of your answer is the same as that given bydF/ds.

25.15 Use the contour in figure 25.7(c) to show that the function with Laplace transform
s−^1 /^2 is (πx)−^1 /^2.
[ For an integrand of the formr−^1 /^2 exp(−rx) change variable tot=r^1 /^2 .]
25.16 Transverse vibrations of angular frequencyωon a string stretched with constant
tensionTare described byu(x, t)=y(x)e−iωt,where
d^2 y
dx^2

+

ω^2 m(x)

T

y(x)=0.

Here,m(x)=m 0 f(x) is the mass per unit length of the string and, in the general

case, is a function ofx. Find the first-order W.K.B. solution fory(x).

Due to imperfections in its manufacturing process, a particular string has

a small periodic variation in its linear density of the formm(x)=m 0 [1 +

sin(2πx/L)], where1. A progressive wave (i.e. one in which no energy is

lost) travels in the positivex-direction along the string. Show that its amplitude

fluctuates by±^14 of its valueA 0 atx= 0 and that, to first order in, the phase

of the wave is

ωL

2 π

√

m 0

T

sin^2

πx
L
ahead of what it would be if the string were uniform, withm(x)=m 0.
25.17 The equation

d^2 y

dz^2

+

(

ν+

1

2

−

1

4

z^2

)

y=0,

sometimes called the Weber–Hermite equation, has solutions known as parabolic
cylinder functions. Find, to within (possibly complex) multiplicative constants,
the two W.K.B. solutions of this equation that are valid for large|z|. In each case,
determine the leading term and show that the multiplicative correction factor is
of the form 1+O(ν^2 /z^2 ).
Identify the Stokes and anti-Stokes lines for the equation. On which of the
Stokes lines is the W.K.B. solution that tends to zero forzlarge, real and negative,
the dominant solution?
25.18 A W.K.B. solution of Bessel’s equation of order zero,

d^2 y

dz^2

+

1

z

dy

dz

+y=0, (∗)

valid for large|z|and−π/ 2 <argz< 3 π/2, isy(z)=Az−^1 /^2 eiz.Obtainan

improvement on this by finding a multiplier ofy(z) in the form of an asymptotic

expansion in inverse powers ofzas follows.

(a) Substitute fory(z)in(∗) and show that the equation is satisfied to O(z−^5 /^2 ).

(b) Now replace the constantAbyA(z) and find the equation that must be

satisfied byA(z). Look for a solution of the formA(z)=zσ

∑∞

n=0anz

−n,

wherea 0 = 1. Show thatσ= 0 is the only acceptable solution to the indicial

equation and obtain a recurrence relation for thean.

(c) To within a (complex) constant, the expressiony(z)=A(z)z−^1 /^2 eizis the

asymptotic expansion of the Hankel functionH 0 (1)(z). Show that it is a

divergent expansion for all values ofzand estimate, in terms ofz, the value

ofNsuch that

∑N

n=0anz

−n− 1 / (^2) eizgives the best estimate ofH(1)
0 (z).