WKB Homework #2 for Math 605
Due: March 16, 2005
It may be helpful to read Zauderer, Chapter on Asymptotic Methods, on reserve.
1. Assume that the plane wave uI = e−ikx is incident on the parabola
C: x=
1 2 a
y −
2a
2
, and that the total field u = 0 on C. Find an asymptotic expansion of the scattered wave uS
valid fo k 1.
2. Assume that a wave uI = eikx for k 1 is incident on a circular cylinder of radius a
C : x2 + y 2 = a 2
and that the total field satisfies
∇2 u + k 2 u = 0, outside C
with u = 0 on C, and the scattered part uS is outgoing from C.
a) Find and plot the rays x = x(s, τ ), y = y(s, τ ) with C parametrized by x = a cos τ and
y = a sin τ for π/2 < τ < 3π/2. What are the rays for −π/2 < τ < π/2?
b) Show that the caustic of these rays lies inside the circle but that the caustic intersects the
circle at the points x = 0, y = ±a. Plot the caustic.
c) Show that Snell’s Law (αI = αR ) holds for the rays at their intersection with the boundary.
d) Calculate the leading order amplitude for the scattered wave, in terms of s and τ .
e) Show that u = 0 in the dark region, and that the total field in the illuminated region is
u = eikx
v
u
u
−t
a/2 cos τ
eik(2s+a cos τ )
a/2 cos τ − 2s
for π/2 < τ < 3π/2.
3. Sketch the rays, orthogonal to the wave fronts, for the case of a stratified medium in the
case when all the rays issue from the origin. That is, consider
uxx + uyy + k 2 n2 (x)u = 0
for k 1, with



n(0)
x<0
n1 (x) 0 ≤ x ≤ 2 ,
n(x) =


n2
x>2
for n2 < n(0) (both constants) and n1 (x) a monotonically decreasing function. In particular
find an expression for y 0 (x) on the rays, with initial conditions y(0) = 0 and y 0 (0) = tan θ for
0 < θ < π. Use this to determine the behavior of the rays. Indicate illuminated and shadow
regions.