Problem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the time-harmonic electric field is given by E= u(p, o) 2, where u satisfies the Helmholtz wave equation 8² u 1 du 1 8²u + + ap² pap p² 86² u(0, 0) is finite, u(p, 0) = u(p, 2π), where k is a constant. Solve it. Hint: the unknown coefficient. + k²u = 0, u(a,0) = eika cos 0 < p
Problem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the time-harmonic electric field is given by E= u(p, o) 2, where u satisfies the Helmholtz wave equation 8² u 1 du 1 8²u + + ap² pap p² 86² u(0, 0) is finite, u(p, 0) = u(p, 2π), where k is a constant. Solve it. Hint: the unknown coefficient. + k²u = 0, u(a,0) = eika cos 0 < p
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Transcribed Image Text:Problem 8: In electromagnetic scattering by an infinite cylinder of radius a, under certain conditions, the
time-harmonic electric field is given by E = u(p, ø) î, where u satisfies the Helmholtz wave equation
1 ди
1 02u
+ k?u = 0,
0 < 0< 27
0 <p< a,
p dp
ika cos o
u(0, ø) is finite,
u (α, φ ) -
= e
u(p, 0) = u(p, 27T),
us(p, 0) = us(p,2m)
where k is a constant. Solve it. Hint: Use the generating function of the Bessel function when obtaining
the unknown coefficient.
Transcribed Image Text:Problem 4: Solve the nonhomogeneous wave equation
curr + F (x, t) = utt,
0 < x < L,
t > 0,
u(0, t) = 0,
u(L, t) = 0
u(x, 0) = f(x),
ut(x, 0) =
Problem 5: The problem of the vibration of an elastic membrane, deoted by u(p, ø, t), fixed on a circular
frame can be described by
1 ди
1 02u
0 < p< a,
0 < ¢ < 27,
t > 0,
Op?' pdp ' p2 d62
u(0, 6, t) is finite,
u (а, ф, t) %3D 0
u(р,0, t) 3 и(р, 2т, t),
Us(p, 0, t) = us(e, 27, t)
u(p, 0,0) = f(p, ),
Ut(p, 0,0) = 0
Solve it.
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