HOMEWORK 4
(1) Solve utt = c2 uxx , u(x, 0) = ex , ut (x, 0) = sin(x).
(2) Solve utt = c2 uxx , u(x, 0) = log(1 + x2 ), ut (x, 0) = 4 + x.
(3) The midpoint of a piano string of tension T , density ⇢, and length l is hit by a hammer whose
head diameter is 2a. A flea sitting at a distance from l/4 from one end. Assume that a < l/4.
How long does it take for the disturbance to reach the flea?
(4) If both and are odd functions of x, show that the solution of u(x, t) of the wave equation is
also odd in x for all t.
(5) Solve uxx 3uxt 4utt = 0, u(x, 0) = x2 , ut (x, 0) = ex .
(6) Use the energy conservation of the wave equation to prove that the only solution with = 0 and
= 0 is u = 0.
(7) The three dimensional wave equation is
utt
where u = u(x, y, z, t) and
=
c2 u = 0,
2
@2
@2
+ @ + @z
2 is the Laplacian operator. For waves with
@x2p @y 2
= x2 + y 2 + z 2 . In this special case, the Laplacian is
spherical
symmetry u = u(⇢, t), where ⇢
given by
2
u = u⇢⇢ + ⇢ u⇢ . By change of variables U = ⇢u, show that the general solution for the spherically
symmetric wave equation
2
utt = c2 (u⇢⇢ + u⇢ )
⇢
is u = ⇢1 (F (⇢ ct) + G(⇢ + ct)). Why do you think an outward-moving wave u = F (⇢
decays in amplitude? Give a physical interpretation.
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