[SOLVED] Amath 569 homework assignment 5

1. For the 1-dimensional heat equation for conduction in a copper rod:
(a) solve using separation of variables. Show that the time-dependence for the nth standing
mode is
.
(b) For , find the mode that dominates the solution, and thus write down the
approximate solution (in time and in space). Describe in words how an initial condition,
which may not look like a sine wave, becomes the sine shape with the largest wavelength
fitting in between the boundaries.2. Sound waves in a box satisfies
PDE:
BC:
Use separation of variables to solve this problem for the following configurations and find
the quantized frequency of oscillation w, where w appears in the time dependence of the
solution in the form of a sine or cosine of wt.
(a) V is a one-dimensional box: 0<x<L.
(b) V is a two-dimensional box: 0<x<L; 0<y<L.
(c) V is a three-dimensional box: 0<x<L; 0<y<L; 0<z<L.3. (a) Solve:
subject to the boundary condition Consider separately and
and matching across Do not use Fourier transform.
(b) Solve the above equation for the case of , subject to the Sommerfeld radiation
condition. Show that the solution from (a) reduce to this solution as
∂
∂t
u =α2 ∂2
∂x2 u, 0< x < L, t >0.
u(x,t)= 0 at x = 0 and x = L.
u(x,0)= f(x), 0< x < L.
exp(−n2
(t /te )), where te ≡(L/πα)
2
, about 1 hour for a 2m-copper rod
t >te
2
2 2
2 uc u V 0, in
t
¶ – Ñ = ¶
u V = 0 on . ¶
2
2 2
2 0 0 ( / ) ( ) / , , 0, finite. d u k i k cu x y c x y dx
++ = ed e – – -¥ < < ¥ >
u x ® ® 0 as . ±¥ x y < x y > ,
x y = .
e = 0
e ® 0.