[SOLVED] Amath 569 homework assignment 6

1. Consider the sound waves governed by
in a circular cylinder of radius a and length L.
Assume that the sound produced in this tube is symmetric, i.e. no dependence. Find the
lowest three frequencies. Take2. Consider the wave function for an electron of mass in a sphere surrounded by an
infinite potential at a radius a from the nucleus, which just mean that
Find the energy levels for the symmetric case, where does not depend on . Your
answer should be exact and in terms of parameters given.3. Consider the Legendre’s equation:
Compute the first four coefficients in the Legendre expansion (similar to Fourier sine or
cosine series expansion):
for
Plot the approximation of the sum consisting of one, two, three and four terms along with
the original function
∂2
∂t
2ψ = c
2
∇2
ψ, ∇2 = 1
r
∂
∂r
(r
∂
∂r
)+
1
r
2
∂2
∂θ2 +
∂2
∂z
2
ψ = 0 at r = a; ψ = 0 at z = 0,L.
θ
c = 300m /s, a = 1cm, L = 0.5m.
ψ µ
ψ = 0 at r = a.
i! ∂
∂t
ψ = − !2
2µ
∇2
ψ.
ψ θ and φ
d
dx (1− x
2
) d
dx y(x) ⎡
⎣
⎢ ⎤
⎦
⎥ + n(n +1)y(x) = 0, −1 ≤ x ≤1,
with the condition that y(±1) are bounded. The solutions are the Legendre polynomials, Pn (x),
which are given by the Rodrigue’s formula:
Pn (x) = 1
2n
n!
dn
dxn (x
2 −1)
n
.
For example P2 (x) = 1
2
(3x
2 −1).
f (x) = an
n=0
∞
∑ Pn (x), where an = 2n +1
2 f (x)Pn (x)dx −1
1
∫
f (x) = 0 for −1 < x < 0
x for 0 < x <1
⎧
⎨
⎩
f (x).