MATH 402 - Assignment #4
Due on Monday March 7, 2011
Name —————————————–
Student number —————————
1
Problem 1:
Prove that the set of coefficients An , Bn in the general solution for the
vibrating string are evaluated as
Z L
Z L
1
σφn wt (x, 0) dx,
An =
σφn w(x, 0) dx, Bn =
λn 0
0
where the initial shape w(x, 0) and initial velocity wt (x, 0) are prescribed arbitrarily.
2
Problem 2:
Consider the vibrating string with the uniform mass density σ = σ0 , a
constant.
(a) Solve the related eigenvalue-eigenfunction problem explicitly, considering all combinations of fixed and vibrating end points at x = 0
and x = L.
(b) Write down the general solution for the string.
3
Problem 3:
Consider the vibrating string with the mass density σ(x) = (1 + x)−2 ,
tension τ = 1, and fixed end point conditions. Assume the initial string
shape is given and the string starts from the rest.
(a) Solve the related eigenvalue-eigenfunction problem. (Hint: Start
by searching a solution of the form (1 + x)a .)
(b) Write down the general solution for the string.