6. (a) The vertical displacement of a string u(x, t) satisfies the wave equation ∂ 2u(x, t) ∂x2 = 1 c 2 ∂ 2u(x, t) ∂t2 .

[answerhappygod]

6. (a) The vertical displacement of a string u(x, t) satisfies
the wave equation ∂ 2u(x, t) ∂x2 = 1 c 2 ∂ 2u(x, t) ∂t2 . Consider
a string on 0 ≤ x ≤ L, where the ends of the string are fixed. Use
separation of variables to solve this boundary value problem and
show that u(x, t) = X∞ n=1 Cn sin(ωnx) cos(ωnct) + Dn sin(ωnx)
sin(ωnct) , where ωn = nπ L . (You can ignore exponential and
linear growth solutions).
(b) Find the particular solution for a string of length L = π,
with the initial conditions u(x, 0) = π − x, and ut(x, 0) = 0, for
0 < x < π. (Ignore initial discontinuities at the ends).