Consider a model of a damped, oscillating string of length L,
u_u = -2(lambda)(u_e) + (c^2)(u)_zz over 0 <= x <= L,
where u(x, t) is the displacement, lambda describes the damping and c is the natural (undamped) wave speed. Suppose that the ends of the string are fixed at u = 0, and that the string is initially at rest, but has a displacement profile, f(x). In other words,
u(0, t) = u(L, t) = 0 with u(x, 0) = f(x) and (u_t)(x, 0) = 0
a) Solve for u(x,t) using separation of variables. Show that the damping shifts the frequency for each mode and that the solution decays exponentially in time. You may assume that 0 < lambda < pi*e / L. (Hint: One wat to check if you answer is correct is to set lambda = 0 and see if you recover the usual undamped solution.)
b) Suppose f(x) = sin(pi8x/L). Compute u(x, t) when 0 < lambda < pi*e/L and lambda > pi*e/L. What is the difference between the two cases?
This shows how to solve an equation using separation of variables in an attached Word document.