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How do you solve for u(x,t) using separation of variables?

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 way 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?


Solution Preview

See attached Word file that uses Microsoft Equation editor (problem28044.doc).

Following the separation of variables method, seek solutions in the product form

Separate the boundary conditions
Plug the form (1) for u into the given homogeneous conditions:

Note that the results to the right of the arrows follow because we are interested only in nontrivial (or nonzero) solutions u(x,t) which mandates that neither X(x) nor T(t) can be identically zero. We have "separated" the three homogeneous conditions for u into two boundary conditions for X and one initial condition for T. We will attach the two boundary conditions to the X differential equation we obtain below, and the one initial condition to the T differential equation we obtain below.

Separate the PDE and solve the homogeneous eigenvalue problem
Plug the form (1) for u into the given partial differential equation, to get

Divide the above equation through by the solution form XT, to get

Isolate all the t-dependence on the left hand-side, isolate all the x-dependence on the right-hand side, and set the results equal to an arbitrary ...

Solution Summary

This solution shows how to solve an equation using separation of variables in an attached Word document.