- #1

RJLiberator

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## Homework Statement

Consider the homogeneous Neumann conditions for the wave equation:

U_tt = c^2*U_xx, for 0 < x < l

U_x(0,t) = 0 = U_x(l, t)

U(x,0) = f(x), U_t(x,0) = g(x)

Using the separation of variables, find a nontrivial solution of (1).

## Homework Equations

Separation of variables

## The Attempt at a Solution

So I've been working away at this problem. First, the instructor gave us a semi-example of a very similar problem in class. Next, I've been using this link to help me ( ).

First, we suppose u(x,t) = X(x)T(t)

then, U_tt = X(x)T''(t) and U_xx = X''(x)T(t)

so, U_tt = c^2*U_xx => X(x)T''(t) = c^2X''(x)T(t)

we can rewrite it and get it into solvable separate equations as follows:

T''(t) = λT(t)

X''(x) = (λ/c^2)X(x)

I think I've done everything right up to this stage.

**Now is where my understanding begins to get hazy**

So from here, I would think the best idea is to separate them into cases where λ =0, λ >1, λ<1 as was done in the instructional YouTube video I posted earlier.

However, in my notes, the instructor just did turned the λ I had in my equations into -λ^2.

He mentioned he did this to include the complex numbers that may be in the solution.

Further a long in the instructional video, I think in the case λ < 1 the video replaces λ with ω^2, but it's still not quite the same.

I'm wondering if a) my instructor only did part of the problem, and if so, was it because of the boundary conditions.

b) Do I need to do all three cases? IF so, will they all intertwine in the end with my boundary conditions?

b) If in this particular problem, where my boundary conditions come into play? MY thinking is that when I solve the various cases I will get general solutions and then use the boundary conditions to get a nontrivial solution.

So my current plan is to solve the general solution for each case, and then hope that I can use the boundary conditions to find a nontrivial solution.

Am I thinking in the right way?