D'Alembert Problem for 1-D wave equation

[rexasaurus]

1.
For the 1-D wave equation, the d’Alembert solution is u(t, x) = f (x + ct) + g(x − ct) where f , g are each a function of 1 variable.

Suppose c = 1 and we know f (x) = x^2 and g(x) = cos 2x for x > 0.

Find u(t, x) for al l t, x ≥ 0 if you are also given the BC: u ≡ 1 at x = 0, all t.

2. See problem statement

3.

Do we simply "plug" into the D'alembert solution?

I started with:
u(t, x) = f (x + ct) + g(x − ct)
so: f(x)=x^2 or f(x)=(x+t)^2
so: g(x)=cos2x or g(x)=cos(2x-2t)

next: plug f(x), g(x) into u(t,x)
u(t,x)=x²+2xt+t²+cos(2x-2t)

Is this all that needs to be done? I am not sure where to go from here or what to do with the BC of u(t,0)=1. Please help.

 

[lippyka]

Thank you.Yes, the next step is to use the boundary condition to solve for t. Since u(t, 0) = 1, we have:1 = x^2 + 2xt + t^2 + cos(2x - 2t)This is a quadratic equation in t that can be solved for t. Once you have the value of t, you can plug it back into the original expression for u(t, x).