The 2D wave-equation: d'Alemberts method

[Niles]

Homework Statement

I am given an initial- and boundary value problem for the two dimensional wave function, and I must solve it by d'Alembert's method.

It is all good, but I am given the initial velocity: $g(x)=x(\pi-x)$.

I have to find the odd extension of this on the interval $-\pi$ to $\pi$. I don't like using Fourier series, because I must integrate the odd extension. Is there any way to do this?

Thanks in advance,

sincerely,
Niles.

 

[lippyka]

Homework EquationsThe equation for the two dimensional wave function is given by: u_t = c^2u_xx, u(x,0)=f(x), u(a,t)=g(a,t)The odd extension on the interval -\pi to \pi of g(x)=x(\pi-x) is given by: g(x)= -x(-\pi-x) = -\pi x-x^2The Attempt at a SolutionI solved the initial- and boundary value problem for the two dimensional wave function using d'Alembert's method. The solution is given by: u(x,t) = \frac{1}{2}[f(x+ct)+f(x-ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)ds.Now, I need to find the odd extension of g(x)=x(\pi-x) on the interval -\pi to \pi. The odd extension of this is given by: g(x)= -x(-\pi-x) = -\pi x-x^2. Thus, the solution becomes: u(x,t) = \frac{1}{2}[f(x+ct)+f(x-ct)] - \frac{1}{2c}\int_{x-ct}^{x+ct} (-\pi s-s^2)ds.