# Questions tagged [wave-equation]

For questions related to solutions and analysis of the wave equation.

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### Finding solution to the wave equation by the characteristics method

Given the wave equation in one spatial dimension, $\phi_{tt} -c^2\phi_{xx} = 0,$ setting $u:=\frac{\partial \phi}{\partial t}, v := \frac{\partial \phi}{\partial x},$ we get the matrix representation ...
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### Given the wave equation and $u_{yw}=0$, conclude $u(x,t)=\phi(x-\alpha t)+\psi(x+\alpha t)$.

I know the wave equation is $\alpha^2u_{xx}=u_{tt}$. I am given the result from a previous problem, $u_{yw}(x,t)=0$ where we let $y=x-\alpha t$ and $w=x+\alpha t$. For this, $\phi$ and $\psi$ are ...
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### General solution to finite interval simple wave equation

I have a wave equation with fixed interval with the following initial and boundary conditions. $$u_{tt} = u_{xx}$$ $$u(x, 0) = \phi(x) = x(1-x)$$ $$u_t(x, 0) = x^3 - x$$ $$u(0, t) = u(1, t) = 0$$ ...
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