If we assume that ##\psi## has a Fourier transform ##\hat{\psi}##, so that ##\psi(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{\psi}(x,\omega)e^{i\omega t}\mathrm{d}\omega##, then the wave equation reduces to ##-\rho\omega^2\hat{\psi}(x,\omega)=E\frac{\partial^2 \hat{\psi}(x,\omega)}{\partial...