6.4: Waves in 3D Space

The wave equation can be generalized to three spatial dimensions by replacing $f(x,t)$ with a wavefunction that depends on three spatial coordinates, $f(x,y,z,t)$. The second-order derivative in $x$ is then replaced by second-order derivatives in each spatial direction: $$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2}\right) \; f(x,y,z,t) = 0. \label{3dwave}$$ This PDE supports complex plane wave solutions of the form $$f(x,y,z,t) = A \, e^{i(\vec{k} \cdot \vec{r} - \omega t)},$$ where $$\vec{k} = \begin{bmatrix}k_x\\k_y\\k_z\end{bmatrix}, \;\;\; \vec{r} = \begin{bmatrix}x\\y\\z\end{bmatrix}, \;\;\;\frac{\omega}{\sqrt{k_x^2 + k_y^2 + k_z^2}} = v.$$ Again, we can verify that this is a solution by direct substitution. We call $\vec{k}$ the wave-vector, which generalizes the wavenumber $k$. The direction of the wave-vector specifies the spatial direction in which the wave travels.