Equation of Plane Wave is Particular Solution of Wave Equation

Theorem

Direction Cosine Form

Let $\phi$ be a plane wave propagated with velocity $c$ in a Cartesian $3$-space.

Let $\phi$ be expressed as:

$\map \phi {x, y, z, t} = \map f {l x + m y + n z - c t}$

where $l$, $m$ and $n$ are the direction cosines of the normal to $P$.

Then $\phi$ satisfies the wave equation.

This article is complete as far as it goes, but it could do with expansion. In particular: To be expressed in a more convenient form. Propagating along the $x$ axis plus a coordinate transformation may be better for immediate comprehensibility. Simplicity is good. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.