# Equation of Plane Wave is Particular Solution of Wave Equation/Direction Cosine Form

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## Theorem

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.

## Proof

The wave equation is expressible as:

- $\dfrac 1 {c^2} \dfrac {\partial^2 \phi} {\partial t^2} = \dfrac {\partial^2 \phi} {\partial x^2} + \dfrac {\partial^2 \phi} {\partial y^2} + \dfrac {\partial^2 \phi} {\partial z^2}$

This needs considerable tedious hard slog to complete it.In particular: Need to know how to differentiate a vector given in direction cosines first, as in, need to understand what they mean.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 `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1955: C.A. Coulson:
*Waves*(7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 5$