Equation of Plane Wave/Direction Cosine Form

Theorem

Let $\phi$ be a plane wave propagated with velocity $c$.

Let the direction of propagation of $\phi$ be expressed as:

$x : y : z = l : m : n$

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

Then $\phi$ can be expressed as:

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

Proof

By Equation of Wavefront of Plane Wave, the equation of the wavefront of $\phi$ is given by:

$l x + m y + n z = K$

Hence it is clear that:

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

is a function which fulfils all the requirements to be a plane wave.

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Hence $\phi$ as defined represents a plane wave propagated with velocity $c$ in the direction given.

Sources

• 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 4$: $(13)$