Equation of Wavefront of Plane Wave/Direction Cosine Form

Theorem

Let $\phi$ be a plane wave.

Let an arbitrary wavefront of $\phi$ be denoted $P$.

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 $P$.

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Then:

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

where $K$ is constant for a given plane wave $\phi$.

Proof

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Sources

• 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 4$: $(12)$