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

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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.


The wave equation is expressible as:

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