# Plane Waves in Three Dimensions

• Eugen Skudrzyk

## Abstract

The three-dimensional wave equation in cartesian coordinates
$$\begin{gathered}{\nabla ^2}p = \frac{1}{{{c^2}}}\frac{{{\partial ^2}p}}{{\partial {t^2}}} \hfill \\ {\nabla ^2} = \frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}} \hfill \\ \end{gathered}$$
(1)
is solved by plane progressive waves
$$p = f(ct \pm \overrightarrow n \cdot \overrightarrow r ),$$
(2)
where $$\overrightarrow n$$ is the direction of propagation, $$\vec{n}\cdot \vec{r} = {{n}_{x}}x + {{n}_{y}}y + {{n}_{z}}z$$ and
$${n_{{x^2}}} + {n_{{y^2}}} + {n_{{z^2}}} = 1$$
(3)
so that the wave equation is satisfied. The quantities n x ,ny, nz can be interpreted as the direction cosines of the angles between the direction of propagation and the x, y, and z axes, respectively:
$$\begin{gathered} {n_x} = \cos \left( {n,x} \right) \hfill \\ {n_y} = \cos \left( {n,y} \right) \hfill \\ {n_z} = \cos \left( {n,z} \right). \hfill \\ \end{gathered}$$
(4)

## Keywords

Incident Wave Sound Pressure Acoustic Impedance Transmitted Wave Direction Cosine

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