Plane Waves in Three Dimensions

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 ,n_y, n_z 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)