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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag/Wien 1971

Authors and Affiliations

  • Eugen Skudrzyk
    • 1
  1. 1.Ordnance Research Laboratory and Physics DepartmentThe Pennsylvania State UniversityUniversity ParkUSA

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