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Plane Waves in Air

Figure B.9 shows a 2D $ xy$ cross-section of a snapshot (in time) of the sinusoidal plane wave

$\displaystyle p(x,y,z) = p_0 + \cos(k_x x + k_y y)
$

for $ k_x = 2\pi 5$ and $ k_y=2\pi 5/2$ , with $ x$ and $ y$ in the range $ [0,1)$ .

Figure: Gray-scale density plot of the $ xy$ cross-section of a sinusoidal plane wave $ p(t,\underline{x}) = \cos\left(\omega t -
\underline{k}^T\underline{x}\right)$ , at $ t=0$ with vector wavenumber $ \underline{k}^T=[10\pi, 5\pi, 0]$ .
\includegraphics[width=\twidth]{eps/planewave}

Figure B.10 depicts a more mathematical schematic of a sinusoidal plane wave traveling toward the upper-right of the figure. The dotted lines indicate the crests (peak amplitude location) along the wave.

Figure: Wave crests of the sinusoidal traveling plane wave $ p(t,\underline{x}) = \cos\left(\omega t -
\underline{k}^T\underline{x}\right)$ , for some fixed time $ t$ and $ \underline{x}$ in the $ (x,y,0)$ plane.
\includegraphics{eps/planewaveangle}

The direction of travel and spatial frequency are indicated by the vector wavenumber $ \underline{k}$ , as discussed in the following section.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA