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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Properties of Gases
          Plane Waves in Air

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Plane Waves in Air

Figure F.3 shows a 2D $ xy$ cross-section 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 F.4 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.
\begin{figure}\input fig/planewaveangle.pstex_t \end{figure}

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

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.