Plane Waves in Air

Figure E.3 shows a 2D $xy$ cross-section of the sinusoidal plane wave

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

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 $\underline{k}^T=[10\pi, 5\pi, 0]$.

Figure E.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.

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