Plane Waves in Air

Figure B.9 shows a 2D $xy$ cross-section of a snapshot (in time) 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 B.9: 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 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 B.10: 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 the following section.