Proof that Mode Density Grows as Freq. Squared

The resonant modes of a rectangular room are given by¹

$$k^2(l,m,n) = k^2_x(l) + k^2_y(m) + k^2_z(n)$$

• $k_x(l) = l\pi/L_x$ = $l$ th harmonic of the fundamental standing wave in the $x$
• $L_x$ = length of the room along $x$
• Similarly for $y$ and $z$
• Mode frequencies map to a uniform 3D Cartesian grid indexed by $(l,m,n)$
• Grid spacings are $\pi/L_x$ , $\pi/L_y$ , and $\pi/L_z$ in $x$ ,$y$ , and $z$ , respectively.
• Spatial frequency $k$ of mode $(l,m,n)$ = distance from the (0,0,0) to $(l,m,n)$
• Therefore, the number of room modes having a given spatial frequency grows as $k^2$