Formula (5.7.10) does not include the standing waves for which one or two of the mode numbers \(l, m, n\) are zero. Establish the more accurate formulat $$ N=\frac{\omega_{\max }^{3}}{6 \pi^{2} c_{f}^{2}} V+\frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} A+\frac{\omega_{\max }}{16 \pi c_{f}} L $$ where \(V=a b c, A=2(a b+a c+b c)\), and \(L=4(a+b+c)\) †P. M. Morse,"Vibration and Sound," \(2 \mathrm{~d}\) ed, chap. 8, McGraw-Hill Book Company, New York, \(1948 .\) ± See Morse, op. cit., p. 394 .

The original formula for the number of standing wave modes is: $$ N=\frac{\omega_{\max }^{3}}{6 \pi^{2} c_{f}^{2}} V $$ where \(N\) is the number of standing wave modes, \(\omega_{\max}\) is the maximum frequency, \(c_f\) is the speed of sound, and \(V\) is the volume of the room.

To account for the modes with one zero mode number, there are 2 cases for each of the mode numbers: Case 1: \(l=0\), \(m\neq 0\), \(n\neq 0\) \(\rightarrow\) \(N_1 = \frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} A_1\) with \(A_1=2(ac+bc)\) Case 2: \(l\neq 0\), \(m=0\), \(n\neq 0\) \(\rightarrow\) \(N_2 = \frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} A_2\) with \(A_2=2(ab+ac)\) Case 3: \(l\neq 0\), \(m\neq 0\), \(n=0\) \(\rightarrow\) \(N_3 = \frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} A_3\) with \(A_3=2(ab+bc)\) Summing up these cases, we get: $$ N_1+N_2+N_3 =\frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} (A_1+A_2+A_3) = \frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} A $$ where \(A=2(ab+ac+bc)\) is the given surface area.

To account for the modes with two zero mode numbers, there are 3 cases: Case 1: \(l=0\), \(m=0\), \(n\neq 0\) \(\rightarrow\) \(N_4 = \frac{\omega_{\max }}{16 \pi c_{f}} L_1\) with \(L_1=4c\) Case 2: \(l=0\), \(m\neq 0\), \(n=0\) \(\rightarrow\) \(N_5 = \frac{\omega_{\max }}{16 \pi c_{f}} L_2\) with \(L_2=4b\) Case 3: \(l\neq 0\), \(m=0\), \(n=0\) \(\rightarrow\) \(N_6 = \frac{\omega_{\max }}{16 \pi c_{f}} L_3\) with \(L_3=4a\) Summing up these cases, we get: $$ N_4+N_5+N_6 = \frac{\omega_{\max }}{16 \pi c_{f}} (L_1+L_2+L_3) = \frac{\omega_{\max }}{16 \pi c_{f}} L $$ where \(L=4(a+b+c)\) is the given linear dimension sum.

Add the results from step 2 and 3 to the original formula from step 1 to account for all missing standing wave modes. The final formula is: $$ N=\frac{\omega_{\max }^{3}}{6 \pi^{2} c_{f}^{2}} V+\frac{\omega_{\max }^{2}}{16 \pi c_{f^{2}}} A+\frac{\omega_{\max }}{16 \pi c_{f}} L $$