Engineering Acoustics. Malcolm J. Crocker

interest to us because we find that in rooms there are solutions only at certain frequencies. It may be of some importance now to mention both the sinusoidal solution and the equivalent solution using complex notation that is very frequently used in acoustics and vibration theory.

      For a one‐dimensional wave, the simple harmonic solution to the wave equation is

(3.88)

      where k = ω/c = 2πf/c (the wavenumber).

The first term in Eq. (3.88) represents a wave of amplitude traveling in the +x‐direction. The second term in Eq. (3.88) represents a wave of amplitude traveling in the −x‐direction.

      The equivalent expression to Eq. (3.88) using complex notation is

(3.89)

where , Re{} means real part; and and are complex amplitudes of the sound pressure; remember k = ω/c; kc = 2πf. Both Eqs. (3.88) and (3.89) are solutions to Eq. (3.86).

For the three‐dimensional case (x, y, and z propagation), the sinusoidal (pure tone) solution to Eq. (3.87) is

(3.90)

Note that there are 2³ (eight) possible solutions given by Eq. (3.90). Substitution of Eq. (3.90) into Eq. (3.87) (the three‐dimensional wave equation) gives (from any of the eight (2³) equations):

(3.91)

      from which the wavenumber k is

(3.92)

and the so‐called direction cosines with the x, y and z directions are cos θ_x = ±k_x /k, cos θ_y = ±k_y /k, and cos θ_z = ±k_z /k (see Figure 3.32).

Figure 3.32 Direction cosines and vector k.

Equations (3.91) and (3.92) apply to the cases where the waves propagate in unbounded space (infinite space) or finite space (e.g. rectangular rooms).

      For the case of rectangular rooms with hard walls, we find that the sound (particle) velocity perpendicular to each wall must be zero. By using these boundary conditions in each of the eight solutions to Eq. (3.87), we find that ω² = (2πf)² and k² in Eqs. (3.91) and (3.92) are restricted to only certain discrete values:

      (3.93)

      or

      Then the room natural frequencies are given by

(3.94)

where A, B, C are the room dimensions in the x, y, and z directions, and n_x = 0, 1, 2, 3,…; n_y = 0, 1, 2, 3,… and n_z = 0, 1, 2, 3, … Note n_x, n_y, and n_z are the number of half waves in the x, y, and z directions. Note also for the room case, the eight propagating waves add together to give us a standing wave. The wave vectors for the eight waves are shown in Figure 3.33.

Figure 3.33 Wave vectors for eight propagating waves.

      There are three types of standing waves resulting in three modes of sound wave vibration: axial, tangential, and oblique modes. Axial modes are a result of sound propagation in only one room direction. Tangential modes are caused by sound propagation in two directions in the room and none in the third direction. Oblique modes involve sound propagation in all three directions.

We have assumed there is no absorption of sound by the walls. The standing waves in the room can be excited by noise or pure tones. If they are excited by pure tones produced by a loudspeaker or a machine that creates sound waves exactly at the same frequency as the eigenfrequencies (natural frequencies) f_E of the room, the standing waves are very pronounced. Figures 3.34 and 3.35 show the distribution of particle velocity and sound pressure for the n_x = 1, n_y = 1, and n_z = 1 mode in a room with hard reflecting walls. See Refs. [23, 24] for further discussion of standing‐wave behavior in rectangular rooms.

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