the so‐called direction cosines with the x, y and z directions are cos θx = ±kx /k, cos θy = ±ky /k, and cos θz = ±kz /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
where A, B, C are the room dimensions in the x, y, and z directions, and nx = 0, 1, 2, 3,…; ny = 0, 1, 2, 3,… and nz = 0, 1, 2, 3, … Note nx, ny, and nz 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) fE 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 nx = 1, ny = 1, and nz = 1 mode in a room with hard reflecting walls. See Refs. [23, 24] for further discussion of standing‐wave behavior in rectangular rooms.
Figure 3.34 Standing wave for nx = 1, ny = 1, and nz = 1 (particle velocity shown).
Figure 3.35 Standing wave for nx = 1, ny = 1, and nz = 1 (sound pressure shown).
Example 3.16
Calculate all the possible natural frequencies for normal modes of vibration under 100 Hz within a rectangular room 3.1 × 4.7 × 6.2 m3.
Solution
Table 3.3 gives all the possible natural frequencies for modes under 100 Hz using Eq. (3.94) and c = 343 m/s.
One can see in Table 3.3 that the frequency spacing becomes smaller with increasing frequency and that there may be degenerate modes present (i.e. when two or more modes have the same characteristic frequency but different values of nx, ny, and nz). Modes which are close to each other in frequency can easily “beat,” while degenerate modes can greatly increase the response of the room at particular frequencies where degeneracy occurs. This can give rise to the “boomy” sensation (at low frequencies) which is often found in regular‐shaped rooms of similar wall dimensions [4].
Table 3.3 Frequencies (less than 100 Hz) for a 3.1 × 4.7 × 6.2 m3 rectangular room, for c = 343 m/s.
| n x | n y | n z | f E |
|---|---|---|---|
| 0 | 0 | 1 | 27.7 |
| 0 | 1 | 0 | 36.5 |
| 0 | 1 | 1 | 45.8 |
| 0 | 0 | 2 | } 55.3 |
| 1 | 0 | 0 | |
| 1 | 0 | 1 | 61.9 |
