Engineering Acoustics. Malcolm J. Crocker

Figure 3.34 Standing wave for n_x = 1, n_y = 1, and n_z = 1 (particle velocity shown).

Figure 3.35 Standing wave for n_x = 1, n_y = 1, and n_z = 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 n_x, n_y, and n_z). 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
0	1	2	} 66.3
1	1	0
1	1	1	71.8
0	2	0	73.0
1	0	2	78.2
0	2	1	78.0
0	0	3	83.0
1	1	2	86.3
0	1	3	90.7
0	2	2	} 91.6
1	2	0
1	2	1	95.7
1	0	3	99.7

      The position of the sound source within the room is also an important parameter, since for many source positions certain types of modes may not be excited. For example, if the source is located in one of the corners of the room, then it is possible to excite every normal mode, while if the source is located at the center of a rectangular room then only the even modes (one eighth of the total number of possible modes) can be excited. Similarly if we keep the position of the source constant and measure the sound pressure throughout the room we see differences in level depending on where we are standing in the room relative to the normal modes. In this way the room superimposes its own acoustical response characteristics upon those of the source. Hence we cannot measure the true frequency response of a sound source (e.g. loudspeaker) in a reverberant room because of the effect of the modal response of the room. This interference can be removed by making all the wall surfaces highly sound‐absorbent. Then all the modes are sufficiently damped so we are able to measure the true output of the source [4]. Such rooms are called anechoic (see Figure 3.22).

3.18 Waveguides

      Waveguides can occur naturally where sound waves are channeled by reflections at boundaries and by refraction. Even the ocean can sometimes be considered to be an acoustic waveguide that is bounded above by the air–sea interface and below by the ocean bottom (see chapter 31 in the Handbook of Acoustics [1]). Similar channeling effects are also sometimes observed in the atmosphere [34]. Waveguides are also encountered in musical instruments and engineering applications. Wind instruments may be regarded as waveguides. In addition, waveguides comprised of pipes, tubes, and ducts are frequently used in engineering systems, for example, exhaust pipes, air‐conditioning ducts and the ductwork in turbines and turbofan engines. The sound propagation in such waveguides is similar to the three‐dimensional situation discussed in Section 3.17 but with some differences. Although rectangular ducts are used in air‐conditioning systems, circular ducts are also frequently used, and theory for these must be considered as well. In real waveguides, airflow is often present and complications due to a mean fluid flow must be included in the theory.

      For low‐frequency excitation, only plane waves can propagate along the waveguide (in which the sound pressure