Engineering Acoustics. Malcolm J. Crocker

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acoustic field, the far field) and integrating Eq. (3.47) over the surface, or by measuring the intensity all over the surface in the near or far acoustic field and integrating over the surface (Eq. (3.41)). We shall discuss source directivity in Section 3.9.

      Example 3.10

      If the sound power level of a source is 120 dB (which is equivalent to 1 acoustical watt), what is the sound pressure level at 50 m (a) for sound radiation to whole space and (b) for radiation to half space?

      Solution

      1 For whole space: I = 1/4π(50)2 = 1/104 π (W/m2), then

       Since we may assume r = 50 m is in the far acoustic field, Lp ≅ LI = 75 dB as well (we have also assumed ρc ≅ 400 rayls).

       For half space: I = 1/2π(50)2 = 2/104 π (W/m2), then

      and L_p ≅ L_I = 78 dB also.

       It is important to note that the sound power radiated by a source can be significantly affected by its environment. For example, if a simple constant‐volume velocity source (whose strength Q will be unaffected by the environment) is placed on a floor, its sound power will be doubled (and its sound power level increased by 3 dB). If it is placed at a floor–wall intersection, its sound power will be increased by four times (6 dB); and if it is placed in a room comer, its power is increased by eight times (9 dB). See Table 3.2. Many simple sources of sound (ideal sources, monopoles, and real small machine sources) produce more sound power when put near reflecting surfaces, provided their surface velocity remains constant. For example, if a monopole is placed touching a hard plane, an image source of equal strength may be assumed.

Table 3.2 Simple source near reflecting surfaces^a.

Intensity	Source	Condition	Number of Images		Power	D	DI
I		Free field	None		W	1	0 dB
4 I		Reflecting plane	1		2W	4	6 dB
16 I		Wall‐floor intersection	3		4W	16	12 dB
64 I		Room corner	7		8W	64	18 dB

^a Q and DI are defined in Eqs. (3.53), (3.58), and (3.60).

3.9 Directivity

      The sound intensity radiated by a dipole is seen to depend on cos² θ (see Figure 3.11). Most real sources of sound become directional at high frequency, although some are almost omnidirectional at low frequency. This phenomenon depends on the source dimension, d, which must be small in size compared with a wavelength λ, so d/λ ≪ 1 for them to behave almost omnidirectionally.

Figure 3.11 Polar directivity plots for the radial sound intensity in the far field of (a) monopole, (b) dipole, and (c) (lateral) quadrupole.

      3.9.1 Directivity Factor (Q(θ, ϕ))

In general, a directivity factor Q_θ,ϕ may be defined as the ratio of the radial intensity 〈I_{θ, ϕ}〉_t (at angles θ and ϕ and distance r from the source) to the radial intensity 〈I_s〉_t at the same distance r radiated from an omnidirectional source of the same total sound power (Figure 3.12). Thus

(3.53)

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