Engineering Acoustics. Malcolm J. Crocker

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)

Figure 3.12 Geometry used in derivation of directivity factor.

      For a directional source, the mean square sound pressure measured at distance r and angles θ and ϕ is p²_rms (θ,ϕ).

      In the far field of this source (r ≫ λ), then

      (3.54)

      But if the source were omnidirectional of the same power W, then

      (3.55)

      where p²_rms is a constant, independent of angles θ and ϕ.

      We may therefore write:

      (3.56)

      and

      (3.57)

      where is the space‐averaged mean‐square sound pressure.

      We define the directivity factor Q as

(3.58)

      the ratio of the mean‐square sound pressure at distance r to the space‐averaged mean‐square pressure at r, or equivalently the directivity Q may be defined as the ratio of the mean‐square sound