3.3 above, the total sound power radiated by the two pure‐tone sources depends on the phasing and separation distance.
Example 3.6
In a certain factory space the noise level with an electric motor in operation is 93 dB. When the motor is turned off the background noise is 85 dB. What is the sound pressure level due to the motor?
Solution
Since 1093/10 = 1085/10 + 10 Lp/10, then Lp = 10log(109.3–108.5) = 92.2 dB.
Figure 3.6 Diagram for combination of two sound pressure levels or two sound power levels of uncorrelated sources.
3.5 Three‐dimensional Wave Equation
In most sound fields, sound propagation occurs in two or three dimensions. The three‐dimensional version of Eq. (3.1) in Cartesian coordinates is
This equation is useful if sound wave propagation in rectangular spaces such as rooms is being considered. However, it is helpful to recast Eq. (3.29) in spherical coordinates if sound propagation from sources of sound in free space is being considered. It is a simple mathematical procedure to transform Eq. (3.29) into spherical coordinates, although the resulting equation is quite complicated. However, for propagation of sound waves from a spherically symmetric source (such as the idealized case of a pulsating spherical balloon known as an omnidirectional or monopole source) (Table 3.1), the equation becomes quite simple (since there is no angular dependence):
Table 3.1 Models of idealized spherical sources: Monopole, Dipole, and Quadrupolea.
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a For simple harmonic sources, after one half‐period the velocity changes direction; positive sources become negative and vice versa, and forces reverse direction with dipole and quadrupole force models.
After some algebraic manipulation Eq. (3.30) can be written as
Here, r is the distance from the origin and p is the sound pressure at that distance.
Equation (3.30) is identical in form to Eq. (3.1) with p replaced by rp and x by r. The general and simple harmonic solutions to Eq. (3.30) are thus the same as Eqs. (3.4) and (3.5) with p replaced by rp and x with r. The general solution is
(3.32)
or
where f1 and f2 are arbitrary functions. The first term on the right of Eq. (3.33) represents a wave traveling outward from the origin; the sound pressure p is seen to be inversely proportional to the distance r. The second term in Eq. (3.33) represents a sound wave traveling inward toward the origin, and in most practical cases such waves can be ignored (if reflecting surfaces are absent).
The simple harmonic (pure‐tone) solution of Eq. (3.31) is
(3.33)
We may now write that the constants A1 and A2 may be written as
3.6 Sources of Sound
The second term on the right of Eq. (3.33), as before, represents sound waves traveling inward to the origin and is of little practical interest. However, the first term represents simple harmonic waves of angular frequency ω traveling outward from the origin, and this may be rewritten as [4]
where Q is termed the strength of an omnidirectional (monopole) source situated at the origin, and Q = 4πA1 /ρck. The mean‐square sound pressure p2rms may be found [4] by time averaging the square of Eq. (3.34) over a period T:
