To evaluate the intensity of a plane wave, we first calculate the energy density as a sum of kinetic and potential energy contributions and then average over space and time. This energy is transported with the wave speed c, s9 we just multiply by c to get the results. Without going into details (Kinsler et al., 1982, p. 110), we find the result
I -- ,.,~-N"/12 -- p2 -pc -- '1'1'11. r-, (6.32)
where p and u are taken as rms quantities (otherwise, a factor of ~ should be inserted into each result).
Analysis for a spherical wave is more complex (Kinsler et al., 1982, p. 112) because much of the kinetic energy in the velocity field near the origin- specifically the part associated with the term 1/jkr in Eq. (6.28)-is not radiated because of its 90° phase shift relative to the acoustic pressure. The radiated intensity is given as in Eq. (6.32) by
I=-, p2 (6.33)
pc
where an rms value of p is implied, but the other forms of the result in Eq. (6.32) do not apply.
The total power P radiated in a spherical wave can be calculated by integrating I(r) over a spherical surface of radius r, giving
p = 47rr2p(r)2 (6.34)
pc
From Eq. (6.26), P is independent of r, as is obviously required. To get some feeling for magnitudes, a source radiating a power of 1 m W as a spherical wave produces an intensity level, or equivalently a sound pressure level, of approximately 79 dB at a distance of 1 m. At a distance of 10 m, assuming no reflections from surrounding walls or other objects, the SPL is 59 dB. These figures correspond to radiation from a typical musical instrument, though clearly a great range is possible. The disparity between this figure and the powers of order 100 W associated with amplifiers and loudspeakers is explained by the facts that the amplifier requires adequate power to avoid overload during transients and the normal operating level is only a few watts, while the loudspeaker itself has an efficiency of only about 1% in converting electrical power to radiated sound.
6.4 Reflection, Diffraction, and Absorption
When a wave encounters any variation in the properties of the medium in which it is propagating, its behavior is disturbed. Gradual changes in
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164 6. Sound Waves in Air
the medium extending over many wavelengths lead mostly to a change in the wave speed and propagation direction-the phenomenon of refraction.
When the change is more abrupt, as when a sound wave in air strikes a solid object, such as a person or a wall, then the incident wave is generally mostly reflected or scattered and only a small part is transmitted into or through the object. That part of the wave energy transmitted into the object will generally be dissipated by internal losses and multiple reflections unless the object is very thin, like a lightweight wall partition, when it may be reradiated from the opposite surface.
It is worthwhile to examine the behavior of a plane pressure wave Ae-ikx moving from a medium of wave impedance z1 to one of impedance z2. In general, we expect there to be a reflected wave Beikx and a transmitted wave Ce-ikx. The acoustic pressures on either side of the interface must be equal, so that, taking the interface to be at x = 0,
A+B =C. (6.35)
Similarly, the displacement velocities must be the same on either side of the interface, so that, using Eq. (6.19) and noting the sign of k for the various waves,
A-B C
= (6.36)
We can now solve Eqs. (6.35) and (6.36) to find the reflection coefficient:
B A and the transmission coefficient:
c A
= (6.37)
(6.38) These coefficients refer to pressure amplitudes. If z2 = z1, then B = 0 and C = A as we should expect. If Z2 > Zt. then, from Eq. (6.37), the reflected wave is in phase with the incident wave and a pressure maximum is reflected as a maximum. If z2 < z1, then there is a phase change of 180°
between the reflected wave and the incident wave and a pressure maximum is reflected as a minimum. If z2 ằ z1 or z2 ~ z1, then reflection is nearly total. The fact that, from Eq. (6.38), the transmitted wave will have a pressure amplitude nearly twice that of the incident wave if z2 ằ z1 is not a paradox, as we see below, since this wave carries a very small energy.
Perhaps even more illuminating than Eqs. (6.37) and (6.38) are the cor- responding coefficients expressed in terms of intensities, using Eq. (6.32). If the incident intensity is Io = A2 jz1 , then the reflected intensity Iris given by
(6.39)
6.4. Reflection, Diffraction, and Absorption 165 and the transmitted intensity It by
(6.40) Clearly, the transmitted intensity is nearly zero if there is a large acoustic mismatch between the two media and either z2 ằ z1 or z2 ô: z1.
These results can be generalized to the case of oblique incidence of a plane wave on a plane boundary (Kinsler et al., 1982, pp. 131-133), and we then encounter the phenomenon of refraction, familiar from optics, with the reciprocal of the velocity of sound Ci in each medium taking the place of its optical refractive index.
All these results can be extended in a straightforward way to include cases where the wave impedances Zi are complex quantities (ri + jxi) rather than real. In particular, the results [Eqs. (6.39) and (6.40)] carry over directly to this more general situation, the reflection and transmission coefficients generally depending upon the frequency of the wave.
If the surface of the object is flat, on the scale of a sound wavelength, and its extent is large compared with the wavelength, then the familiar rules of geometrical optics are an adequate approximation for the treatment of reflections. It is only for large areas, such as the walls or ceilings of concert halls, that this is of more than qualitative use in understanding behavior (Beranek, 1962; Rossing, 1982; Meyer, 1978).
At the other extreme, an object that is small compared with the wave- length of the sound wave involved will scatter the wave almost equally in all directions, the fractional intensity scattered being proportional to the sixth power of the size of the object. When the size of the object ranges from, for example, one-tenth of a wavelength up to 10 wavelengths, then scattering behavior is very complex, even for simply shaped objects (Morse, 1948, pp. 346-356; Morse and Ingard, 1968, pp. 400-449).
There is similar complexity in the "sound shadows" cast by objects.
Objects that are very large compared with the sound wavelength create well-defined shadows, but this situation is rarely encountered in other than architectural acoustics. More usually, objects will be comparable in size to the wavelength involved, and diffraction around the edges into the shadow zone will blur its edges or even eliminate the shadow entirely at distances a few times the diameter of the object. Again, the discussion is complex even for a simple plane edge (Morse and Ingard, 1968, pp. 449-458). For the purposes of this book, a qualitative appreciation of the behavior will be adequate.
Even in an unbounded uniform medium, such as air, a sound wave is attenuated as it propagates, because of losses of various kinds (Kinsler et al., 1982, Chapter 7). Principal among the mechanisms responsible are viscosity, thermal conduction, and energy interchange between molecules with differing external excitation. If we write
k ---+ w - ja,
c ( 6.41)
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166 6. Sound Waves in Air
so that the wave amplitude decays as e-a:z: for a plane wave or as (1/r)e-ar for a spherical wave, then a, or rather the quantity 8686a corresponding to attenuation in decibels per kilometer, is available in standard tables (Evans and Bass, 1986). In normal room air with relative humidity greater than about 50%, most of the attenuation is caused by molecular energy exchange with water vapor. The behavior is not simple, but for a frequency J, in hertz, it has roughly the form
a ~ 4 x 10-7 J, 100Hz < f < 1 kHz,
a~ 1 x 10-10j2, 2kHz< f <100kHz. (6.42) If the relative humidity is very low, say less than 20%, then a is increased by as much as a factor 10 over most of this frequency range. For completely dry air, a is increased by nearly a factor 30 below 100Hz but is decreased by a factor of about 4 between 10 kHz and 100 kHz.
For many musical applications in small rooms, this absorption can be neglected, but this is not true of large concert halls, since the absorption at 10kHz amounts to about 0.1 dB m-1• This gives a noticeable reduction in brightness of the sound at the back of the hall.
More important, in most halls, is the absorption of sound upon reflection from the walls, ceilings, furnishings, and audience. If an impulsive sound is made in a hall, then the sound pressure level decays nearly linearly, corresponding to an exponential decay in sound pressure. The time T60 for the level to decay by 60 dB is known as the reverberation time. Details of the behavior are complicated, but to a first approximation no is given by the Sabine equation:
T6o = 0.161V ' (6.43)