5.7 Acoustical Impedance
129
cannot comfortably hear the typical low sounds from a hi-fi audio system playing
an orchestral piece with a dynamic range of 90 dB.
Assuming our ‘small amplitude’ linearization assumptions have some validity for
intense sounds, the loudest sinusoidal sound wave that can be created in air would
have a loudness of
β max = 20 log 10
(δp) 2 /2
/p 0
= 20 log 10
1.01325 × 10
5 N/m
2 /(
√
2 × 20 × 10
−6 N/m
2 )
= 191 dB,
at which level the pressure variations go to zero absolute pressure (p o = 1.01325 ×
10 5 N/m 2 ) on the low side. The denominator factor in the log is the peak thresholdof-hearing pressure,
√
2×(20×10 −6 N/m 2 ), which is
√
2 times the root-mean-square
(RMS) threshold pressure. Such a wave generates a sound intensity of 1.2823 ×
10 7 W/m 2 , more than ten million times the intensity which can damage human
ears, and more than ten thousand times the power delivered by sunlight on a square
meter of the Earth’s surface. Another limit occurs. Being a longitudinal wave, the
amplitude of the wave cannot be greater than half the wavelength. Again with our
‘linearization’ assumptions, this happens when
I = (1/2)π
2 ρ o v
3
= 2.42 × 10
8 W/m
2 ,
which is 204 dB.
However, our linearization for the effects of pressure and density variations in
the dynamics of sound will eventually fail as the sound intensity increases. We enter
the world of non-linear behavior, wherein shock waves and other phenomena can
reach more extreme levels of sound. 14 In liquids and solids, negative pressures can
exist and be far larger than one atmosphere, because layers of liquids and solids can
hold themselves together with adhesive forces. 15
5.7 Acoustical Impedance
The transfer of sound energy from one material to another is important in hearing,
sound insulation, ultrasonics, and other medical applications. The degree to which
a material resists any kind of flow is called its impedance. This term applies to
sound transmission just as well as electric current flow. In the case of sound, three
14 An acoustical shock wave is a sound wave with strong variations in the supporting material
speed over times much shorter than the period of the wave. A dynamical description of a shock
wave involves forces non-linear in the speed variations.
15 We return to this topic in the discussion of liquid cavitation by ultrasound, Sect. 5.24.2.
129
cannot comfortably hear the typical low sounds from a hi-fi audio system playing
an orchestral piece with a dynamic range of 90 dB.
Assuming our ‘small amplitude’ linearization assumptions have some validity for
intense sounds, the loudest sinusoidal sound wave that can be created in air would
have a loudness of
β max = 20 log 10
(δp) 2 /2
/p 0
= 20 log 10
1.01325 × 10
5 N/m
2 /(
√
2 × 20 × 10
−6 N/m
2 )
= 191 dB,
at which level the pressure variations go to zero absolute pressure (p o = 1.01325 ×
10 5 N/m 2 ) on the low side. The denominator factor in the log is the peak thresholdof-hearing pressure,
√
2×(20×10 −6 N/m 2 ), which is
√
2 times the root-mean-square
(RMS) threshold pressure. Such a wave generates a sound intensity of 1.2823 ×
10 7 W/m 2 , more than ten million times the intensity which can damage human
ears, and more than ten thousand times the power delivered by sunlight on a square
meter of the Earth’s surface. Another limit occurs. Being a longitudinal wave, the
amplitude of the wave cannot be greater than half the wavelength. Again with our
‘linearization’ assumptions, this happens when
I = (1/2)π
2 ρ o v
3
= 2.42 × 10
8 W/m
2 ,
which is 204 dB.
However, our linearization for the effects of pressure and density variations in
the dynamics of sound will eventually fail as the sound intensity increases. We enter
the world of non-linear behavior, wherein shock waves and other phenomena can
reach more extreme levels of sound. 14 In liquids and solids, negative pressures can
exist and be far larger than one atmosphere, because layers of liquids and solids can
hold themselves together with adhesive forces. 15
5.7 Acoustical Impedance
The transfer of sound energy from one material to another is important in hearing,
sound insulation, ultrasonics, and other medical applications. The degree to which
a material resists any kind of flow is called its impedance. This term applies to
sound transmission just as well as electric current flow. In the case of sound, three
14 An acoustical shock wave is a sound wave with strong variations in the supporting material
speed over times much shorter than the period of the wave. A dynamical description of a shock
wave involves forces non-linear in the speed variations.
15 We return to this topic in the discussion of liquid cavitation by ultrasound, Sect. 5.24.2.