5.1 The Nature of Sound Waves
117
pV
γ
= constant
(5.11)
where the so-called polytropic exponent, 4 γ = c p /c v , is the ratio of the specific
heat at constant pressure to the specific heat at constant volume. In any material, the
adiabatic approximation during sound transmission is a good assumption. 5
To estimate the speed of sound in air, we use v =
√
dp/dρ applied to an ideal
gas acting adiabatically to get
v G =
γp o
ρ o
(5.12)
=
γ RT
M
(5.13)
= 331.45
m
s
+ 0.607
m
s
T c
◦ C
(5.14)
where M is the average mass per mole of gas, i.e. the molecular weight, and
T c = T − 273.15 ◦ K. The third expression applies to air at atmospheric pressure
for temperatures near 0 ◦ C.
If you breathe helium gas instead of air, the speed of sound in your throat is
almost three times that of air (being that He is about 1/7.2 the average molecular
weight of air, and its γ is 5/3 while nitrogen is 7/5). This makes you sound like
Mickey Mouse when you speak after inhaling from a helium balloon (but be careful,
because helium balloons contain no oxygen).
The speeds of longitudinal (compressional) sound waves and transverse (shearing) sound waves in a solid turn out to be
v C =
(B + 4n s /3)/ρ ,
(5.15)
v S =
n s /ρ ,
(5.16)
where B is the material’s bulk modulus, n s is its shear modulus, and ρ its
density. (Seismologists call these two kinds of waves ‘primary’ and ‘secondary’,
because, after an earthquake, the compressional waves arrive at detectors before the
transverse shearing waves.)
Liquids do not easily support a shearing wave. When only the compressional
wave survives, it will have a speed given by
v L =
B/ρ .
(5.17)
4 For an ideal gas, γ = 1 + 2/f , where f , the ‘degrees of freedom’, is three for a monatomic gas,
one for each translational degree of freedom, and is five for a diatomic gas, three translational, two
rotational degrees of freedom.
5 In thermodynamics, an adiabatic process requires that the local entropy S remain constant, so that
the speed of sound is written v =
√
(∂p/∂ρ) s .
117
pV
γ
= constant
(5.11)
where the so-called polytropic exponent, 4 γ = c p /c v , is the ratio of the specific
heat at constant pressure to the specific heat at constant volume. In any material, the
adiabatic approximation during sound transmission is a good assumption. 5
To estimate the speed of sound in air, we use v =
√
dp/dρ applied to an ideal
gas acting adiabatically to get
v G =
γp o
ρ o
(5.12)
=
γ RT
M
(5.13)
= 331.45
m
s
+ 0.607
m
s
T c
◦ C
(5.14)
where M is the average mass per mole of gas, i.e. the molecular weight, and
T c = T − 273.15 ◦ K. The third expression applies to air at atmospheric pressure
for temperatures near 0 ◦ C.
If you breathe helium gas instead of air, the speed of sound in your throat is
almost three times that of air (being that He is about 1/7.2 the average molecular
weight of air, and its γ is 5/3 while nitrogen is 7/5). This makes you sound like
Mickey Mouse when you speak after inhaling from a helium balloon (but be careful,
because helium balloons contain no oxygen).
The speeds of longitudinal (compressional) sound waves and transverse (shearing) sound waves in a solid turn out to be
v C =
(B + 4n s /3)/ρ ,
(5.15)
v S =
n s /ρ ,
(5.16)
where B is the material’s bulk modulus, n s is its shear modulus, and ρ its
density. (Seismologists call these two kinds of waves ‘primary’ and ‘secondary’,
because, after an earthquake, the compressional waves arrive at detectors before the
transverse shearing waves.)
Liquids do not easily support a shearing wave. When only the compressional
wave survives, it will have a speed given by
v L =
B/ρ .
(5.17)
4 For an ideal gas, γ = 1 + 2/f , where f , the ‘degrees of freedom’, is three for a monatomic gas,
one for each translational degree of freedom, and is five for a diatomic gas, three translational, two
rotational degrees of freedom.
5 In thermodynamics, an adiabatic process requires that the local entropy S remain constant, so that
the speed of sound is written v =
√
(∂p/∂ρ) s .