5.4 Energy Carried by Sound
125
The restriction v s ≤ c applies even in the case of the so-called ‘quantum information
transport’ and more fanciful ‘quantum teleportation.’ Within the formalism of
standard quantum theory, there are instantaneous non-local elements. However, as
soon as a measurement is made, information cannot be transferred faster than the
speed of light. This must be so, because quantum theory satisfies Einstein’s relativity
postulates.
The relationship ω = ω(κ) between the component frequencies in a wave and
their wave numbers is called a ‘dispersion relation’, because the relation determines
how the component waves of individual wavelengths separate from each other in
speed and, during refraction, separate in direction. Since ω = v p κ, if the speed of
a sound wave, v p =
√
dp/dρ does not depend on wavelength of the sound, then
v g = dω/dκ = v p , i.e. the phase velocity and the group velocity of the wave
match. Quantum electron waves in relativity theory satisfy ω 2 = c 2 κ 2 + (mc/ ¯
h) 2 ,
so v g = c 2 /v p .
5.4 Energy Carried by Sound
Because sound causes material compression and rarefaction as well as oscillatory
motion of material mass, sound energy has both potential and kinetic forms. This
energy may be transported through the sequence of interactions of each layer of
material with the adjacent layer as the sound propagates. We receive a small amount
of energy each time we hear a sound. For any fluid, the energy per unit volume
is given by u = δE/δV = 1/2ρv 2 + ρρ, where is the internal energy of the
fluid element per unit mass. As sound passes through the fluid, we will assume only
small variations in the density, δρ, and internal energy, δδ, will occur. If the fluid
was initially at rest, the speed is due solely to the vibrational motion. Expanding the
energy of the fluid elements per unit volume in terms of the density variations will
give
u =
1
2
ρv
2
+ δρ
∂(ρ ρ)
∂ρ
+
1
2
(δρ)
2 ∂ 2 (ρ ρ)
∂ρ 2 ,
(5.28)
in which terms of third order and higher in the density variation δρ are dropped.
Usually, as the sound moves through the fluid, there is little time for heat energy
to be transported from the instantaneously pressurized regions to the rarefied regions
of the sound before the wave moves half a wavelength. (The heat diffusion constant
is usually much smaller than the wavelength of the wave times its speed.) Zero
heat transfer defines an adiabatic process, and means that the entropy of each fluid
mass element is unchanged. Thermodynamically, the internal energy of the fluid per
unit mass increases only by heat input or work input, expressed as dd = T ds +
(p/ρ 2 )dρ, where ds is the change in the entropy per unit mass in the fluid element,
and T ds is the heat input. For our adiabatic processes, ds = 0. This makes dd =
(p/ρ 2 )dρ. We have, at constant entropy per unit mass, ∂(ρρ)/∂ρ = + p/ρ, and
125
The restriction v s ≤ c applies even in the case of the so-called ‘quantum information
transport’ and more fanciful ‘quantum teleportation.’ Within the formalism of
standard quantum theory, there are instantaneous non-local elements. However, as
soon as a measurement is made, information cannot be transferred faster than the
speed of light. This must be so, because quantum theory satisfies Einstein’s relativity
postulates.
The relationship ω = ω(κ) between the component frequencies in a wave and
their wave numbers is called a ‘dispersion relation’, because the relation determines
how the component waves of individual wavelengths separate from each other in
speed and, during refraction, separate in direction. Since ω = v p κ, if the speed of
a sound wave, v p =
√
dp/dρ does not depend on wavelength of the sound, then
v g = dω/dκ = v p , i.e. the phase velocity and the group velocity of the wave
match. Quantum electron waves in relativity theory satisfy ω 2 = c 2 κ 2 + (mc/ ¯
h) 2 ,
so v g = c 2 /v p .
5.4 Energy Carried by Sound
Because sound causes material compression and rarefaction as well as oscillatory
motion of material mass, sound energy has both potential and kinetic forms. This
energy may be transported through the sequence of interactions of each layer of
material with the adjacent layer as the sound propagates. We receive a small amount
of energy each time we hear a sound. For any fluid, the energy per unit volume
is given by u = δE/δV = 1/2ρv 2 + ρρ, where is the internal energy of the
fluid element per unit mass. As sound passes through the fluid, we will assume only
small variations in the density, δρ, and internal energy, δδ, will occur. If the fluid
was initially at rest, the speed is due solely to the vibrational motion. Expanding the
energy of the fluid elements per unit volume in terms of the density variations will
give
u =
1
2
ρv
2
+ δρ
∂(ρ ρ)
∂ρ
+
1
2
(δρ)
2 ∂ 2 (ρ ρ)
∂ρ 2 ,
(5.28)
in which terms of third order and higher in the density variation δρ are dropped.
Usually, as the sound moves through the fluid, there is little time for heat energy
to be transported from the instantaneously pressurized regions to the rarefied regions
of the sound before the wave moves half a wavelength. (The heat diffusion constant
is usually much smaller than the wavelength of the wave times its speed.) Zero
heat transfer defines an adiabatic process, and means that the entropy of each fluid
mass element is unchanged. Thermodynamically, the internal energy of the fluid per
unit mass increases only by heat input or work input, expressed as dd = T ds +
(p/ρ 2 )dρ, where ds is the change in the entropy per unit mass in the fluid element,
and T ds is the heat input. For our adiabatic processes, ds = 0. This makes dd =
(p/ρ 2 )dρ. We have, at constant entropy per unit mass, ∂(ρρ)/∂ρ = + p/ρ, and