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5 Acoustics in Biology and Medicine
5.10 Non-linear Sound Waves
The linear wave equation (5.22) applies when the pressure amplitude is small
compared to the ambient pressure, the variation of the material density is small
compared to the ambient density, and the speed of the fluid as it moves in response
to a sound wave is small compared to the speed of sound in the material.
Phenomena such as surface water waves breaking at the beach shore, shock
waves produced by whips and supersonic aircraft, bumble bee flight, and other fluid
turbulence produced by sound can be explained by the non-linear effects coming
from the (v · ∇)v term in the Navier-Stokes equation (4.42).
If density variations are not small, such as in fluids containing bubbles, the
non-linear effects again become important. As sound passes, the pressure variation
changes the volume of bubbles, and thus the density of the medium. Non-linear
effects cause harmonics in the wave as it propagates. A wave in bubble water
initially sinusoidal can evolve into a saw-tooth shaped wave as the wave progresses.
5.11 Absorption of Sound
The source of sound attenuation may be baffling.
—wcp
Besides impedance mismatch causing sound reflection, sound also tends to
reflect better from hard surfaces than from soft ones, because during the reflection
from a soft material, some sound energy may be lost by conversion into heat. For
example, a piece of cloth will absorb energy when the fibers within the cloth flex and
heat while responding to the incoming sound. The sound ‘absorptivity’ of a surface
is defined as the fractional power dissipation when sound encounters that surface.
Even within one material, sound scattering and viscous loss causes sound energy
to be dissipated. The loss of sound intensity as a sound wave propagates is referred
to as ‘sound attenuation’ of the wave. For a traveling wave, scattering and viscous
energy loss means that the further the wave goes, the more it loses kinetic and elastic
potential energies. Since the loss over a short distance is proportional to the distance
traveled and the energy present, the intensity through a homogeneous material must
satisfy 18
18 This follows from dI ∝ I dx, or
dI/I = −α
dx, so ln I = −αx + C or I = I 0 exp (−αx).
The relationship is quite ‘universal’. The only important assumptions are that the absorbing
material is homogeneous along the direction x and that the loss of intensity is proportional to
the intensity. We will use this same behavior when considering the attenuation of light traveling
through a lossy material (as in Eq. (7.14) for microwaves and in the Beer-Lambert Law Eq. (7.19)).
There are exceptions, such as the intensity loss with distance for a beam of charged particles sent
through tissue, because the loss depends non-linearly on the beam energy flux.
5 Acoustics in Biology and Medicine
5.10 Non-linear Sound Waves
The linear wave equation (5.22) applies when the pressure amplitude is small
compared to the ambient pressure, the variation of the material density is small
compared to the ambient density, and the speed of the fluid as it moves in response
to a sound wave is small compared to the speed of sound in the material.
Phenomena such as surface water waves breaking at the beach shore, shock
waves produced by whips and supersonic aircraft, bumble bee flight, and other fluid
turbulence produced by sound can be explained by the non-linear effects coming
from the (v · ∇)v term in the Navier-Stokes equation (4.42).
If density variations are not small, such as in fluids containing bubbles, the
non-linear effects again become important. As sound passes, the pressure variation
changes the volume of bubbles, and thus the density of the medium. Non-linear
effects cause harmonics in the wave as it propagates. A wave in bubble water
initially sinusoidal can evolve into a saw-tooth shaped wave as the wave progresses.
5.11 Absorption of Sound
The source of sound attenuation may be baffling.
—wcp
Besides impedance mismatch causing sound reflection, sound also tends to
reflect better from hard surfaces than from soft ones, because during the reflection
from a soft material, some sound energy may be lost by conversion into heat. For
example, a piece of cloth will absorb energy when the fibers within the cloth flex and
heat while responding to the incoming sound. The sound ‘absorptivity’ of a surface
is defined as the fractional power dissipation when sound encounters that surface.
Even within one material, sound scattering and viscous loss causes sound energy
to be dissipated. The loss of sound intensity as a sound wave propagates is referred
to as ‘sound attenuation’ of the wave. For a traveling wave, scattering and viscous
energy loss means that the further the wave goes, the more it loses kinetic and elastic
potential energies. Since the loss over a short distance is proportional to the distance
traveled and the energy present, the intensity through a homogeneous material must
satisfy 18
18 This follows from dI ∝ I dx, or
dI/I = −α
dx, so ln I = −αx + C or I = I 0 exp (−αx).
The relationship is quite ‘universal’. The only important assumptions are that the absorbing
material is homogeneous along the direction x and that the loss of intensity is proportional to
the intensity. We will use this same behavior when considering the attenuation of light traveling
through a lossy material (as in Eq. (7.14) for microwaves and in the Beer-Lambert Law Eq. (7.19)).
There are exceptions, such as the intensity loss with distance for a beam of charged particles sent
through tissue, because the loss depends non-linearly on the beam energy flux.