Sound is a vibration that travels through an
elastic medium as a
wave. The
speed of sound
describes how far this wave travels in a given amount of time. In
dry air at , the speed of sound is . This equates to , or about one
kilometre in three seconds and about one mile in five seconds. This
figure for
air (or any given
gas) increases with gas temperature (equations are given
below), but is nearly independent of
pressure or
density for a
given gas. For different gases, the speed of sound is dependent on
the mean
molecular weight of the
gas, and to a lesser extent upon the ways in which the
molecules of the gas can store
heat energy from
compression (since sound in gases is a type of
compression).
Although "the speed of sound" is commonly used to refer
specifically to the speed of sound waves in
air, the speed of sound can be measured
in virtually any substance. Sound travels faster in
liquids and non-porous
solids
(5,120 m/s in iron) than it does in air, traveling about 4.3 times
faster in
water (1,484 m/s) than in air at 20
degrees Celsius.
Additionally, in solids, there occurs the possibility of two
different types of sound waves: one type (called "longitudinal
waves" when in solids) is associated with compression (the same as
all sound waves in fluids) and the other is associated with
shear stresses, which cannot occur in
fluids. These two types of waves have
different speeds, and (for example in an
earthquake) may thus be initiated at the same
time but arrive at distant points at appreciably different times.
The speed of compression-type waves in all media is set by the
medium's
compressibility and
density, and the speed of
shear waves in solids is set by the material's
rigidity, compressibility and
density.
Basic concept
The transmission of sound can be illustrated by using a
toy model consisting of an array of balls
interconnected by springs. For real material the balls represent
molecules and the springs represent the bonds between them. Sound
passes through the model by compressing and expanding the springs,
transmitting energy to neighboring balls, which transmit energy to
their springs, and so on. The speed of sound through the
model depends on the stiffness of the springs (stiffer springs
transmit energy more quickly). Effects like dispersion and
reflection can also be understood using this model.
In a real material, the stiffness of the springs is called the
elastic modulus, and the mass
corresponds to the
density. All other things
being equal, sound will travel more slowly in denser materials, and
faster in stiffer ones. For instance, sound will travel faster in
iron than uranium, and faster in hydrogen than nitrogen, due to the
lower density of the first material of each set. At the same time,
sound will travel faster in solids than in liquids and faster in
liquids than in gases, because the internal bonds in a solid are
much stronger than the bonds in a liquid, as are the bonds in
liquids compared to gases.
Some textbooks mistakenly state that the speed of sound increases
with increasing density. This is usually illustrated by presenting
data for three materials, such as air, water and steel, which also
have vastly different compressibilities which more than make up for
the density differences. An illustrative example of the two effects
is that sound travels only 4.3 times faster in water than air,
despite enormous differences in compressibility of the two media.
The reason is that the larger density of water, which works to
slow sound in water relative to air, nearly makes up for
the compressibility in the two media.
General formula
In general, the speed of sound
c is given byc =
\sqrt{\frac{C}{\rho}}where
- C is a coefficient of
stiffness (or the modulus of bulk elasticity for gas
mediums),
- \rho is the density
Thus the speed of sound increases with the stiffness (the
resistance of an elastic body to deformation by an applied force)
of the material, and decreases with the density.For general
equations of state, if classical mechanics is used, the speed of
sound c is given byc^2=\frac{\partial p}{\partial\rho}where
differentiation is taken with respect to adiabatic change.
If
relativistic effects are
important, the speed of sound may be calculated from the
relativistic Euler
equations.
In a
non-dispersive medium sound speed is
independent of sound frequency, so the speeds of energy transport
and sound propagation are the same. For audible sounds air is a
non-dispersive medium. But air does contain a small amount of
CO
2 which
is a dispersive medium, and it
introduces dispersion to air at
ultrasonic frequencies (> 28
kHz).
In a
dispersive medium sound speed is a function
of sound frequency, through the
dispersion relation. The spatial and
temporal distribution of a propagating disturbance will continually
change. Each frequency component propagates at its own
phase velocity, while the energy of the
disturbance propagates at the
group
velocity. The same phenomenon occurs with light waves; see
optical
dispersion for a description.
Dependence on the properties of the medium
The speed of sound is variable and depends on the properties of the
substance through of which the wave is travelling. In solids, the
speed of longitudinal waves depend on the stiffness to tensile
stress, and the density of the medium. In fluids, the medium's
compressibility and density are the important factors.
In gases, compressibility and density are related, making other
compositional effects and properties important, such as temperature
and molecular composition. In low
molecular weight gases, such as
helium, sound propagates faster compared to heavier
gases, such as
xenon (for monatomic gases the
speed of sound is about 68% of the mean speed that molecules move
in the gas). For a given
ideal gas the
sound speed depends only on its
temperature. At a constant temperature, the
ideal gas
pressure has no effect on the
speed of sound, because pressure and
density
(also proportional to pressure) have equal but opposite effects on
the speed of sound, and the two contributions cancel out exactly.
In a similar way, compression waves in solids depend both on
compressibility and density—just as in liquids—but in gases the
density contributes to the compressibility in such a way that some
part of each attribute factors out, leaving only a dependence on
temperature, molecular weight, and heat capacity (see derivations
below). Thus, for a single given gas (where molecular weight does
not change) and over a small temperature range (where heat capacity
is relatively constant), the speed of sound becomes dependent on
only the temperature of the gas.
In non-ideal gases, such as a
van
der Waals gas, the proportionality is not exact, and there is a
slight dependence of sound velocity on the gas pressure.
Humidity has a small, but measurable effect on sound speed (causing
it to increase by about 0.1%-0.6%), because
oxygen and
nitrogen molecules
of the air are replaced by lighter molecules of
water. This is a simple mixing effect.
Implications for atmospheric acoustics
In the
Earth's atmosphere, the
most important factor affecting the speed of sound is the
temperature (see
Details below). Since temperature and
thus the speed of sound normally decrease with increasing altitude,
sound is
refracted upward, away from
listeners on the ground, creating an
acoustic shadow at some distance from the
source. The decrease of the sound speed with height is referred to
as a negative
sound speed
gradient. However, in the
stratosphere, the speed of sound increases with
height due to heating within the
ozone
layer, producing a positive sound speed gradient.
Practical formula for dry air
The approximate speed of sound in dry (0% humidity) air, in meters
per second (
m·s−1), at temperatures
near 0 °C, can be calculated from:c_{\mathrm{air}} = (331{.}3
+ (0{.}606^{\circ}\mathrm{C}^{-1} \cdot \vartheta)) \ \mathrm{m
\cdot s^{-1}}\,where \vartheta is the temperature in degrees
Celsius (°C).
This equation is derived from the first two terms of the
Taylor expansion of the following more
accurate equation:
- c_{\mathrm{air}} = 331.3 \mathrm{m \cdot s^{-1}}
\sqrt{1+\frac{\vartheta}{273.15^{\circ}\mathrm{C}}}\
The value of 331.3 m/s, which represents the 0 °C speed, is
based on theoretical (and some measured) values of the
heat capacity ratio, \gamma, as well as
on the fact that at 1
atm real air
is very well described by the ideal gas approximation. Commonly
found values for the speed of sound at 0 °C may vary from
331.2 to 331.6 due to the assumptions made when it is calculated.
If ideal gas \gamma is assumed to be 7/5 = 1.4 exactly, the
0 °C speed is calculated (see section below) to be 331.3 m/s,
the coefficient used above.
This equation is correct to a much wider temperature range, but
still depends on the approximation of heat capacity ratio being
independent of temperature, and will fail, particularly at higher
temperatures. It gives good predictions in relatively dry, cold,
low pressure conditions, such as the Earth's
stratosphere. A derivation of these equations
will be given in the following section.
Details
Speed in ideal gases and in air
For a gas,
K (the
bulk modulus
in equations above, equivalent to C, the coefficient of stiffness
in solids) is approximately given byK = \gamma \cdot p
thus
c = \sqrt{\gamma \cdot {p \over \rho}}
Where:
- \gamma is the adiabatic index
also known as the isentropic expansion factor. It is the
ratio of specific heats of a gas at a constant-pressure to a gas at
a constant-volume(C_p/C_v), and arises because a classical sound
wave induces an adiabatic compression, in which the heat of the
compression does not have enough time to escape the pressure pulse,
and thus contributes to the pressure induced by the
compression.
- p is the pressure.
- \rho is the density
Using the
ideal gas law to replace p with
nRT/
V, and replacing
ρ with
nM/
V, the equation for an ideal gas
becomes:
c_{\mathrm{ideal}} = \sqrt{\gamma \cdot {p \over \rho}} =
\sqrt{\gamma \cdot R \cdot T \over M}= \sqrt{\gamma \cdot k \cdot T
\over m}
where
- c_{\mathrm{ideal}} is the speed of sound in an ideal gas.
- R (approximately 8.3145 J·mol−1·K−1) is
the molar gas constant.[22866]
- k is the Boltzmann
constant
- \gamma (gamma) is the adiabatic
index (sometimes assumed 7/5 = 1.400 for diatomic molecules
from kinetic theory, assuming from quantum theory a temperature
range at which thermal energy is fully partitioned into rotation
(rotations are fully excited), but none into vibrational modes.
Gamma is actually experimentally measured over a range from 1.3991
to 1.403 at 0 degrees Celsius, for air. Gamma is assumed from
kinetic theory to be exactly 5/3 = 1.6667 for monoatomic molecules
such as noble gases).
- T is the absolute temperature in kelvins.
- M is the molar mass in kilograms per
mole. The mean molar mass for dry air is
about 0.0289645 kg/mol.
- m is the mass of a single molecule in kilograms.
This equation applies only when the sound wave is a small
perturbation on the ambient condition, and the certain other noted
conditions are fulfilled, as noted below. Calculated values for
c_{\mathrm{air}} have been found to vary slightly from
experimentally determined values.
Newton famously considered the speed of
sound before most of the development of
thermodynamics and so incorrectly used
isothermal calculations instead of
adiabatic. His result was missing the
factor of \gamma but was otherwise correct.
Numerical substitution of the above values gives the ideal gas
approximation of sound velocity for gases, which is accurate at
relatively low gas pressures and densities (for air, this includes
standard Earth sea-level conditions). Also, for diatomic gases the
use of \ \gamma\, = 1.4000 requires that the gas exist in a
temperature range high enough that rotational heat capacity is
fully excited (i.e., molecular rotation is fully used as a heat
energy "partition" or reservoir); but at the same time the
temperature must be low enough that molecular vibrational modes
contribute no heat capacity (i.e., insigificant heat goes into
vibration, as all vibrational quantum modes above the
minimum-energy-mode, have energies too high to be populated by a
significant number of molecules at this temperature). For air,
these conditions are fulfilled at room temperature, and also
temperatures considerably below room temperature (see tables
below). See the section on gases in
specific heat capacity for a more
complete discussion of this phenomenon.
If temperatures in degrees
Celsius(°C) are
to be used to calculate air speed in the region near 273
kelvins, then Celsius temperature \vartheta = T -
273.15 may be used.
- c_{\mathrm{ideal}} = \sqrt{\gamma \cdot R \cdot T} =
\sqrt{\gamma \cdot R \cdot (\vartheta +
273.15\;^{\circ}\mathrm{C})}
- c_{\mathrm{ideal}} = \sqrt{\gamma \cdot R \cdot 273.15} \cdot
\sqrt{1+\frac{\vartheta}{273.15\;^{\circ}\mathrm{C}}}
For dry air, where \vartheta\, (theta) is the temperature in
degrees
Celsius(°C).
Making the following numerical substitutions:\ R =
R_*/M_{\mathrm{air}}, where \ R_* = 8.314510 \cdot \mathrm{J \cdot
mol^{-1}} \cdot K^{-1} is the molar gas constant, \
M_{\mathrm{air}} = 0.0289645 \cdot \mathrm{kg \cdot mol^{-1}}, and
using the ideal diatomic gas value of \ \gamma\, = 1.4000
Then:
- c_{\mathrm{air}} = 331.3 \ \mathrm{m \cdot s^{-1}}
\sqrt{1+\frac{\vartheta^{\circ}\mathrm{C}}{273.15\;^{\circ}\mathrm{C}}}
Using the first two terms of the Taylor expansion:
c_{\mathrm{air}} = 331.3 \ \mathrm{m \cdot s^{-1}} (1 +
\frac{\vartheta^{\circ}\mathrm{C}}{2 \cdot
273.15\;^{\circ}\mathrm{C}}) \,
c_{\mathrm{air}} = ( 331{.}3 + 0{.}606\;^{\circ}\mathrm{C}^{-1}
\cdot \vartheta)\ \mathrm{ m \cdot s^{-1}}\,
The derivation includes the two approximate equations which were
given in the introduction.
Effects due to wind shear
The speed of sound varies with temperature. Since temperature and
sound velocity normally decrease with increasing altitude, sound is
refracted upward, away from listeners on
the ground, creating an
acoustic
shadow at some distance from the source. Wind shear of 4
m·s
−1·km
−1 can produce refraction equal to a
typical temperature
lapse rate of
7.5 °C/km. Higher values of wind gradient will refract sound
downward toward the surface in the downwind direction, eliminating
the acoustic shadow on the downwind side. This will increase the
audibility of sounds downwind. This downwind refraction effect
occurs because there is a wind gradient; the sound is not being
carried along by the wind.
For sound propagation, the exponential variation of wind speed with
height can be defined as follows:
- \ U(h) = U(0) h ^ \zeta
- \ \frac {dU} {dH} = \zeta \frac {U(h)} {h}
where:
- \ U(h) = speed of the wind at height \ h, and \ U(0) is a
constant
- \ \zeta = exponential coefficient based on ground surface
roughness, typically between 0.08 and 0.52
- \ \frac {dU} {dH} = expected wind gradient at height h
In the
1862 American Civil War Battle of Iuka
, an acoustic shadow, believed to have been enhanced
by a northeast wind, kept two divisions of Union soldiers out of
the battle, because they could not hear the sounds of battle only
six miles downwind.
Tables
In the
standard
atmosphere:
- T0 is 273.15 K (= 0 °C = 32 °F),
giving a theoretical value of 331.3 m·s−1 (=
1086.9 ft/s = 1193 km·h−1 = 741.1 mph =
644.0 knots). Values ranging from
331.3-331.6 may be found in reference literature, however.
- T20 is 293.15 K (= 20 °C =
68 °F), giving a value of 343.2 m·s−1 (=
1126.0 ft/s = 1236 km·h−1 = 767.8 mph =
667.2 knots).
- T25 is 298.15 K (= 25 °C =
77 °F), giving a value of 346.1 m·s−1 (=
1135.6 ft/s = 1246 km·h−1 = 774.3 mph =
672.8 knots).
In fact, assuming an
ideal gas, the speed
of sound
c depends on temperature only,
not on the
pressure or
density (since these change
in lockstep for a given temperature and cancel out). Air is almost
an ideal gas. The temperature of the air varies with altitude,
giving the following variations in the speed of sound using the
standard atmosphere -
actual conditions may vary.
Given normal atmospheric conditions, the temperature, and thus
speed of sound, varies with altitude:
Altitude |
Temperature |
m·s−1 |
km·h−1 |
mph |
knots |
Sea level |
15 °C (59 °F) |
340 |
1225 |
761 |
661 |
11 000 m−20 000 m
(Cruising altitude of commercial jets,
and first supersonic flight)
|
−57 °C (−70 °F) |
295 |
1062 |
660 |
573 |
29 000 m (Flight of X-43A) |
−48 °C (−53 °F) |
301 |
1083 |
673 |
585 |
|
Effect of frequency and gas composition
The medium in which a sound wave is travelling does not always
respond adiabatically, and as a result the speed of sound can vary
with frequency.
The limitations of the concept of speed of sound due to extreme
attenuation are also of concern. The attenuation which exists at
sea level for high frequencies applies to successively lower
frequencies as atmospheric pressure decreases, or as the
mean free path increases. For this reason,
the concept of speed of sound (except for frequencies approaching
zero) progressively loses its range of applicability at high
altitudes.: The standard equations for the speed of sound apply
with reasonable accuracy only to situations in which the wavelength
of the soundwave is considerably longer than the mean free path of
molecules in a gas.
The molecular composition of the gas contributes both as the mass
(M) of the molecules, and their heat capacities, and so both have
an influence on speed of sound. In general, at the same molecular
mass, monatomic gases have slightly higher sound speeds (over 9%
higher) because they have a higher \gamma (5/3 = 1.66...) than
diatomics do (7/5 = 1.4). Thus, at the same molecular mass, the
sound speed of a monatomic gas goes up by a factor of
{ c_{\mathrm{gas: monatomic}} \over c_{\mathrm{gas: diatomic}} } =
\sqrt{{{{5 / 3} \over {7 / 5}}}} = \sqrt{25 \over 21} =
1.091...
This gives the 9 % difference, and would be a typical ratio for
sound speeds at room temperature in
helium
vs.
deuterium, each with a molecular
weight of 4. Sound travels faster in helium than deuterium because
adiabatic compression heats helium more, since the helium molecules
can store heat energy from compression only in translation, but not
rotation. Thus helium molecules (monatomic molecules) travel faster
in a sound wave and transmit sound faster. (Sound generally travels
at about 70% of the mean molecular speed in gases).
Note that in this example we have assumed that temperature is low
enough that heat capacities are not influenced by molecular
vibration (see
heat capacity).
However, vibrational modes simply cause gammas which decrease
toward 1, since vibration modes in a polyatomic gas gives the gas
additional ways to store heat which do not affect temperature, and
thus do not affect molecular velocity and sound velocity. Thus, the
effect of higher temperatures and vibrational heat capacity acts to
increase the difference between sound speed in monatomic vs.
polyatomic molecules, with the speed remaining greater in
monatomics.
Mach number
Mach number, a useful quantity in aerodynamics, is the ratio of air
speed to the local speed of sound. At
altitude, for reasons explained, Mach number is a function of
temperature.
Aircraft
flight instruments,
however, operate using pressure differential to compute Mach
number; not temperature. The assumption is that a particular
pressure represents a particular altitude and, therefore, a
standard temperature. Aircraft flight instruments need to operate
this way because the stagnation pressure sensed by a
Pitot tube is dependent on altitude as well as
speed.
Assuming air to be an
ideal gas, the
formula to compute Mach number in a subsonic compressible flow is
derived from
Bernoulli's
equation for
M<1:></1:>
-
{M}=\sqrt{5\left[\left(\frac{q_c}{P}+1\right)^\frac{2}{7}-1\right]}
where
- M is Mach number
- q_c is dynamic pressure and
- P is static pressure.
The formula to compute Mach number in a supersonic compressible
flow is derived from the
Rayleigh
Supersonic Pitot equation:
-
{M}=0.88128485\sqrt{\left[\left(\frac{q_c}{P}+1\right)\left(1-\frac{1}{[7M^2]}\right)^{2.5}\right]}
where
- M is Mach number
- q_c is dynamic pressure measured behind a normal shock
- P is static pressure.
As can be seen,
M appears on both sides of the equation.
The easiest method to solve the supersonic
M calculation
is to enter both the subsonic and supersonic equations into a
computer spreadsheet such as
Microsoft
Excel,
OpenOffice.org Calc, or
some equivalent program. First determine if
M is indeed
greater than 1.0 by calculating
M from the subsonic
equation. If
M is greater than 1.0 at that point, then use
the value of
M from the subsonic equation as the initial
condition in the supersonic equation. Then perform a simple
iteration of the supersonic equation, each time using the last
computed value of
M, until
M converges to a
value—usually in just a few iterations.
Experimental methods
A range of different methods exist for the measurement of sound in
air.
The first man to successfully measure the speed of Sound was
William Derham
Single-shot timing methods
The simplest concept is the measurement made using two
microphones and a fast recording device such as a
digital storage scope. This method uses the
following idea.
If a sound source and two microphones are arranged in a straight
line, with the sound source at one end, then the following can be
measured:
1. The distance between the microphones (
x), called
microphone basis.2. The time of arrival between the signals (delay)
reaching the different microphones (
t)
Then
v =
x /
t
An older method is to create a sound at one end of a field with an
object that can be seen to move when it creates the sound. When the
observer sees the sound-creating device act they start a stopwatch
and when the observer hears the sound they stop their stopwatch.
Again using
v = x /
t you can calculate the speed
of sound. A separation of at least 200 m between the two
experimental parties is required for good results with this
method.factors like wind speed and direction have to be taken into
account with this method
Other methods
In these methods the
time measurement has been
replaced by a measurement of the inverse of time (
frequency).
Kundt's tube is an example of an
experiment which can be used to measure the speed of sound in a
small volume. It has the advantage of being able to measure the
speed of sound in any gas. This method uses a powder to make the
nodes and
antinodes visible to the human eye. This is an
example of a compact experimental setup.
A
tuning fork can be held near the mouth
of a long
pipe which is dipping into
a barrel of
water. In this system it is the
case that the pipe can be brought to resonance if the length of the
air column in the pipe is equal to
({1+2n}λ/4) where
n is an integer. As the
antinodal
point for the pipe at the open end is slightly outside the mouth of
the pipe it is best to find two or more points of resonance and
then measure half a wavelength between these.
Here it is the case that
v =
fλ
Non-gaseous media
Speed of sound in solids
In a solid, there is a non-zero stiffness both for volumetric and
shear deformations. Hence, it is possible to generate sound waves
with different velocities dependenton the deformation mode. Sound
waves generating volumetric deformations (compressions) and shear
deformations are called longitudinal waves and shear waves,
respectively. In
earthquakes, the
corresponding seismic waves are called
P-waves and
S-waves,
respectively. The sound velocities of these two type wavesare
respectively given by:
c_{\mathrm{l}} = \sqrt {\frac{K+\frac{4}{3}G}{\rho}} = \sqrt
{\frac{E (1-\nu)}{\rho (1+\nu)(1 - 2 \nu)}}
c_{\mathrm{s}} = \sqrt {\frac{G}{\rho}},
where
K and
G are the
bulk modulus and
shear
modulus of the elastic materials, respectively.
E is
the
Young's modulus, and \nu is
Poisson's ratio.
For example, for a typical steel alloy,
K = 170 GPa,
G = 80 GPa and \rho = 7700 kg/m
3, yielding
a longitudinal velocity
cl of6000 m/s.This is
in reasonable agreement with
cl=5930 m/s
measured experimentally for a (possibly different) type of
steel.
The shear velocity
cs is estimated at 3200 m/s
using the same numbers.
Speed of sound in liquids
In a fluid the only non-zero stiffness is to volumetric deformation
(a fluid does not sustain shear forces).
Hence the speed of sound in a fluid is given byc_{\mathrm{fluid}} =
\sqrt {\frac{K}{\rho}}
where
- K is the bulk modulus of
the fluid
Water
The speed of sound in water is of interest to anyone using
underwater sound as a tool, whether in
a laboratory, a lake or the ocean. Examples are
sonar,
acoustic
communication and
acoustical
oceanography. See
Discovery of Sound in the Sea for other examples of
the uses of sound in the ocean(by both man and other animals). In
fresh water, sound travels at about 1497 m/s at 25 °C.
See
Technical Guides - Speed of Sound in Pure Water for an
online calculator.
Seawater
In salt water that is free of air bubbles or suspended sediment,
sound travels at about 1560 m/s. The speed of sound in
seawater depends on pressure (hence depth), temperature (a change
of 1 °C ~ 4 m/s), and
salinity (achange of 1‰ ~ 1 m/s),
and empirical equations have been derived to accurately calculate
sound speed from these variables. Other factors affecting sound
speed are minor. For more information see Dushaw et al.
A simple empirical equation for the speed of sound in sea water
with reasonable accuracy for the world's oceans is due to
Mackenzie:
- c(T, S, z) =
a1 + a2T +
a3T2 +
a4T3 +
a5(S - 35) +
a6z +
a7z2 +
a8T(S - 35) +
a9Tz3
where
T,
S, and
z are temperature in
degrees Celsius, salinity in parts per thousand and depth in
metres, respectively. The constants
a1,
a2, ...,
a9 are:
- a1 = 1448.96,
a2 = 4.591,
a3 = -5.304×10-2,
a4 = 2.374×10-4,
a5 = 1.340,
a6 = 1.630×10-2,
a7 = 1.675×10-7,
a8 = -1.025×10-2,
a9 = -7.139×10-13
with check value 1550.744 m/s for
T=25 °C,
S=35‰,
z=1000 m. This equation has a standard
error of 0.070 m/s for salinities between 25 and 40
ppt. See
Technical Guides - Speed of Sound in Sea-Water for an
online calculator.
Other equations for sound speed in sea water are accurate over a
wide range of conditions, but are far more complicated, e.g., that
by V. A. Del Grosso and the Chen-Millero-Li Equation.
Speed in plasma
The speed of sound in a
plasma for
the common case that the electrons are hotter than the ions (but
not too much hotter) is given by the formula (see
here)
- c_s = (\gamma ZkT_e/m_i)^{1/2} = 9.79\times10^3\,(\gamma
ZT_e/\mu)^{1/2}\,\mbox{m/s}
In contrast to a gas, the pressure and the density are provided by
separate species, the pressure by the electrons and the density by
the ions. The two are coupled through a fluctuating electric
field.
Gradients
When sound spreads out evenly in all directions, the intensity
drops in proportion to the inverse square of the distance. However,
in the ocean there is a layer called the 'deep sound channel' or
SOFAR channel which can confine sound
waves at a particular depth, allowing them to travel much further.
In the SOFAR channel, the speed of sound is lower than that in the
layers above and below. Just as light waves will refract towards a
region of higher
index, sound waves
will
refract towards a region where their
speed is reduced. The result is that sound gets confined in the
layer, much the way light can be confined in a sheet of glass or
optical fiber.
A similar effect occurs in the atmosphere.
Project Mogul successfully used this effect to
detect a nuclear explosion at a considerable distance.
See also
References
- Applied Physics Laboratory – University of Washington,
1994
External links