Fluids such as water or air have mass
density and volume elasticity thus they can be approximated with multi degree
of freedom systems for example chain of masses and springs. The elasticity or
volume elasticity helps the fluid to resist being compressed and when the
external forces act on fluid, a fluid tends to return to its original state.
At equilibrium fluid has density ρ [kg/m^2], has a uniform temperature T [K] and
uniform pressure P [Pa, atm, N/m^2] .
These three quantities are connected by an equation of state, which can be
given in two forms. These forms are explicit equation of state which connects
these three quantities in explicit form
or in terms of partial
derivatives, assuming density as function of T and P.
The equation of state is usually
expressed in terms of volume V that occupied by n moles of the fluid. If nM kg
of the fluid occupies volume V the density of the fluid is obviously ρ=nM/V.
The two partial derivatives which are usually employed to measure the equation
of state of the fluid near the equilibrium state are:
$$\begin{align}
& {{\kappa }_{T}}=-\frac{1}{V}{{\left( \frac{\partial V}{\partial P} \right)}_{T}}=\frac{1}{\rho }{{\left( \frac{\partial \rho }{\partial P} \right)}_{T}} \\
& \beta =\frac{1}{V}{{\left( \frac{\partial V}{\partial T} \right)}_{P}}=-\frac{1}{\rho }{{\left( \frac{\partial \rho }{\partial T} \right)}_{P}} \\
\end{align}$$
Quantity
κt the fraction rate of change of volume or density with pressure at constant temperature
is called isothermal compressibility of the fluid.
Quantity
β is the fractional change in volume with temperature at constant pressure is
called the coefficient of thermal expansion of the fluid.
Both
κt and β are functions of P and T though usually their rates of change are not
very large. From κt and β we can compute the other partial derivatives of ρ, P
and T near the equilibrium state of the fluid.
If
the P, β and κt are expressed as functions of ρ and T, then
$${{\left( \frac{\partial P}{\partial T} \right)}_{\rho }}=\frac{\beta }{{{\kappa }_{T}}}$$
Kinetic theory indicates that the
pressure of a perfect gas is caused by its molecular motion and that P=1/3 * ρ
where is the mean square molecular velocity and ro is
the gas density. The compressibility of a perfect gas is thus inversely
proportional to the mean kinetic energy of the gas molecules. The velocity
propagation of the spring-mass system is:
$$v=\sqrt{\frac{1}{\kappa \varepsilon }}$$
Where
k is the compressibility of the spring, and ε is the mass per unit length of
the chain. In acoustics the velocity of propagation of sound waves in a fluid
can be determined from:
$$c=\sqrt{\frac{1}{\kappa \rho }}$$
Where
k is the compressibility of the air and ρ is density. For perfect or ideal
gases
$${{\kappa }_{T}}=\frac{1}{P}=\frac{3}{\rho \left\langle {{v}^{2}} \right\rangle }$$
And
the speed of sound is:
$$c=\sqrt{\frac{1}{{{\kappa }_{T}}\rho }}\simeq \sqrt{\frac{1}{\frac{3}{\rho \left\langle {{v}^{2}} \right\rangle }\rho }}=\sqrt{\frac{\left\langle {{v}^{2}} \right\rangle }{3}}$$
So
we can see from previous equation that the speed of sound in perfect or ideal
gas is approximately equal to the root mean square of the molecular speed.
In
liquids and solid the speed of sound is much larger than the root mean square
speed of vibration of an atom about its equilibrium position. The c or speed of
sound in solids is roughly equal to the mean speed a molecule would have if its
amplitude of vibration were equal to the mean distance between molecules. In
reality the actual amplitudes are much smaller than the mean distance between
molecules so the speed of sound is not equal to a mean speed of a molecule. In
case that the mean speed of molecule is roughly equal to the speed of sound the
solids would melt or vaporize.
Presence
of a sound wave causes the change in density, pressure and temperature in the
fluid and each change is being proportional to the amplitude of the wave. The
pressure changes are most easily measurable, though some sound detectors
measure the motion of the fluid, caused by the sound. You can also measure the
changes in temperature of density but the results are not particularly
accurate; so they are usually computed from the measured pressure change.
In
fluids with very large thermal conductivity the temperature of the fluid is
practically unchanged by the passage of the sound wave. In this case the
isothermal compressibility kt is directly applicable. If the pressure change
caused by the sound p and the equilibrium pressure is P, so that the total
pressure in the presence of sound is P+p then the density of the fluid is:
$$\begin{align}
& \rho +\delta =\rho +{{\left( \frac{\partial \rho }{\partial P} \right)}_{T}}p=\rho +\rho {{\kappa }_{T}}p \\
& \rho =\rho {{\kappa }_{T}}p \\
\end{align}$$
Where
ρ is the equilibrium pressure and δ is the small increase in density produced
by the sound.
If
the frequency of sound is smaller than 10^9 cps or so in gases it is better
approximation to assume that the compression is adiabatic than the entropy content
of the gas remains constant during compression. In the case of adiabatic
compression we need to formulate differential relations between density and
pressure and between temperature and pressure and they are:
$$\begin{align}
& {{\left( \frac{\partial \rho }{\partial P} \right)}_{s}}=\rho {{\kappa }_{s}}=\frac{1}{\gamma }{{\left( \frac{\partial \rho }{\partial P} \right)}_{T}} \\
& {{\kappa }_{s}}=\frac{{{\kappa }_{T}}}{\gamma } \\
& {{\left( \frac{\partial T}{\partial P} \right)}_{s}}=\frac{\gamma -1}{\gamma }{{\left( \frac{\partial T}{\partial P} \right)}_{\rho }}=\left( \gamma -1 \right)\frac{{{\kappa }_{s}}}{\beta } \\
& \rho +\delta =\rho +\rho {{\kappa }_{s}}p \\
& T+\tau =T+\left( \gamma -1 \right)\frac{{{\kappa }_{s}}}{\beta }p \\
\end{align}$$
γ
is the ration between specific heat at constant pressure of the gas and the
specific heat at constant volume. τ is the small change in temperature caused
by the sound wave. For a perfect:
-
Monatomic
gas (Helium) γ=5/3,
-
Diatomic
gas (hydrogen or air) γ=7/5,
-
Polyatomic
gases γ=4/3.
Just
for remainder for the perfect gas the state equation is MP=RTρ.
$$\begin{align}
& {{\kappa }_{s}}=\frac{1}{\gamma P} \\
& {{\left( \frac{\partial \rho }{\partial P} \right)}_{s}}=\frac{\rho }{\gamma P}=\frac{M}{\gamma RT} \\
& {{\left( \frac{\partial T}{\partial P} \right)}_{s}}=\frac{\gamma -1}{\gamma }\frac{T}{P}=\left( \gamma -1 \right)\frac{M}{\gamma R\rho } \\
\end{align}$$
If
the sound is present in fluid than as we know the pressure is P+p, the density
and the temperature are:
$$\begin{align}
& \rho +\delta =\rho +\left( \frac{\rho }{\gamma P} \right)p \\
& \delta =\rho {{\kappa }_{s}}p=\frac{M}{\gamma RT}p \\
& T+\tau =T+\frac{\left( \gamma -1 \right)T}{\gamma P}p=T+\frac{\left( \gamma -1 \right)M}{\gamma R\rho }p \\
& \tau =\frac{\left( \gamma -1 \right)M}{\gamma R\rho }p \\
\end{align}$$
for
adiabatic compression, for a perfect gas.
We
can see that the ratio between the density change and the acoustic pressure p
is independent of the equilibrium pressure but are inversely proportional to
the equilibrium temperature of the gas. Also the ratio between the change in
temperature and sound pressure is independent of temperature but inversely
proportional to the equilibrium density. All these relationships are valid only
to the first order in the small quantity p/P
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