10 CHAPTER 2. ADIABATIC THEORY
where is the density of the uid, is the shear viscosity, and b is the bulk viscosity2. It
is inherent in the physics of acoustics that we need to consider the uid to be compressible,
thus the divergence of the velocity eld is non-zero, in contrast to most basic uid dynamic
problems. This means that the density should be considered a eld just as the velocity
and the pressure, contrary to being just a material parameter. We also need to include
the bulk viscosity, which is a material parameter related to the viscous dissipation due to
compression of the uid.
As the density is now also a eld variable, the equations (2.1) contains three variables,
v, p, and , but only two equations, thus a third equation is needed to determine the
problem. This third equation is an equation of state, which should describe the state of
the uid i.e. an equation that relates p and . One example of an equation of state is the
ideal gas law, which relates the pressure, volume, and temperature of an ideal gas. For a
uid in general, it is not possible to write down an analytical equation of state, however,
we can use thermodynamics to relate the changes in the thermodynamic variables for small
perturbations to a known equilibrium state. These relations will be described in detail in
Chapter 3, for now we will postulate the result in the case of the adiabatic approximation,
d
0
= s dp: (2.2)
This equation states that the relative change in the local density d
equals the change
in pressure times the isentropic compressibility of the uid s, which is valid for small
perturbations d and dp to an equilibrium state. Equation (2.2) follows directly from the
denition of the isentropic compressibility
s
1
0
d
dp
s
: (2.3)
Here the subscript s implies that the dierentiation is evaluated for constant entropy
ds = 0, which is the adiabatic approximation. Since the isentropic speed of sound is
dened as c2s
��dp
d
s, the isentropic compressibility can be expressed as s = 1=(0c2s
),
and the adiabatic equation of state can thus be written in the customary form
dp = c2s
d: (2.4)
The equation of state Eq. (2.4) allows us to eliminate the pressure eld from the governing
equations (2.1), by substituting rp = c2s
r. The NavierStokes equation and the
continuity equation are now expressed in terms of v and only,
@tv + (v r) v = ��c2s
r + r2v +
1
3 + b
r(r v) ; (2.5a)
@t = ��r (v) : (2.5b)
These two equations describe the relation between the oscillating density and velocity
elds.
2The bulk viscosity is sometimes referred to as the second viscosity and represented by the letter 17.
Its value is often incorrectly taken to be zero, however, for water the values of and b have the same
order of magnitude.