Introduction to Acoustics

temperature,
q =−κ∇T , (3.16)
where κ is the coefficient of thermal conduction.
Transport Properties of Air
Regarding the values of the viscosities and the thermal
conductivity that enter into the above expressions, the
values for air are approximated [3.16–18]by
� �3/2 T T0 + TS
µS = µ0
, (3.17)
T0 T + TS
µB = 0.6µS , (3.18)
� �3/2 T T0 + TA e
κ = κ0
−TB/T0
. (3.19)
T + TA e−TB/T T0
Here TS is 110.4K, TA is 245.4K, TB is 27.6K,
T0 is 300 K, µ0 is 1.846 × 10 −5 kg/(ms), and κ0 is
2.624 × 10 −2 W/(mK).
Transport Properties of Water
The corresponding values of the viscosities and the thermal
conductivity of water depend only on temperature
andaregivenby[3.19, 20]
µS = 1.002 × 10 −3 e −0.0248∆T , (3.20)
µB = 3µS , (3.21)
κ = 0.597 + 0.0017∆T − 7.5×10 −6 (∆T ) 2 .
(3.22)
The quantities that appear in these equations are understood
to be in MKS units, and ∆T is the temperature
relative to 283.16 K (10 ◦ C).
3.2.5 Navier–Stokes–Fourier Equations
The assumptions represented by (3.12, 13, 15), and
(3.16) cause the governing continuum mechanics equations
for a fluid to reduce to
∂ρ
+∇·(ρv) = 0 , (3.23)
∂t
ρ Dv
=−∇p +∇(µB∇·v)
Dt
+ � ∂
(µφij) + gρ, (3.24)
ij
ij
ei
∂x j
ρT Ds
Dt = 1 �
2 µ φ 2 ij + µB (∇·v) 2 +∇·(κ∇T) ,
(3.25)
Basic Linear Acoustics 3.2 Equations of Continuum Mechanics 31
which are known as the Navier–Stokes–Fourier equations
for compressible flow.
3.2.6 Thermodynamic Coefficients
An implication of the existence of an equation of state
of the form of (3.12) is that any thermodynamic variable
can be regarded as a function of any other two (independent)
thermodynamic variables. The pressure p, for
example, can be regarded as a function of ρ and T,orof
s and ρ. In the expression of differential relations, such
as that which gives dp as a linear combination of ds
and dρ, it is helpful to express the coefficients in terms
of a relatively small number of commonly tabulated
quantities. A standard set of such includes:
1. the square of the sound speed,
c 2 � �
∂p
= , (3.26)
∂ρ
s
2. the bulk modulus at constant entropy,
BV = ρc 2 � �
∂p
= ρ , (3.27)
∂ρ
3. the specific heat at constant pressure,
� �
∂s
cp = T , (3.28)
∂T
p
4. the specific heat at constant volume,
� �
∂s
cv = T , (3.29)
∂T
ρ
5. the coefficient of thermal expansion,
� �
∂(1/ρ)
β = ρ
. (3.30)
∂T
p
(The subscripts on the partial derivatives indicate the
independent thermodynamic quantity that is kept fixed
during the differentiation.) The subscript V on BV is included
here to remind one that the modulus is associated
with changes in volume. For a fixed mass of fluid the
decrease in volume per unit volume per unit increase in
pressure is the bulk compressibility, and the reciprocal
of this is the bulk modulus.
The coefficients that are given here are related by the
thermodynamic identity,
γ − 1 = Tβ 2 c 2 /cp , (3.31)
where γ is the specific heat ratio,
γ = cp
. (3.32)
cv
s
Part A 3.2