Introduction to Acoustics

84 Part A Propagation of Sound
Part A 3.13
Rectangular Piston
If the piston is rectangular, with dimensions a and b,and
centered at the origin, the far-field pattern is given by
F(θ,φ) = −iωρv0 sin [(1/2)a sin θ cos φ]
ab
2π (1/2)a sin θ cos φ
× sin [(1/2)b sin θ sin φ]
. (3.526)
(1/2)b sin θ sin φ
Circular Piston
If the piston is circular of radius a, the corresponding
expression is independent of the azimuthal angle and
given by
F(θ) = −iωρv0
2π
Ir (θ)
Ir (0) θ
�a
0
0.2 0.4 0.6 0.8
�
0
2π
a) ka =0 b)
c) ka =4 d)
Ir (θ)
Ir (0)
θ
0.2 0.4 0.6 0.8
Node at 73.3°;
0.001 at 90°
e −ikw0 sin θ cos φ0 dφ0 w0 dw0.
ka =2
Ir (θ)
Ir (0) θ
0.2 0.4 0.6 0.8
ka =8
0.4 0.6 0.8
Nodes at 28.6°
and 72.2°;
0.017 at 39.6°;
0.003 at 90°
(3.527)
Fig. 3.37 Far-field radiation patterns of a vibrating circular
piston in an otherwise rigid baffle for various values of ka.
The quantity plotted is the intensity at a large radial distance
r relative to that when the polar angle θ is zero, which is
[2J1(ξ)/ξ] 2 , with ξ = ka sin θ
The integral over the azimuthal angle on the piston yields
a Bessel function, so one has
F(θ) =−iωρv0
�a
0
J0(kw0 sin θ)w0 dw0 . (3.528)
The recursion relation (3.472) implies that
d
dη J1(η) = J0(η) − 1
η J1(η) , (3.529)
so
ηJ0(η) = η d
dη J1(η) + J1(η) = d
dη (ηJ1(η)) .
(3.530)
Consequently the integration in (3.528) can be performed
in terms of the Bessel function of the first order,
with the result
F(θ) =−iωρv0a 2
� �
J1(ka sin θ)
. (3.531)
ka sin θ
3.13.5 Thermoacoustic Sources
The differential equation (3.516) arises when one suddenly
adds heat to a fluid [3.71,72], as with a laser or by
combustion, so that the entropy s per unit mass changes
according to the thermodynamic relation
ρTDs/Dt = h , (3.532)
so that, to first order, the equation of state (3.87) yields
∂p ′ ∂ρ′
= c2
∂t ∂t + c2β h . (3.533)
cp
Here h is the heat added per unit time and unit volume,
β is the volume expansion coefficient defined by (3.30),
and cp is the specific heat at constant pressure.
Equation (3.533) changes the basic linear acoustic
equations (3.71)and(3.72)to
∂p
∂t + ρc2∇·v = c2β h , (3.534)
cp
∂v
ρ0 +∇p = 0 , (3.535)
∂t
so the energy conservation corollary of (3.273) becomes
∂w β
+∇·I = ph , (3.536)
∂t ρcp
and the wave equation becomes
∇ 2 p − 1
c 2
∂2 p ∂h
=−β . (3.537)
∂t2 cp ∂t