Introduction to Acoustics

of radiation modes and their link with sound power and
efficiency is also presented in this section [22.24, 25].
In the case of structure–cavity coupling, the effects
of the cavity field are different from those of
the free field. Strong structural–acoustic modes can appear,
which may contribute to a substantial modification
of the radiated spectrum. These features are illustrated
in Sect. 22.5.3 for the example of a 1-DOF oscillator
coupled to a finite tube.
The concept of coincidence (or critical) frequency
is essential in the radiation of sound by structures. By
comparing the dispersion equations for the structure and
fluid, respectively, one can make a distinction between
frequency domains where the radiation efficiency of the
structure is strong and where it is weak. This is treated
in Sect. 22.5.4 for an isotropic plate.
22.5.1 Longitudinally Vibrating Bar
Coupled to an External Fluid
Model and Modal Projection
We consider here the simple case of a longitudinally
vibrating bar of length L coupled at one end to a 1-D
infinite tube filled with air, which presents a resistive
loading Ra = ρcS at the end of the bar. It is assumed
throughout this section that the bar has a constant crosssectional
area S and that it is clamped at one end (x = 0)
and free at the other (x = L). ρs is the density of the bar,
E its Young’s modulus, cL = √ E/ρs is the longitudinal
wave speed and ξ(x, t) the longitudinal displacement
at a point M at position x along the bar (0 ≤ x ≤ L).
Similarly, ρ denotes the air density, c the speed of sound
and p(x, t) the sound pressure in the tube (L < x < ∞)
(Fig. 22.14). In the absence of an exciting force (free
vibration), the equations of the problem are:
⎧
ρsS
⎪⎨
⎪⎩
∂2ξ ∂t2 = ES∂2 ξ
− Sp(L, t)δ(x − L)
∂x2 for 0 ≤ x ≤ L ,
p(L, t) = ρc˙ξ(L, t) ,
ξ(0, t) = 0 ,
p(x, t) = ρc˙ξ � L, t − x−L
.
�
c
for L < x < ∞
(22.203)
We look for the solution ξ(x, t) expanded in terms of
the in vacuo modes φn(x) of the bar. Both sides of the
first equation in (22.203) are multiplied by any eigenfunction
and integrated over the length of the bar. This
Structural Acoustics and Vibrations 22.5 Structural–Acoustic Coupling 927
gives
�L
0
L
�
−
0
L
0
�
�
�
ρsS φm(x) ¨qm(t) φn(x)dx
m
�
�
ES φm ′′ �
(x)qm(t) φn(x)dx
m
� �
�
�
= Sρc φm(L) ˙qm(t) φn(x)δ(x − L)dx .
m
(22.204)
This equation can be rewritten in a simpler form, by
using the definition of the modal mass
�L
mn = ρsSφ 2 n (x)dx . (22.205)
0
Because of the mass orthogonality property of the eigenfunctions,
the terms of the series in the first integral on
the left-hand side of (22.204) are zero for m �= n,sothat
the integral reduces to mn ¨qn(t). The second integral can
be rewritten:
�L
−
0
ES ω2 m
c 2 L
� �
m
�
φm(x)qm(t) φn(x)dx (22.206)
which, due to stiffness orthogonality properties of the
eigenfunctions, reduces to −ω2 nmnqn(t). Finally, due to
the properties of the delta function, the third integral can
be rewritten:
−Raφn(L) �
φm(L) ˙qm(t) , (22.207)
m
where Ra = ρcS is the acoustic radiation resistance.
Remark. In the case of the clamped–free bar, we would
get φn(L) = (−1) n+1 and, similarly, φm(L) = (−1) m+1 .
However, in what follows, we choose to retain the more
general formulation of (22.207) so that the equations
remain general.
For a 1-D bar of length L coupled to a semi-infinite
tube, the displacement is given by
ξ(x, t) = �
φn(x)qn(t) , (22.208)
n
where the functions of times qn(t) obey the set of coupled
equations
mn ¨qn(t) + mnω 2 nqn(t) =−Raφn(L) �
φm(L) ˙qm(t) . (22.209)
m
Part G 22.5