ABSTRACT
As an expression for sound insulation performance, we may use transmission loss (TL), which is defined
as (also see Chapter 17 and Chapter 18)
TL ¼ 10 log
t
¼ 10 log
I
I
ð19:2Þ
Consider a plane sound wave incident on a impermeable infinite plate at angle u, which is placed in a
uniform air space as shown in Figure 19.1. The sound pressure of the incident, reflected, and transmitted
given by
p
¼ P
e
p
¼ P
e
ð19:3Þ
p
¼ P
e
where P
; P
; and P
are the sound pressure
amplitudes of incident, reflected, and transmitted
waves, respectively; v is angular frequency; k is the
wave number of the sound wave; c is the speed
of sound, respectively in the air. Assuming
that the plate is sufficiently thin compared
with the wavelength of the incident sound wave,
the vibration velocities on the incident and
transmitted surfaces of the plate are equal. Then vibration velocity, u, of the plate in the x direction is
equal to the particle velocity of the incident and transmitted sound waves, and we obtain relations
u ¼ 2
jvr
›ðp
þ p
Þ
›x
¼ 2
jvr
›p
›x
ð19:4Þ
p
þ p
2 p
u
¼ Z
ð19:5Þ
where r is the air density and Z
is the mechanical impedance of the plate per unit area. From these
equations, the transmission coefficient, t
; and then the transmission loss, TL
; at the incident angle, u; are
obtained according to
TL
¼ 10 log
t
¼ 10 log
p
p
¼ 10 log 1þ
Z
cos u
2rc
ð19:6Þ
19.1.2.1 Coincidence Effect
Consider the vibration of the plate in the x-y plane shown in Figure 19.1. Denoting by m the surface
density, and by B the bending stiffness per unit length of the plate, the equation of motion of the plate is
given by
m
›
j
›t
þ Bð1þ jhÞ
›
j
›y
¼ p
þ p
2 p
; B ¼
Eh
12ð12 n
Þ
ø
Eh
ð19:7Þ
where
j ¼ displacement in the x direction
E ¼ Young’s modulus of the plate
h ¼ thickness of the plate
h ¼ loss factor of the plate
n ¼ Poisson’s ratio of the plate
The plane sound wave of angular frequency, v; and of incidence angle, u; causes a bending wave in the
plate where displacement is assumed to be j ¼ j
e
; as a solution of Equation 19.7. Hence, the
mechanical impedance per unit area is obtained:
Z
¼
p
þ p
2 p
›j=›t
¼ h
Bk
v
þ j vm2
Bk
v
{ !
