## 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

{ !