4.5 CHANGES IN MEDIA WITHNORMAL INCIDENCE

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We will analyze the transmission of sound from one material to another

material in this section. As shown in Fig. 4-3, when a sound wave moving in

one fluid strikes the surface or interface of a different material, a portion of

the acoustic energy is reflected, and a portion is transmitted into the second

medium. Let us consider the case of transmission from one material into

another one in which the sound wave strikes the interface at normal incidence

or with zero angle between the direction of the sound wave and the

normal drawn to the interface.

From the previous discussion of plane sound waves, we know that the

acoustic pressure may be written in the following form for a wave moving in

material 1:

p1рx; tЮ ј A1 ejр!t_k1xЮ ю B1 ejр!tюk1xЮ (4-80Ю

рIncident waveЮ ю рReflected waveЮ

Transmission of Sound 91

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Similarly, assuming no reflections in material 2 or that material 2 is very

large in extent, the instantaneous acoustic pressure for the transmitted wave

in material 2 may be written as follows:

p2рx; tЮ ј A2 ejр!t_k2xЮ (4-81)

The instantaneous particle velocities in the two materials may be written

from Eq. (4-47), noting that the reflected wave is traveling in the _x-direction:

u1рx; tЮ ј р1=Z1ЮЅA1 ejр!t_k1xЮ _ B1 ejр!tюk1xЮ_ (4-82)

u2рx; tЮ ј р1=Z2ЮA2 ejр!t_k2xЮ (4-83)

The quantities Z1 ј _1c1 and Z2 ј _2c2 are the characteristic impedances

for materials 1 and 2, respectively.

At the interface between the two materials, the instantaneous acoustic

pressure in material 1 must be equal to the instantaneous acoustic pressure

in material 2. Using this fact, at x ј 0, we find the following relation

between the coefficients:

A1 ю B1 ј A2 (4-84)

Similarly, the instantaneous particle velocities must be the same in each

media at the interface рx ј 0):

92 Chapter 4

FIGURE 4-3 Transmission of sound from one material into another for normal

incidence of the sound wave.

Copyright © 2003 Marcel Dekker, Inc.

рA1 _ B1Ю

Z1 ј

A2

Z2

(4-85)

We may use Eqs (4-84) and (4-85) to solve for the ratio of the two constants:

A2

A1 ј

2Z2

Z1 ю Z2

(4-86)

We note that the magnitudes of the rms acoustic pressure for the

transmitted wave and for the incident wave are given by:

ptr ј A2=

ffiffiffi

2 p and pin ј A1=

ffiffiffi2p (4-87)

Therefore, A2=A1 ј ptr=pin.

The sound power transmission coefficient at is defined as the ratio of the

transmitted acoustic power to the incident acoustic power. This is a significant

parameter in selecting the materials for controlling sound transmission:

at _

Wtr

Win ј

SItr

SIin ј р ptrЮ2=Z2

р pinЮ2=Z1 ј

A22

Z1

A21

Z2

(4-88)

Making the substitution for the coefficient ratio from Eq. (4-86), we obtain

the following expression for the sound power transmission coefficient for a

sound wave in material 1 striking material 2:

at ј

4Z1Z2

рZ1 ю Z2Ю2 (4-89)

An alternative way of expressing the transmission of acoustic energy

from one material to another is in terms of the transmission loss TL. The

transmission loss expresses the sound power transmission coefficient in

decibel units:

TL _ 10 log10рWin=WtrЮ ј 10 log10р1=atЮ (4-90)

We note from Eq. (4-89) that the sound power transmission coefficient

is unity if the characteristic impedances of the two materials are the same,

i.e., the impedances are matched. This result means that all of the acoustic

energy is transmitted through the interface and none is reflected. On the

other hand, if the acoustic impedances are quite different from each other,

then the sound power transmission coefficient will be small. This result

means that little acoustic energy is transmitted through the interface and

most of the energy is reflected.

Another term that is not as widely used in noise control work as is the

sound power transmission coefficient is the sound power reflection coefficient

ar:

ar _ Wr=Win (4-91)

Transmission of Sound 93

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For this case, the acoustic energy is either reflected or transmitted, so

Win ј Wtr юWr, and the sound power reflection coefficient is related to

the sound power transmission coefficient by the following relation:

ar ј 1_at (4-92)

Making the substitution from Eq. (4-89), we obtain the following:

ar ј

Z2 _Z1

Z1 юZ2

_ _2

(4-93)

Example 4-1. A sound wave in air at 258C (778F) strikes a concrete wall at

normal incidence, as shown in Fig. 4-4. The intensity level of the incident

sound wave is 90dB. Determine the transmission loss, and the sound pressure

level for the transmitted wave.

We find the following values for the characteristic impedance for air

and concrete in Appendix B:

air Z1 ј 409:8 rayl

concrete Z2 ј 7:44 _ 106 rayl

The sound power transmission coefficient is found from Eq. (4-89):

at ј р4Юр409:8Юр7:44Юр106Ю

р409:8 ю 7:44 _ 106Ю2 ј 2:203 _ 10_4

94 Chapter 4

FIGURE 4-4 Physical system for Example 4-1.

Copyright © 2003 Marcel Dekker, Inc.

The transmission loss is found from Eq. (4-90):

TL ј 10 log10р1=2:203 _ 10_4Ю ј 36:6dB

The intensity of the incident wave is found from the following:

LI;in ј 10 log10рIin=Iref Ю

Iin ј р10_12Ю1090=10 ј 0:00100W=m2 ј 1:00mW=m2

The intensity of the transmitted wave is given by Eq. (4-88):

Itr ј atIin ј р2:203Юр10_4Юр0:00100Ю ј 0:2203 _ 10_6W=m2

ј 0:2203 mW=m2

The intensity level of the transmitted wave is given by:

LI;tr ј 10 log10р0:2203 _ 10_6=10_12Ю ј 53:4dB

We note that, in this case, we could also have calculated the intensity level

for the transmitted wave from:

LI;tr ј LI;in _ TL ј 90 _ 36:6 ј 53:4dB

The intensity and rms acoustic pressure magnitude are related by the

following:

I ј

p2

_c ј

p2

Zo

The acoustic pressure for the incident and transmitted waves is found, as

follows:

pin ј рZ1IinЮ1=2 ј Ѕр409:8Юр0:00100Ю_1=2 ј 0:640 Pa

ptr ј рZ2ItrЮ1=2 ј Ѕр7:44Юр106Юр0:2203Юр10_6Ю_1=2 ј 1:280 Pa

The sound pressure levels are found from the definition of sound pressure

level:

Lp ј 20 log10р p=pref Ю

For the incident sound wave,

Lp;in ј 20 log10р0:640=20 _ 10_6Ю ј 90:1dB

For the transmitted wave,

Lp;tr ј 20 log10р1:280=20 _ 10_6Ю ј 96:1dB

Although the acoustic pressure of the transmitted wave is greater than

the acoustic pressure of the incident wave, in this example, there is no

Transmission of Sound 95

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violation of any physical principle. The acoustic energy is conserved (the

conservation of energy principle is valid), because we find that Iin ј Itr ю Ir.

The acoustic pressures are different because the media in which the two

waves (incident and transmitted waves) are transmitted are different. The

intensity of the reflected wave is:

Ir ј 1:00 _ 10_3 _ 0:2203 _ 10_6 ј 0:999780 _ 10_3W=m2

In this example, most of the energy is reflected and only about 0.02% is

transmitted, because of the large difference in the characteristic impedances

of the two materials.

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