4.6 CHANGES IN MEDIA WITH OBLIQUE INCIDENCE

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In the previous section, we examined the case in which the sound wave

strikes the interface at normal incidence. In most cases in noise control,

we find that sound waves may strike a surface at various angles of incidence.

Let us now consider the case of a plane sound wave which strikes the interface

between two materials at an angle of incidence _i with the normal to the

interface, as shown in Fig. 4-5.

96 Chapter 4

FIGURE 4-5 Transmission of sound from one material into another for oblique

incidence of the sound wave.

Copyright © 2003 Marcel Dekker, Inc.

The expression for a wave moving at an angle _ with the normal to the

interface may be written in the following form:

pрx; y; tЮ ј Aejр!t_kLЮ (4-94)

The quantity L is related to the coordinates by the following expression, as

illustrated in Fig. 4-6:

L ј xcos_юysin_ (4-95)

The corresponding expression for the acoustic wave in medium 1 may be

written as follows:

p1рx; y; tЮ ј A1 ejЅ!t_k1рxcos _iюysin_iЮ_ юB1 ejЅ!tюk1рxcos _i_ysin_iЮ_ (4-96)

We have used the fact that the angle of reflection is equal to the angle of

incidence, or _r ј _i. The expression for the wave moving in medium 2 is:

p2рx; y; tЮ ј A2 ejЅ!t_k2рxcos _tюysin_tЮ_ (4-97)

The angle of transmission _t is related to the angle of incidence _i

through Snell’s law. For the sound wave to remain a plane sound wave,

the wave must travel the distance L1 in the same time as it travels the

distance L2, as illustrated in Fig. 4-7:

d sin_i ј L1 ј c1 _t

d sin_t ј L2 ј c2 _t

By dividing the first expression by the second, we obtain Snell’s law:

Transmission of Sound 97

FIGURE 4-6 Relationship between the x- and y-coordinates and the coordinate L in

the direction of propagation of an oblique sound wave.

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sin _i

sin _t ј

c1

c2 ј

k2

k1

(4-98)

The expressions for the particle velocity may be written as follows:

u1рx; y; tЮ ј рA1=Z1Ю ejЅ!t_k1рx cos _iюy sin _i Ю_ _ рB1=Z1Ю ejЅ!tюk1рx cos _i_y sin _iЮ_

(4-99)

u2рx; y; tЮ ј рA2=Z2Ю ejЅ!t_k2рx cos _tюy sin _tЮ_ (4-100)

The acoustic pressure at the interface рx ј 0Ю is the same in each medium:

p1р0; y; tЮ ј p2р0; y; tЮ

The normal component (x-component) of the particle velocities is also the

same in each medium at the interface:

u1р0; y; tЮ cos _i ј u2р0; y; tЮ cos _t

If we use the previous expressions for the sound pressure and particle velocity

in these two conditions, we obtain the following expression for the

pressure magnitude ratio:

A2

A1 ј

ptr

pin ј

2Z2 cos _i

Z1 cos _t ю Z2 cos _i

(4-101)

The sound power transmission coefficient for oblique incidence may

be found from its definition and Eq. (4-101):

98 Chapter 4

FIGURE 4-7 Wave front striking an interface at oblique incidence.

Copyright © 2003 Marcel Dekker, Inc.

at ј

Wtr

Win ј

ItrStr

IinSin

(4-102)

As shown in Fig. 4-7, the area through which the incidence and transmitted

waves travel is related to the surface area of the interface S by the expressions:

Sin ј S cos _i

Str ј S cos _t

Using the expression for the intensity of the plane wave, I ј p2=Zo, Eq.

(4-102) may be written in the form:

at ј

p2

trZ1 cos _t

p2

inZ2 cos _i

(4-103)

If we make the substitution for the pressure ratio from Eq. (4-101) into Eq.

(4-103), we obtain the final expression for the sound power transmission

coefficient for oblique incidence:

at ј

4Z1Z2 cos _i cos _t

рZ1 cos _t ю Z2 cos _iЮ2 (4-104)

From the Snell law expression, Eq. (4-98), we note that there can be a

critical angle of incidence _cr for which the transmitted wave will make an

angle of 908 with the normal to the interface. For this condition,

sin _t ј sin 908 ј 1. The expression for the critical angle of incidence may

be found by making this substitution into the Snell law expression:

sin _cr ј c1=c2 (4-105)

A critical angle of incidence exists only if рc1 < c2Ю. If the actual angle of

incidence is equal to or greater than the critical angle of incidence, then no

acoustic energy will be transmitted into the second material.

at ј 0 for _i        _cr (4-106)

Example 4-2. A sound wave having a sound pressure level of 70 dB is

incident at an angle of 458 with the normal to the interface between oil

and water. The properties of the oil and water are as follows:

oil (material 1) _1 ј 850 kg=m3; c1 ј 1350m=s;

Z1 ј 1:148 _ 106 rayl

water (material 2) _2 ј 998 kg=m3; c2 ј 1481m=s;

Z2 ј 1:478 _ 106 rayl

Transmission of Sound 99

Copyright © 2003 Marcel Dekker, Inc.

Determine the sound pressure level for the transmitted wave in the water.

The angle of transmission may be found from Snells’ law, Eq. (4-98):

sin _t ј рc2=c1Ю sin _i ј р1:481=1:350Ю sinр458Ю ј 0:7757

_t ј 50:878

We note that there is a critical angle of incidence. In this case:

sin _cr ј c1=c2 ј р1:350Ю=р1:481Ю ј 0:9115

_cr ј 65:728

If the angle of incidence were greater than 65.728, there would be total

reflection and no acoustic energy transmission.

The sound power transmission coefficient is found from Eq. (4-104):

at ј р4Юр1:478Юр106Юр1:148Юр106Ю cosр458Ю cosр50:878Ю

Ѕр1:478Юр106Ю cosр458Ю ю р1:148Юр106Ю cosр50:878Ю_2

at ј 0:9649

The transmission loss is fairly small in this example:

TL ј 10 log10р1=0:9649Ю ј 0:2dB

The magnitude of the incident pressure wave is found from the sound

pressure level:

pin ј р20Юр10_6Ю1070=20 ј 0:0632 Pa ј 63:2 mPa

The intensity of the incident wave is:

Iin ј

p2

in

Z1 ј р0:0632Ю2

р1:148Юр106Ю ј 3:484 _ 10_9W=m2 ј 3:484nW=m2

The intensity of the transmitted wave is found from Eq. (4-102):

Itr ј

cos _i

cos _t

atIin ј

cosр458Ю

cosр50:878Ю р0:9649Юр3:484Ю ј 3:767nW=m2

Although the intensity of the transmitted wave is somewhat greater than the

intensity of the incident wave, there is no violation of any physical principle.

The energy of the transmitted wave is ‘‘squeezed’’ into a smaller area than

that of the incident wave, so the intensity of the transmitted wave is

increased. This phenomenon is analogous to the fact that flowing water

will experience an increase in velocity (volume flow per unit area) when

the flow enters a smaller size pipe.

The sound pressure for the transmitted wave may be found from the

intensity of the wave:

100 Chapter 4

Copyright © 2003 Marcel Dekker, Inc.

ptr ј рZ2ItrЮ1=2 ј Ѕр1:478Юр106Юр3:767Юр10_9Ю_1=2

ptr ј 0:0746 Pa ј 74:6mPa

The sound pressure level for the transmitted wave is:

Lp;tr ј 20 log10р0:0746=20 _ 10_6Ю ј 71:4dB

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