• Author / Uploaded
• hashmisahib48

Citation preview

TEXT FLY WITHIN THE BOOK ONLY

00

(E
m

OSMANIA UNIVERSITY LIBRARY /

Call No.

f>

3 ^

/

^

"^

c being the velocity in

1

2

See Aigner, Unterwasserschalltechnik, 1922, p. 46. Smithsonian Tables.

ACOUSTICS

12

of sound in

If the time

air.

C

from

B

to

be

r,

then r

=

=

CB' /c

BB'/Ui. To obtain the wave front in the upper region we note u 2 r. The position that in time r the upper medium moves A A' of the wave is hence the same as if it had originated at A and in is at D, where /I'D = cr; time r the disturbance originally at DB can be shown by taking intermediate points to be the wave '

A

AD

is the direction of front in the upper region and propagation. It is interesting to note that in neither region is the direction of propagation normal to the wave front, a fact due to the motion

medium with respect to the observer. From Fig. 1*6, the relation between 0i and

of the

refraction,

AB +

A'B A'D that

f

2>

i.e.,

the law of

For we have simply:

be readily obtained.

may

^_

B'B - AA' cr

""

cr

c

1

is,

csc 6 2

-

= '^~~^

csc 0!

(1-5)

c

If the

"ray"

directions are given

by

\l/\

and fa respectively we can and 0%. For if we draw

also ascertain their values in terms of 0i

DN normal

to

AB> Z

A'DN =

tan* 2 There

is

=

2,

tan0 2

and

it

follows that

^

2

+

an exactly similar eauation, with

all

.

(1-6)

the subscripts unity,

for the incident ray. If the values are such that

then, since there reflection

which

esc

is

must be 2

=

no value of

total.

The

esc

2

critical

that

is

angle

less

is,

than unity, the

of course, that for

i.

emphasized that the above discussion is by no is no sharp boundary regions of differing wind velocity. The variation is

It should be

means complete, between

air

since in the actual case there

always a more or less gradual one. Nevertheless a more detailed study can follow the above lines by dividing the medium into thin strata. 1

We

shall derive a

(Atmospheric Acoustics).

more general equation (Sec Sec. 12-2.)

for

sound refraction

in

Chapter XII

SOME PROPERTIES OF ACOUSTIC WAVES

13

Keeping in mind the discussion in this section the reader have no difficulty in understanding why sound in air will often pass more readily in one direction between two points than in the other, and also why elevated sources are very advantageous in transIt must be borne in mind, however, that mitting to windward. as the above we do not have a beam of such discussions in any sound as in light. For example, one could not secure a plane sound wave sharply limited to a given area of cross section, and conseAs has been noted before quently could not secure total reflection. will

and due

as will be

of a beam of sound, emphasized again, the production

a difficult matter. length, is in general strata of air with bountwo Let Refraction. 1-8. Temperature at temperatures /i and rest and at assumed be AB'

wave

to its long

dary

(Fig. 1-7)

respectively, with corresponding sound velocities c\ and c^ y

/2

where ci>c\. front

is

AB

perature

dence front

is

is

/i

0i.

The

in the

incident

wave

stratum of tem-

and the angle of

The

refracted

A'B' with angle of

inci-

wave

refrac'

IC *

*

7

"

The construction of the refracted wave front follows at once on the application of Huyghens' = c*/c\ and in this case the rays are normal principle, for AA' JBB' law of refraction is The fronts. to the wave tion

2.

sin0 2 /sin O l

=

c z /Ci,

(1-7)

the ordinary law of Snell in optics.

Naturally, in practice continuous variation there is in temperature with a corresponding continuous bending of the wave front. Usually the effects of both wind and temperature occur simultaneously. From the discussion above it is seen that a negative temperature

which

is

no sharp boundary but a more or

less

obtains near gradient (i.e., coldest air nearest the ground), such as a clear after hours in the the surface of the earth early morning the toward fronts wave of sound night, tends to produce a bending

The ordinary earth, increasing the range of sound transmission. day time condition is a positive gradient and in this case the wave away from the earth, considerably reducing the Here again wind introduces a complicating factor.

fronts are bent

range.

I

ACOUSTICS

4

In view 1*9. Scattering by Selective Reflection and Refraction. of the discUvSsion of the preceding two sections it is not surprising that atmospheric conditions exercise an important influence on the propagation of sound in the air. The various strata of the atmos-

phere are not always horizontal or even plane. In the case of prominent irregularities, we should then expect a great deal of scattering

by

reflection

should he selective,

and

i.e.,

refraction.

Moreover,

greater for short

this scattering for

wave lengths than

long wave lengths.

This has actually been found to be true in on airplane detection under what may be termed "poor listening" conditions. In these tests the sound from an airplane at the greatest hearing distance was found to be limited l

experiments

to the lowest frequencies in the emitted complex sound. The effect of the scattering on the decay of sound intensity was also well illustrated by the same experiments. As will be shown later, the falling off in intensity of a

sound wave

in a

homogeneous

fluid is

proportional to the inverse square of the distance from the source. The listening apparatus used in the experiments iust mentioned

had an amplification factor of 100, so that under the best conditions one should have heard a sound 10 times the distance it could be heard with the ear alone. As a matter of fact, on bright sunny days with cumulus clouds forming, the airplane noise range was only twice that of the unaided ear. Even under the best night conditions the range was only three times that of the ear alone. Selective scattering also plays an important role in submarine transmission as will be noted later (Chap. X). The interesting silence areas observed during the propagation of an explosive wave may also be traced to meteorological conditions

of wind and temperature. Reference on this point may be made to the interesting theory of Esclangon. 2 Wiechert 3 has developed a theory to account for the same phenomenon by postulating a reflecting layer some 50 km. above the earth's surface and the interference of the normal direct wave and the reflected wave (as witness the analogous phenomenon in radio wave transmission with the

W. Stewart, Phys. Rev., 14, 376 (1919). E. Esclungon, Comptes Rendus, 178, 1892, 1924. For further observations on silence zones, see the work of Dufour, Deslandres, Villard, Maurian, Collignon in 1

See G.

2

volumes 178 and 179 of the Comptes Rendus. 3

E. Wicchert, Meteorolog. Zeitschrift, 43, 81, 1926. For further German work in this field see J. Kolzer, Meteorolog. Zeitschrift, 42, 457, 1925 and W. J. Witkiewitsch,

Meteorolog. Zeitschrift, 43, 91, 1926.

SOME PROPERTIES OF ACOUSTIC WAVES

15

casts postulation of the Heaviside layer). Esclangon (loc. cit.) on this explanation. grave doubt, however, More recent investigations l seem to indicate that the most

of audibility is probable single explanation for the abnormal zone to be found in the reversal of the temperature gradient, producing at heights above 30 km. temperatures which are of the same order It is evident that such a state of affairs as that at the surface.

would be

wave producing the necessary bending of the

effective in

fronts to account for the distant audible zone.

The

presence of

such a high temperature layer has already been indicated by the is mainstudy of meteors and it is believed that the temperature tained by the absorption of solar energy by the ozone layer whose center of gravity

is in

the neighborhood of 45 to 50

km.

Sound Shadows.

Any change in the direction of propagation of sound waves, not caused by a variation in of the medium but by the bending of the waves about the i-io. Diffraction.

properties

FIG. 1-8.

obstacles,

is

ascribed to diffraction, which

important acoustic

phenomenon.

On

its

is

thus an extremely

occurrence depends in

sounds from all directions, and which does not a technical application of acoustics hardly One of the most interesting and in some way involve diffraction. that of the sound shadow cast by typical diffraction problems is the human head, whether of speaker or auditor. This is most large there

1

measure our

ability to hear

is

See, for example, E.

H. Gowan, Nature, 124, 452, 1919 (Sept.

2i).

ACOUSTICS

i6

by the discussion of the

ideal case of a rigid sphere. circle the Consulting represent the cross section of the rigid sphere. (The word rigid is to be interpreted to mean that the sphere does not vibrate under the influence of any sound

easily treated

Fig. 1-8, let

which may happen to fall on it.) A is a point source on the sphere, and the problem is to determine the relative intensity of sound at points which, like P and P' y are equidistant from the sphere but

79.0 I

.75

2 A=120 cm. 3 Jl=l20cm. 4 fel20cm.

.

r = 1910 cm.

r* 477cm. r= 19.1cm"

10.6

15

8-

30

45

75

90

120

105

135

150

165

180

Degrees

FIG.

1-9.

at different azimuths with respect to the source.

The complete

work theory has been presented by Stewart, following the initial his article to be made should and reference and of Rayleigh others, for details. Briefly stated, the method of calculation consists in 1

setting

up the general equation of wave motion

(eq. (i'i6) of Sec.

1-12) using the assumption that coordinates of a particle of the medium, Uy Vy Wy component velocities of a particle of the py

medium,

density,

py pressure in the medium, Cy velocity of propagation of the disturbance, Sy

the

condensation,

=

,

where

defined

by

po is the constant

the

relation

s

=

-

Po

mean density at any point,

Po py the velocity potential, ,

fiy

1

f , the

component

particle displacements along #,

See Lamb, Dynamical Theory of Sound, 1925, p. 244.

y

y

z axes.

SOME PROPERTIES OF ACOUSTIC WAVES

21

Of

the above quantities all except c are functions of x y y, z and /. have now to define the velocity potential, the most important single quantity in the study of the irrotational motion of fluids.

We

This

is

done

in the following equation:

l

Let us consider a small volume element of the and influx of the medium

difference between the efflux

ment

fluid.

The

in this ele-

by the so-called principle of continuity, to the time of mass in the element. growth The most simple method of obtaining the mathematical expression of this principle is by the consideration of the elemental paralis

equal,

rate of

lelepiped of dimensions A#, Aj, Az. By considering the influx and efHux through each pair of faces respectively, the difference between

the latter and the former for the whole cube

found to be:

is

Moreover, the rate of growth of mass in the cube is clearly dp/dt X AtfA^Az. Equating these two quantities we get the following relation:

dp

d( P v) "

d(pu) -

If in this equation

we make

the substitution p

=

p (i

+

j),

we

have:

du

Now 1

.

dv

.

dw\

in acoustics the condensation

In the definition of

as for example

Lamb

there

is

/ ,

6V

ds ,

is

ds ,

a very small quantity com-

no general agreement as

and Crandall, preferring the negative sign

to sign, (i.e.,

u

some authors,

=

d

/p, where for air 7 1.41. It can also be shown that of the order of icr7 , 0,

known

=

is

5p/po are usually of the order of IO""1 to io~~2

w

.

cm

per sec.

Moreover,

wave lengths are so long that u y v y w and s change very little with #, y, z. Hence du/dx y ds/dx y etc., are very small quanWe can therefore neglect terms like s(du/dx) and u(ds/dx) tities. in comparison with du/dx and to this approximation the continuity acoustical

equation becomes: ds

du

.

^+d-X + Substituting from (1-8)

.

dv d-y

dw

+ Tz = ,

f '




we have:

ds

ay

e*y

""

1

a/^d^^a/"

ay az2

'

or

o

(i-ii)

more compact notation. replace (i-n) by an equation in which