P. HILLION

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1206

1ar r

A

(19) and, if the body executes harmonic oscillations of fre-

quency

, I is proportional to

6.

Then, it is shown that the velocity v =

is when the

emitting body undergoes pulsations during which its

volume V changes

Thus from a theoretical viewpoint, it is possible to

generate sound waves becoming a sequence of elemen-

tary unit pulses at large distances of an oscillating small

cube.

24π

ttrc c vn r (20)

Sound waves may also be generated with the help of

pulsed electromagnetic beams [6-8] The pulsed sound

waves investigated here have simple analytical expres-

sions making them particularly suitable for computer

simulation of the processes in which they are involved.

in which n is a unit vector in the direction of r and 2

tV

the variation of volume.

Then, the mean value of the total energy emitted per

unit time is

2dIcv

(21) 5. References

where the integration is taken on a surface surrounding

the origin. So, substituting (20) into (21) and taking as

surface a sphere of radius r we have finally

[1] C. Boyd-Brewer, “Vibro Acoustic the Rapy: Sound Vi-

brations in Medicine,” Alternative and Complementary

Therapies, Vol. 9, 2005, pp. 257-263.

2

24π

t

IV

(22) [2] Sonar Site Internet (Wikipedia.)

[3] Ch. Hellier, “Ultrasonic testing in Hand-book of Nonde-

structive Evaluation,” Mac Graw Hill, New York, 2003.

If the body executes harmonic pulsations of frequency

, the intensity of emission is proportional to

4. [4] L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” Per-

gamon, London, 1959.

The situation is still different for a body oscillating

without changes of volume. Then one has to deal with a

dipole emission [4] characterized by a dipole vector A [5] B. Van Der Pol and H. Bremmer, “Operational Calculus,”

University Press, Cambridge, 1959.

ttrc c

n (23) [6] B. J. Van Gutfeld and R. L. Melcher, “20 MH Acoustic

Waves from Pulsed Thermoelastic Expansion of Con-

strained Surface,” Applied Physics Letters, Vol. 30, No. 6,

1977, pp. 257-259. doi:10.1063/1.89375

and

2

t

vnn A

(24) [7] A. Puskarev, J. Isakova, G. Kholudnaya and R. Sazonov,

“Sound Waves Due to the Absorption of a Pulsed Elec-

tronic Beams,” Advances in Sound Localization Intech,

2011.

so that

2

32

t

c

nA r

(25)

[8] C. Bacon, E. Guillorit, B. Hosten and D. Chimenti,

“Acousric Waves by Pulsed Microwaves in Viscoelastic

Rods,” Journal of the Acoustical Society of America, Vol.

110, 2001, pp. 1396-1407. doi:10.1121/1.1391241

Taking the surface of integration to be a sphere of ra-

dius r and using spherical coordinates with the polar axis

in the direction of the vector A, we finally have

2

2

4πt

I

A (26)