 Applied Mathematics, 2011, 2, 1204-1206 doi:10.4236/am.2011.210167 Published Online October 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Pulsed Sound Waves in a Compressible Fluid Pierre Hillion Institut Henri Poincaré, Paris, France E-mail: pierre.hillion@wanadoo .fr Received July 22, 2011; revised August 23, 2011; accepted August 30, 2011 Abstract The propagation along oz of pulsed sound waves made of sequences of elementary unit pulses U (sin) where U is the unit step function and  = kz  t is analyzed using the expansion of U (sin) and of the Dirac dis-tribution  (sin) in terms of   nπ where n is an integer. Their properties and how these pulsed sound waves could be generated are discussed. Keywords: Sound Wave, Unit Step Function, Travelling Pulses, Compressible Fluid 1. Introduction In a compressible fluid, waves of compression with small amplitudes, sound waves, can propagate causing alter-nate compression and rarefaction at each point of the fluid. These travelling perturbations are periodic in op-posite to noise, characterized as being aperiodic, that is having a non repetitive pattern. Pulsed sound waves are widely used in different fields such as medicine , un-derwater detection  or non-destructive evaluation . Sound waves are generally described by a Fourier se-ries of harmonic plane waves . Here, we consider in-stead a sequence of elementary unit pulses. They come from the expansion  of the function U (sin) in which U is the unit step function and  = kz  t, (assuming propagation along oz) in terms of   nπ where n is an integer. A similar expansion exists for the Dirac distribu-tion  (sin). We discuss the properties of these pulsed sound waves and how they could be generated. 2. Isentropic Compressible Fluid For an isentropic fluid in which viscosity and thermal conductivity can be neglected, the equations of motion are  with the pression, density, velocity p,  0tvv (1a) 1t  vv v0, (1b) A sound wave is generated by a small disturbance of the fluid so that 00000,,,cppppppp0, 0 are the pression and density in absence of distur-bances and c the velocity of sound. Taking into account (2), in (1b) can be ne-glected, as well as the quantities of second order so that Equations (1a, 1b) reduce to vv00tv (3a) 00tp v (3b) with 2pc (4) Substituting (4) into (3a) gives 200tpcv (5) and we get from (3b) with v =  0tp (6) We then obtain from (5) the wave equation 22 0tc (7) To sum up, once p' known from (6), (7), we get for the density and the velocity 20,cpvpc (8) while for the density of energy E and momentum j of the sound wave, it comes  22020011+22Evcvc0jvn (9) ,  v (2) n is a unit vector in the direction of v and 0tE v. P. HILLION 1205When the disturbance propagates in the z-direction, the wave Equation (7) becomes 2220ztc  (10) with the plane wave solutions in which f is an arbitrary function and k2 = 2c2  ,ztf kzt (11) and in particular the harmonic plane waves ,expztikz tt (11a) used to perform the Fourier transform of the f functions. 3. Pulsed Sound Waves We now suppose that the functions  (z, t) has the form with kc =   ,sin,uAuUuu kz 1 (12) in which U is the unit step function and A a constant am-plitude with the dimension  2ALT. Then since u (u) = 0 where  (u) = dU(u)/du is the Dirac distribu-tion, we get ,costududtU dudtdu dtdu dddtddt   (13) and with the signum function sgn() = 1 for  > 0 and  for  < 0 sgnddt (13a) so that according to (13) and (13a)   22 2sgncossin ,sin sincossinttUU     (14) a similar calculation gives   22 2sgncossin ,sinsincos sinzzkUkU   (14a) It is checked at once that  is solution of the wave Equation (10) and taking into account (14) we get from (6) and (8) 0202sgn cossinsgn cossinsgn cossinpA UAc UvAc U     (15) while for travelling waves the energy and momentum densities have the simple form 200,01xyzEvjjjcWe now come to the important property of these dis-turbances. Since sin = 0 for  = nπ where n is an arbi-trary integer, the unit function U (sin) and the Dirac dis- tribution  (sin) have the expansions  ,n (15a)   sin1 π,sin πnnnUUn   (16) So, these disturbances are pulsed sound waves made of a sequence of elementary unit step functions. Remark: The relation  = ck is valid for a sound wave propagating in a medium at rest and, in particular, in a frame K' relative to a fluid moving with a velocity V so that in the fixed frame K we get the Doppler formula in which  is the angle of V with oz. 1coskc cV (17) 4. Discussion An important question concerns the possibility to gener-ate sound waves of the elementary unit pulse type. Sound waves are produced by oscillating bodies with as proto-types a sphere pulsating in any manner and a cylinder oscillating perpendicularly to its axis . The velocity potential , taken as the fundamental quantity, is solution of the wave Equation (7) with the boundary condition v = Vn on the surface of the body and it satisfies the Som-merfeld radiation condition at infinity. We follow closely  to get the energy of sound waves emitted by oscillat-ing bodies. We assume here that the body oscillating with a pulsa-tion , in the z-direction, is a small cube parallel to the coordinate axis with a characteristic dimension l so that its volume and the area of its lateral faces are respec-tively l3 and l2. Then,  = 2πc being the wave length of the emitted wave, we consider two asymptotic situa-tions 1 and 1 in which the actual shape and the dimensions of the oscillating body do not intervene at distances far from this body. So, we may assume spheri-cal waves as in Sec. 73 of . For 1, the mean energy I emitted per unit time in the form of sound waves is  22zIcl V (18) in which Vz is the vertical velocity of the x, y planes, (the usual notation: a bar over a quantity to denote its mean value being not available here, we use the symbol < >). The situation is more intricate for1 but at large distances r where r is the distance from an origin anywhere inside the oscillating body, the velocity poten-tial  is a solution of the Laplace equation ∆ = 0 with the solutions Copyright © 2011 SciRes. AM P. HILLION Copyright © 2011 SciRes. AM 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. 24πttrc 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 2tV 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  C. Boyd-Brewer, “Vibro Acoustic the Rapy: Sound Vi-brations in Medicine,” Alternative and Complementary Therapies, Vol. 9, 2005, pp. 257-263. 224πtIV (22)  Sonar Site Internet (Wikipedia.)  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.  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  characterized by a dipole vector A  B. Van Der Pol and H. Bremmer, “Operational Calculus,” University Press, Cambridge, 1959. ttrc c An (23)  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 2tvnn A (24)  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 232tIcnA r (25)  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 224πtIA (26)