Applied Mathematics, 2011, 2, 1204-1206
doi:10.4236/am.2011.210167 Published Online October 2011 (http://www.SciRP.org/journal/am)
Pulsed Sound Waves in a Compressible Fluid
Pierre Hillion
Institut Henri Poincaré, Paris, France
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 [1], un-
derwater detection [2] or non-destructive evaluation [3].
Sound waves are generally described by a Fourier se-
ries of harmonic plane waves [4]. Here, we consider in-
stead a sequence of elementary unit pulses. They come
from the expansion [5] 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 [4] with the pression, density, velocity p,

0
t

 vv (1a)

1
t
   vv v0, (1b)
A sound wave is generated by a small disturbance of
the fluid so that

00
0
00
,,
,
cpppp
pp

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

vv
00 t




v (3a)
00 tp



 v (3b)
with
2
pc


  (4)
Substituting (4) into (3a) gives
2
00
tpc



v (5)
and we get from (3b) with v =  
0t
p




 (6)
We then obtain from (5) the wave equation
22 0
t
c 


 (7)
To sum up, once p' known from (6), (7), we get for the
density and the velocity
2
0
,cpvpc



 
(8)
while for the density of energy E and momentum j of the
sound wave, it comes [4]
22
0
2
00
11
+
22
Evc
vc
0



 
jvn
(9)
,



   



v (2) n is a unit vector in the direction of v and


0
tE



 v.

P. HILLION
1205

When the disturbance propagates in the z-direction,
the wave Equation (7) becomes
222
0
zt
c 

  (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
 
2
A
LT

. Then since
u
 (u) = 0 where  (u) = dU(u)/du is the Dirac distribu-
tion, we get
,
cos
tu
dudtU dudt
du 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 2
sgncossin ,
sin sincossin
t
t
U
U
  

 


 
 
(14)
a similar calculation gives
 
 
22 2
sgncossin ,
sinsincos sin
z
z
kU
kU


 


 
 
(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)



0
2
0
2
sgn cossin
sgn cossin
sgn cossin
pA U
Ac U
vAc U

 

  

 





(15)
while for travelling waves the energy and momentum
densities have the simple form

2
0
0
,0
1
xy
z
Evjj
jc





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 [5]
,
n
(15a)
 
 
sin1 π,
sin π
n
n
n
UU
n

 
 


 (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 [4]. 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
[4] 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 [4].
For 1

, the mean energy I emitted per unit time in
the form of sound waves is [4]
22
z
I
cl 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

P. HILLION
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
  
A
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
I
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)