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Linearized supersonic motion under a Lorentz-like transformation
Jounml of Sound and Vibration (1982) 82(3), 391--400
LINEARIZED SUPERSONIC M O T I O N U N D E R A LORENTZ-LIKE T R A N S F O R M A T I O N T. S. SHANKARA and S. N. MAJHI
Department of Atathematics, Indian Institute of Technology, Madras-600036, India (Received 29 July 1980, and h, revised [orm 6 October 1981) The symmetry of the Lorentz transformation is shown to be inherent in the acoustics of sources in subsonic motion. The symmetry in the supersonic case is expressed in terms of a Lorentz-like transformation. The linearized wave equation describing the motion of a point source and a thin airfoil in supersonic motion is discussed by using this new transformation. Results in the paper suggest that there is a unified relativistic foundation for the acoustics of both subsonic and supersonic sources.
1. INTRODUCTION The acoustic wave motion from actual sources moving in a fluid is a complicated function of several variables like the amplitude and the frequency of the wave, the thermal properties of the boundary of the medium and many such others. Even the assumption of the wave speed c being independent of the amplitude is possible only as long as the sound pressure is small in comparison with the equilibrium pressure. Hence, real acoustic problems invariably involve non-linear equations which are amenable only to numerical solutions. But the disadvantage of the numerical solution is that it hides the physical features of the phenomena involved. On the other hand, if one works in a linear approximation, it is possible to obtain a better perception of the underlying physics. In a large class of problems of practical interest, linearization does not even involve sacrifice of accuracy, since the analytic solutions seldom have inadequate accuracy. Therefore one can often make the usual assumptions necessary to linearize the wave propagation problem, such as considering air at rest in thermodynamic equilibrium, and others, and arrive at the wave equation as the equation of motion. For subsonic motion, in the linear approximation, Kiissner and later others [1-3] resorted to the Lorentz transformation (LT) to go from the frame fixed in the air to one fixed in the source. This was done in preference to the Galilean transformation (GT) since the wave operator a~x + 0yy + a~z - (1/c2)a,, is invariant under LT. But this convenient transformation is not available for supersonic motion since the L T is defined only for values of M a t h number less than 1. In the supersonic case, one may start from the Green functions of the wave equation and obtain the solution by using the theory of generalized functions. Morse and Ingard have computed the field from a point source in supersonic motion on rather heuristic arguments [4]: from geometry they obtained two equivalent virtual sources E+ and E_ (section 3) and the solution was taken as the sum of the solutions for E+ and E_, each of these being borrowed from results for subsonic motion. Miles introduced a modified L T to obtain the solution for an oscillating airfoil in supersonic flow and bypassed the invariance of the wave equation [5]. In what follows here, the solutions to both these problems are rederived via Kiissner's procedure by introducing an appropriate L T to handle the supersonic case. This analysis shows that 391 0022-460x/82/110391 + 10 $03.00/0 9 1982 Academic Press Inc. (London) Limited
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T.S. S|IANKARA AND S. N. MAJHI
application of the mathematics of special relativity to problems of linearized aerodynamics can provide a unified treatment of the kinematics of both subsonic and supersonic motions. Philosophically speaking, it appears that acoustics is relativity with respect to the sound signal [6]. The matter is discussed section-wise as follows. Results and comments on the kinematics of a point souce in subsonic motion are summarized in section 2. In section 3 the LT for supersonic motion is derived. By using this LT, the solution for a point source and an oscillating airfoil in supersonic motion are rederived in sections 4 and 5, respectively. 2. SUBSONIC MOTION OF A POINT SOURCE Let the source be moving with a velocity v < c along the x-axis of a frame fixed in the medium (see Figure 1). If the source strength is q(t) and the velocity potential is r then
V2 d/ -- (1/ C2)021~/ 0t 2 = - q ( t ) 8(x - vt) 8(y) 8(z).
(1)
P (x,y,z, l )
vt - - - ~ C
~vt=MtT~ S
L, I i . ~ . ~
.~x
I I
Figure 1. Kinematics of subsonic motion of a point source.
On using the LT, x' = y ( x - vt),
y' = y,
z' = z,
3' = (1 - m2) -1/2,
t' = y ( t - vx/c2),
m = v / c < 1,
(2)
this equation takes the form
V'2d/- (I/C2)O2~/Ot '2= - y q ( y t ' ) 5(x') ~i(y') ~(z'),
(3)
whose solution is r
t') = y q ( y t ' • y R ' / c ) / 4 ~ r R ' ,
(4)
where R '2= x'2+ y , 2 + Z,2. Therefore the solution of equation (1) in the unprimed variables is obtained by expressing the primed variables in equation (4) in terms of them with the help of equations (2). Writing R = (x2+ y 2 + 22)1/2 one has ~b(R, t) = q ( t - R + / c ) / 4 r
gl=[(x-l)t)2+(l-m2)(y2+z2)]
1/2,
R~'=[m(x-vt)+Rl]/(1-m2).
(5) (6,7)
Only the root R + is valid in equation (5) for the case of subsonic motion since the other root is negative. Further, from Figure 1 one has x = R cos 0, v t = m R a n d y 2 + z 2 R 2 sin 2 0. Substituting these values in equation (7) gives R + = R. Therefore equation (7)
A LORENTZ-LIKE TRANSFORMATION
393
can be rewritten as (1 - m2)t/2R = [ m (x - vt) + R 1]/(1 - m2)1( 2.
(8)
Now, substituting the spatial part of the LT (2) in equation (6) gives R 1 / ( 1 - m2) 1/2= R ' so that equation (8) becomes R = (rex' + R ' ) / ( 1 - m 2)1/2.
(9)
In analogy with R = ct one has R ' = ct' and therefore this gives
(10)
t = (t'+ v x ' / c 2 ) / ( 1 - m2) 1/2,
which is the inverse of the temporal part of equations (2). Now by mutual substitutions of equation (10) and the spatial part of equations (2) into one another, the whole set of transformations (2) and its inverse follow. This inherent symmetry in the space-time co-ordinates of the laboratory frame and the frame moving with the source demonstrates the appropriateness of a full use of the framework of special relativity with respect to the sound signal. In the nomenclature of Rosen [6], sonic relativity becomes a c'-relativity: i.e., special relativity in which the light signal is replaced by another signal of an invariant speed. This raises the following question: to what extent can this framework be pushed in the study of linearized acoustics? In this context the following notes on the Doppler formula in subsonic motion may be relevant. Consider harmonic time dependence for the source strength so that q(t') = qo sin toot', where too is the frequency in its rest frame. At large distances this gives rise to the pressure field given by equation (4): i.e., p,_
1
4=g,
d
r[,
R'\'l
TqLk, --;-)j =
qotoo
R'
cos [too(,'--;-)].
(11)
In the calculations leading to equation (5), it is implied that the phase of this pressure field ~(t') = too(t'-R'/c) is an invariant under the LT. Since too(t - R / c ) = ytoo(t'-R'/c), this invariance gives the phase in the laboratory frame as ~p(t)= y - l t o o ( t - R / c ) . Therefore, in the unprimed frame, the frequency defined as the derivative of g,(t) is given by to = d ~ / d t = y-ltoo(1 - d R / c
dt).
(12)
Since it may be shown that d R / c dt = - m cos 0 / ( 1 - m cos 0) by differentiating equation (8), this leads to to = too(1 - m2)1/2/(1 - m cos 0)
(13)
as the relation between the frequency too of the source in its rest frame (primed) and the frequency in the unprimed frame. This is the "sonic relativity" Doppler formula. In a classical calculation of the Doppler formula the multiplication factor (1 - m2) 1/2 is missing, since the GT on which it is based cannot recognize the difference in the source strengths q(t) and q(yt') brought about by the relative motion; therefore the source strengths on the right-hand sides of equations (1) and (3) are left unaltered under the GT. (Notice that the GT does not give the invariance of c for a point source moving in a fixed medium; but in computations involving moving sources, sometimes one changes c according to the G T and somehow sometimes keeps it unchanged in accordance with the LT!) Such inadequacy of the GT leads to irreconcilable situations like the Doppler formula showing up an asymmetry: in a fixed medium the two cases of the observer moving towards the source and the source moving towards the observer have different
394
T. S. S | I A N K A R A
A N D S. N. M A J H I
Doppler shifts. Clearly such an asymmetry violates the basic reciprocity that is assumed between the two frames in all moving source problems of acoustics. In the background of the foregoing analysis, it may be advisable to check the Doppler formula. Its experimental verification would confirm or deny sonic relativity which is implied in existing literature. Of course, a crucial experiment would be to verify whether oJ = ~ O o ( 1 - - t / | 2 ) 1/2 is true along the line 0 = ~ / 2 at large distances. The most important aspect in an experimental set-up would obviously be that it should conform to linear acoustics in a fixed medium. Therefore it appears inevitable that the surface of a streamlined rotating body be chosen to place the point source. Next in importance would be exactness in the alignment since, even a small difference can mask the second order term in m. 3. LT FOR SUPERSONIC MOTION When the motion is supersonic, the situation is as follows [4]. Now both values of R • in equation (7) are physically valid. The actual source does not directly contribute to the pressure field at a point, but instead the contribution comes from two effective sources
x
~
ut-- x
u/= MR
Figure 2. Kinematics of supersonic motion of a point source.
E § and E - on the line of motion corresponding to the distances R § and R - (see Figure 2). Therefore the velocity potential is the algebraic sum of the potential due to each one of them: i.e., ~b(R, t) = [ q ( t - R + / c ) / 4 1 r R l ] + [ q ( t - R - / c ) / 4 7 r R i ] . (14) This may be immediately verified by writing down the corresponding Green function involving the effective sources and computing the integral according to the operational rules of generalized functions [7]. However, in the light of the discussion of the foregoing section, it will be demonstrated in the following section that the solution to equation (14) can be obtained on the basis of sonic relativity. This would ensure a unification of the kinematics of both subsonic and supersonic motions of sources. For this purpose, one can construct an appropriate LT for the supersonic case. It is better to begin by first recapitulating the nature of the LT [8] for the subsonic motion. This is conveniently done when the line of motion is chosen as the x-axis. The space-time diagram is then the trajectory (of the source) in the x-ct plane (see Figure 3). The two bisector lines x • = 0 represent the trajectory of a sound signal emanating from the origin of this plane. The trajectory of a source moving with subsonic speed lies inside the cone containing the ct-axis since x 2= v2t2< c2t 2. The LT is characterized by its invariant action on the quadratic forms s 2 = x 2 - c 2 t 2 < O , i.e., x ' : - c 2 t ' 2 = x 2 - c 2 t 2, where the primed quantities are defined by equations (2).
A LORENTZ-LIKE TRANSFORMATION
r
395
,Ig
Figure 3. Space-time diagram.
For supersonic motion it is obvious that the trajectory lies outside the "ct-cone" and the quadratic form is o ' 2 = x 2 - c 2 t 2 > O , whose signature is opposite of s 2. Now the trajectory of a stationary observer is the ct-axis. Therefore a transportation from the inside of the ct-cone to the outside is necessarily discontinuous (Jacobian-- -1). Indeed the modified LT x' = (x - u t ) / [ ( u 2 / c 2) - 1] 1/2, ct' = (ct - .xlc)l[(u2/c
y' = y,
2) -
1] 1/2,
z' = z,
u > c,
(15)
of Miles is exactly this, reference [5], and it cannot be physically realized in terms of actual observations by using sound signals: i.e., the fixed observer and the moving observer of equations (15) cannot communicate information acoustically. It transforms the wave operator axx+ayy4-azz-(1/c2)O, to -ax,x,+~y,r or, equivalently, the quadratic form x 2 + y2 + z 2 _ c2t 2 into - x 2 + y2 + z 2 + c2t 2. This shows that the rotational symmetry in (x, y, z)-space is lost in supersonic motion. This gives rise to the Mach cone. Actually the physical meaning of the discontinuity of the transformation (15) is that the Mach cone divides the medium into two acoustically disconnected regions. Now in order to retrieve the wave equation in the region of disturbance (viz., inside the Mach cone), at first sight it appears just sufficient to replace x ' and t' in equations (15) by ix' and it' respectively. In that case, these quantities will lose the meaning of physical distance and time for the moving observer. Also the generalized function ~ ( x - u t ) in the source term will then contain an imaginary argument whose mathematical definition disallows its presence. Therefore, instead, one can interchange x' and ct' to retrieve the wave operator and the resulting transformation is
X'=(X--C2I/U)I(1--C2/U2) 112,
y'=y,
Z'=Z,
t'=(t--X/lt)/(1--C2/li2) i/2, (16)
appropriate for the supersonic case [9]. Notice that equations (16) have been obtained by putting c 2 / u in place of v in equations (2). The Mach number o / c = x / c t < 1 is the parameter in equations (2) which connects two points on the hyperboloid x 2 - c 2 t 2 = s 2 inside the "ct-cone" x 2 - c2t 2 = O. In the region outside the "ct-cone", the interchange of space and time co-ordinates has given c / u < 1 as the parameter in the transformation (16). (The reason is that this change has, in effect, put supersonic motion back into the ct-cone.) As the inside and the outside of the ct-cone are two disjoint regions (i.e., there does not exist a LT with Jacobian +1 connecting them), it is natural that the parameters are different for them. However, since both the parameters are <1, in what sense is the LT (16) a transformation for supersonic motion? This is a pertinent question as the
396
T. S, S H A N K A R A A N D S. N. M A J | I I
origin O' of the primed frame in (16) moves with velocity c2/u < c and does not coincide with the source S which is moving with speed u > c. The meaning may be seen in the following way. As shown in Figure 4, where O is the origin and S is the supersonic source, the triangles OPS and O 0 ' P are similar, from which one has x / ( y 2 + Z2) 1/2 = R / ( l t 2 t
2 - - R 2 ) 1/2 =
1/(M 2 -- 1) 1/2.
(17)
S
o
.,
J
Figure 4. Observersin the LT (16).
Further, since P lies on the surface of the Mach cone the two roots of equation (7) coincide at P. Therefore from equation (6) one has 1/(M 2 - 1 ) 1/2 = ( y 2 + z 2 ) l / 2 / ( u t - x ) .
(18)
Solving for x from these two equations gives xut = R 2= c2t 2 whence x = c2t/u. This shows that the velocity of O' is c2/u < c and therefore the LT (16) is the one which connects the two observers at O and O'. Coming back to the question of how equations (16) represent a supersonic LT, one can first note the crucial point of the problem that, after all, an observer moving with S cannot gather any information about the sound field by using sound signals and hence the S frame represented by equations (15) is physically irrelevant; S does not even generate the sound field directly. Any physical measurement incorporating sound signals has to take place only within the Mach cone. To find an "effective observer" O'.in this region who is equivalent to one moving with S, one can note that O 0 ' / O S = ( c 2 t / u ) / u t = M -2 and therefore O' is uniquely obtained from S at any given time by a constant contraction mapping. In other words, a unique LT which is physically realizable is associated with every modified LT which is only a mathematical transformation. The moving frame attached to O' is equivalent to the one attached to S and the corresponding transformation between O and O' is a legitimate LT for supersonic motion, meeting both physical and mathematical requirements. As proof of this it is demonstrated in subsequent sections that the new LT (16) does indeed lead to the same solutions as the conventional ones for linearized supersonic problems [4, 5]. 4. KUSSNER'S METHOD FOR A POINT SOURCE IN SUPERSONIC MOTION With the appropriate LT for supersonic motion in hand, it is now possible to make use of the invariance of the wave operator just as in the subsonic case. Consider the wave equation for the velocity potential of a point source moving at speed U > c along the x-axis. It is 02~b/Ox2+c32~b/c3y2+O2~/Oz2- (1/c2)O2~/Ot 2 = - q ( t ) 8(x - Ut) 8(y) 8(z).
(19)
A LORENTZ-LIKE
TRANSFORMATION
397
In the primed frame fixed to the observer O' of the transformation (16) this becomes a2r
+
a2~/ay,2 + a 2 e , / a z, 2
_
(1/c2)a2~,/at 'z =
(r/U)q
(Fx'/U)
a(t') a(y ')(z '),
(20)
where _ F ' = ( 1 - c 2 1 U 2 ) 1/2. It may be noticed that the source term now has a(t') in the transformed variables in place of ~(x - Ut) in the old variables, yielding an initial value problem in contrast to the origin problem which is a boundary value problem. This feature of the transformation will be very advantageous in the oscillating airfoil case (see section 5). Taking the Laplace transformation [10] with respect to t' and the Fourier transform with respect to the other variables in equation (20) gives O(x') = q ( F x ' / U ) .
(21)
X [ Q [ x ' q- (c2/'2 - y ,2 _ z ,2) 1/2] _1_Q [ x ' - (c2t '2 - y,2 _ z '2)]}/(c2t'2 - y,2 _ z ,2)1/2,
(22)
~**#(t~ 2"t-fl2"b l.t2 q - p 2 / c 2 ) ~- - ( F / 2 1 r U ) Q * ( A ),
On inversion this gives ~(x', y', z ', t') = - (F/4~rU)
(23)
(y,2 d" Z'2) 1/2 ~ ct'.
In terms of the unprimed variables the inequality (23) reduces to 2
which is exactly the Mach cone (equation (18)) deduced by considering the geometry of a point source [4] moving with Mach number M. Also the solution (22) reduces to [(U~/c2)-t-R1/c'~ tp(R, t ) =
qt
~
. [ ( U x / c 2 ) - t + R1/c~
)*qt
M2-1
)
4 lr [(Ut - x ) 2 - (M 2 - 1)(y 2 + z2)] 1/2
-- q (t - R §
1 + q (t - R - / c ) / 4 r r R t,
(25)
which is the same as equation (14). 5. OSCILLATING AIRFOIL IN SUPERSONIC FLOW In this section the linearized potential equation for supersonic flow past a thin oscillating wing is considered. This has been investigated by various authors using different methods. In particular, Miles [5] started with the wave equation for the velocity potential r in dimensionless co-ordinates (non-dimensionalized by the characteristic quantities !, length and l/c, time) Cxx + Cyy + ~p~ - ~ r r = 0, (26) where X is measured downstream from the foremost point of the airfoil at time zero, y and z are positive and T is the dimensionless time. The boundary conditions on equation (26) are ~ ]z=o = - v in S,
~p[==o = 0 outside S,
(27)
where S is the projection of the wing surface on the plane z = 0. The variables in those equations are changed according to the transformation
[t;] = fl-l(D 1M)[XTT]' [ff] =f1-1(M-1 -1]//,j,"<'~r"'l
398
T. S. SHANKARA AND S. N. MAJHI
For obvious reasons this transformation is called the modified LT which is the same as transformation (15) in dimensionless form and fixes the co-ordinate frame in S; however, equation (26) is not invariant under the modified LT. The problem can now be approached in the spirit of the previous section. The GT
[7]--(;
_Mx
in dimensionless form fixes the ca-ordinate in 5'. It leaves the time invariant though not the wave equation (21). On the other hand the L L T (16) in dimensionless form, viz.,
M/It'J'
(32, 32')
leaves equation (26) invariant but not the time. Consequently, t' is not the true time. However from equations (32) and (31'), one has
[/]=f_,(M-~)[t]'
[t]=f-'(?l
M/It'l'f2'~rx']
(33,33')
Now, for the case of an oscillating wing for which the prescribed motion (in Galilean co-ordinates) is given by v = Re [5(x, z) exp (ikMt)],
(34)
the potential may be assumed in the form = Re [~5(x, z) exp (ikMt)],
(35)
where k = to//U is the dimensionless frequency (and o~ is the true radian frequency). Then by equations (33) and (33'),
kMt = kMf1-1 ( - x ' + Mt') = 71(Mfl- ix - x'),
(3 6)
with r / = kMfl -~. Hence one can write
v = V(t', z) e i"X', V = 5 exp (ikM2fl-2x),
~p = @(t', z) e inx',
(37)
q~ = ff exp ( i k M 2 f - 2 x ) ,
(38, 39)
with x = fit'. The boundary value problem of equations (26) and (27) for this oscillating airfoil (see Figure 5) now reduces to the initial value problem, q~,','- ~zz + r/2q~ = 0, q,, I,=o = - V(t'),
t'>O,
,/,1,=o= O,
(40) t'~
~_X t
Figure 5. Two-dimensionalairfoil boundary value problem.
(41)
A LORENTZ-LIKE TRANSFORMATION
399
This feature of the transformation (32) is clearly due to the interchange of the roles of x' and ct' that was effected on the modified LT (15) of Miles. Thus equations (32), or equivalently equations (16) not only ensure the invariance of the wave equation inside the Mach cone but also convert the boundary value problem into an initial value problem. Under the Laplace transform with respect to t' equations (40) and (41) become
~z*~ - A 2 ~ , = 0,
h = (s2+ 2)1/2,
(42)
cO
qb* [ , = o = - V * ( s ) ,
q~*=fo q~(z, t ' ) e - ' " d t ' .
(43,44)
The solution of equations (42) which vanishes at infinity and satisfies condition (43) is cb* = h -l V(s) e - ~ ,
z > 0.
(45)
Applying the inversion theorem corresponding to the transform (44) one obtains q~ = Jo
J~
d~r
(46)
Returning to the original co-ordinates reduces this to the same expression as given by Miles. 6. CONCLUSION In 1952 Rosen [6] gave the following result: for a homogeneous and isotropic medium with which a limiting signal speed c' can be associated, it is possible to set up a c'-relativity, i.e., a theory analogous to Einstein's c-relativity, but with c' replacing c. K/issner's [1] application of LT to iinearized subsonic motion is indeed an example of c'-relativity. But with the LT being defined only for Mach numbers M < I , it might appear that a relativistic approach is not available for supersonic motion. It is possible, however, to construct an appropriate LT for Mach numbers M > 1 as follows. First one can note that the rotational symmetry in the physical (x, y, z) space is lost when a source is moving supersonically. In contradistinction to the rest of the medium, it is only the interior of the Mach cone which is the active zone and here the wave equation continues to be invariant, therefore a physically valid and unique L L T can be obtained from Miles' modified LT by interchanging the roles of space and time variables. This has the further advantage that a boundary value problem is changed into an initial value problem which is simpler to solve in many cases. In this paper, we have first outlined in section 2 the inherent symmetry of the LT in subsonic kinematics. In section 3 the L L T has been constructed from a detailed study of the topology of the medium when a source is in supersonic motion. In section 4 it has been applied to a supersonically moving source. In the moving frame, this becomes an initial value problem and, in addition, the solution at once exhibits the two effective sources and the Mach cone with no extra effort. The potential of the new transformation has been shown further in section 5 dealing with an oscillating airfoil. It appears that sonic relativity can provide a unified foundation for the acoustics of both subsonic and supersonic sources.
ACKNOWLEDGMENT The authors are grateful to the referees for their critical comments and suggestions.
400
T. S. SHANKARA AND S. N. MAJHI REFERENCES
1. H. G. KUSSNER 1940 NACA TM 979. Allgemeine Tragslachentheorie. 2. R. W. TRUITT 1950 North Carolina State Department of Engineering Research Bulletin No. 47. (See also 1949 Bulletin No. 44 (unpublished).) 3. W. R. SEARS 1954 General Theory of High Speed Aerodynamics, Volume VIC (editor W. R. Sears). Princeton, New Jersey: Princeton University Press. 4. P. M. MORSE and K. U. INGARD 1968 TheoreticalAcoustics. New York: McGraw-Hill. 5. J. W. MILES 1959 T/re Potential Theory of Unsteady Supersonic Flow. London: Cambridge University Press. 6. N. ROSEN 1952 American Journal o/Physics 20, 161-164. Special theories of relativity. 7. 1. M. GELFAND and G. E. SHILOV 1964 Generalized Functions Volume I. New York and London: Academic Press. 8. J. A. WHEELER and E. F. TAYLOR 1966 Spacetime PI,ysics. San Francisco: Freeman. See p. 139. 9. T. S. SHANKARA and K. K. NANDI 1978 Journal of Applied Physics 49, 5783-5789. Transformation of coordinates associated with linearized supersonic motions. 10. H. BATEMAN 1954 Tables ofbztegral Transforms. New York: McGraw-Hill.
LINEARIZED SUPERSONIC M O T I O N U N D E R A LORENTZ-LIKE T R A N S F O R M A T I O N T. S. SHANKARA and S. N. MAJHI
Department of Atathematics, Indian Institute of Technology, Madras-600036, India (Received 29 July 1980, and h, revised [orm 6 October 1981) The symmetry of the Lorentz transformation is shown to be inherent in the acoustics of sources in subsonic motion. The symmetry in the supersonic case is expressed in terms of a Lorentz-like transformation. The linearized wave equation describing the motion of a point source and a thin airfoil in supersonic motion is discussed by using this new transformation. Results in the paper suggest that there is a unified relativistic foundation for the acoustics of both subsonic and supersonic sources.
1. INTRODUCTION The acoustic wave motion from actual sources moving in a fluid is a complicated function of several variables like the amplitude and the frequency of the wave, the thermal properties of the boundary of the medium and many such others. Even the assumption of the wave speed c being independent of the amplitude is possible only as long as the sound pressure is small in comparison with the equilibrium pressure. Hence, real acoustic problems invariably involve non-linear equations which are amenable only to numerical solutions. But the disadvantage of the numerical solution is that it hides the physical features of the phenomena involved. On the other hand, if one works in a linear approximation, it is possible to obtain a better perception of the underlying physics. In a large class of problems of practical interest, linearization does not even involve sacrifice of accuracy, since the analytic solutions seldom have inadequate accuracy. Therefore one can often make the usual assumptions necessary to linearize the wave propagation problem, such as considering air at rest in thermodynamic equilibrium, and others, and arrive at the wave equation as the equation of motion. For subsonic motion, in the linear approximation, Kiissner and later others [1-3] resorted to the Lorentz transformation (LT) to go from the frame fixed in the air to one fixed in the source. This was done in preference to the Galilean transformation (GT) since the wave operator a~x + 0yy + a~z - (1/c2)a,, is invariant under LT. But this convenient transformation is not available for supersonic motion since the L T is defined only for values of M a t h number less than 1. In the supersonic case, one may start from the Green functions of the wave equation and obtain the solution by using the theory of generalized functions. Morse and Ingard have computed the field from a point source in supersonic motion on rather heuristic arguments [4]: from geometry they obtained two equivalent virtual sources E+ and E_ (section 3) and the solution was taken as the sum of the solutions for E+ and E_, each of these being borrowed from results for subsonic motion. Miles introduced a modified L T to obtain the solution for an oscillating airfoil in supersonic flow and bypassed the invariance of the wave equation [5]. In what follows here, the solutions to both these problems are rederived via Kiissner's procedure by introducing an appropriate L T to handle the supersonic case. This analysis shows that 391 0022-460x/82/110391 + 10 $03.00/0 9 1982 Academic Press Inc. (London) Limited
392
T.S. S|IANKARA AND S. N. MAJHI
application of the mathematics of special relativity to problems of linearized aerodynamics can provide a unified treatment of the kinematics of both subsonic and supersonic motions. Philosophically speaking, it appears that acoustics is relativity with respect to the sound signal [6]. The matter is discussed section-wise as follows. Results and comments on the kinematics of a point souce in subsonic motion are summarized in section 2. In section 3 the LT for supersonic motion is derived. By using this LT, the solution for a point source and an oscillating airfoil in supersonic motion are rederived in sections 4 and 5, respectively. 2. SUBSONIC MOTION OF A POINT SOURCE Let the source be moving with a velocity v < c along the x-axis of a frame fixed in the medium (see Figure 1). If the source strength is q(t) and the velocity potential is r then
V2 d/ -- (1/ C2)021~/ 0t 2 = - q ( t ) 8(x - vt) 8(y) 8(z).
(1)
P (x,y,z, l )
vt - - - ~ C
~vt=MtT~ S
L, I i . ~ . ~
.~x
I I
Figure 1. Kinematics of subsonic motion of a point source.
On using the LT, x' = y ( x - vt),
y' = y,
z' = z,
3' = (1 - m2) -1/2,
t' = y ( t - vx/c2),
m = v / c < 1,
(2)
this equation takes the form
V'2d/- (I/C2)O2~/Ot '2= - y q ( y t ' ) 5(x') ~i(y') ~(z'),
(3)
whose solution is r
t') = y q ( y t ' • y R ' / c ) / 4 ~ r R ' ,
(4)
where R '2= x'2+ y , 2 + Z,2. Therefore the solution of equation (1) in the unprimed variables is obtained by expressing the primed variables in equation (4) in terms of them with the help of equations (2). Writing R = (x2+ y 2 + 22)1/2 one has ~b(R, t) = q ( t - R + / c ) / 4 r
gl=[(x-l)t)2+(l-m2)(y2+z2)]
1/2,
R~'=[m(x-vt)+Rl]/(1-m2).
(5) (6,7)
Only the root R + is valid in equation (5) for the case of subsonic motion since the other root is negative. Further, from Figure 1 one has x = R cos 0, v t = m R a n d y 2 + z 2 R 2 sin 2 0. Substituting these values in equation (7) gives R + = R. Therefore equation (7)
A LORENTZ-LIKE TRANSFORMATION
393
can be rewritten as (1 - m2)t/2R = [ m (x - vt) + R 1]/(1 - m2)1( 2.
(8)
Now, substituting the spatial part of the LT (2) in equation (6) gives R 1 / ( 1 - m2) 1/2= R ' so that equation (8) becomes R = (rex' + R ' ) / ( 1 - m 2)1/2.
(9)
In analogy with R = ct one has R ' = ct' and therefore this gives
(10)
t = (t'+ v x ' / c 2 ) / ( 1 - m2) 1/2,
which is the inverse of the temporal part of equations (2). Now by mutual substitutions of equation (10) and the spatial part of equations (2) into one another, the whole set of transformations (2) and its inverse follow. This inherent symmetry in the space-time co-ordinates of the laboratory frame and the frame moving with the source demonstrates the appropriateness of a full use of the framework of special relativity with respect to the sound signal. In the nomenclature of Rosen [6], sonic relativity becomes a c'-relativity: i.e., special relativity in which the light signal is replaced by another signal of an invariant speed. This raises the following question: to what extent can this framework be pushed in the study of linearized acoustics? In this context the following notes on the Doppler formula in subsonic motion may be relevant. Consider harmonic time dependence for the source strength so that q(t') = qo sin toot', where too is the frequency in its rest frame. At large distances this gives rise to the pressure field given by equation (4): i.e., p,_
1
4=g,
d
r[,
R'\'l
TqLk, --;-)j =
qotoo
R'
cos [too(,'--;-)].
(11)
In the calculations leading to equation (5), it is implied that the phase of this pressure field ~(t') = too(t'-R'/c) is an invariant under the LT. Since too(t - R / c ) = ytoo(t'-R'/c), this invariance gives the phase in the laboratory frame as ~p(t)= y - l t o o ( t - R / c ) . Therefore, in the unprimed frame, the frequency defined as the derivative of g,(t) is given by to = d ~ / d t = y-ltoo(1 - d R / c
dt).
(12)
Since it may be shown that d R / c dt = - m cos 0 / ( 1 - m cos 0) by differentiating equation (8), this leads to to = too(1 - m2)1/2/(1 - m cos 0)
(13)
as the relation between the frequency too of the source in its rest frame (primed) and the frequency in the unprimed frame. This is the "sonic relativity" Doppler formula. In a classical calculation of the Doppler formula the multiplication factor (1 - m2) 1/2 is missing, since the GT on which it is based cannot recognize the difference in the source strengths q(t) and q(yt') brought about by the relative motion; therefore the source strengths on the right-hand sides of equations (1) and (3) are left unaltered under the GT. (Notice that the GT does not give the invariance of c for a point source moving in a fixed medium; but in computations involving moving sources, sometimes one changes c according to the G T and somehow sometimes keeps it unchanged in accordance with the LT!) Such inadequacy of the GT leads to irreconcilable situations like the Doppler formula showing up an asymmetry: in a fixed medium the two cases of the observer moving towards the source and the source moving towards the observer have different
394
T. S. S | I A N K A R A
A N D S. N. M A J H I
Doppler shifts. Clearly such an asymmetry violates the basic reciprocity that is assumed between the two frames in all moving source problems of acoustics. In the background of the foregoing analysis, it may be advisable to check the Doppler formula. Its experimental verification would confirm or deny sonic relativity which is implied in existing literature. Of course, a crucial experiment would be to verify whether oJ = ~ O o ( 1 - - t / | 2 ) 1/2 is true along the line 0 = ~ / 2 at large distances. The most important aspect in an experimental set-up would obviously be that it should conform to linear acoustics in a fixed medium. Therefore it appears inevitable that the surface of a streamlined rotating body be chosen to place the point source. Next in importance would be exactness in the alignment since, even a small difference can mask the second order term in m. 3. LT FOR SUPERSONIC MOTION When the motion is supersonic, the situation is as follows [4]. Now both values of R • in equation (7) are physically valid. The actual source does not directly contribute to the pressure field at a point, but instead the contribution comes from two effective sources
x
~
ut-- x
u/= MR
Figure 2. Kinematics of supersonic motion of a point source.
E § and E - on the line of motion corresponding to the distances R § and R - (see Figure 2). Therefore the velocity potential is the algebraic sum of the potential due to each one of them: i.e., ~b(R, t) = [ q ( t - R + / c ) / 4 1 r R l ] + [ q ( t - R - / c ) / 4 7 r R i ] . (14) This may be immediately verified by writing down the corresponding Green function involving the effective sources and computing the integral according to the operational rules of generalized functions [7]. However, in the light of the discussion of the foregoing section, it will be demonstrated in the following section that the solution to equation (14) can be obtained on the basis of sonic relativity. This would ensure a unification of the kinematics of both subsonic and supersonic motions of sources. For this purpose, one can construct an appropriate LT for the supersonic case. It is better to begin by first recapitulating the nature of the LT [8] for the subsonic motion. This is conveniently done when the line of motion is chosen as the x-axis. The space-time diagram is then the trajectory (of the source) in the x-ct plane (see Figure 3). The two bisector lines x • = 0 represent the trajectory of a sound signal emanating from the origin of this plane. The trajectory of a source moving with subsonic speed lies inside the cone containing the ct-axis since x 2= v2t2< c2t 2. The LT is characterized by its invariant action on the quadratic forms s 2 = x 2 - c 2 t 2 < O , i.e., x ' : - c 2 t ' 2 = x 2 - c 2 t 2, where the primed quantities are defined by equations (2).
A LORENTZ-LIKE TRANSFORMATION
r
395
,Ig
Figure 3. Space-time diagram.
For supersonic motion it is obvious that the trajectory lies outside the "ct-cone" and the quadratic form is o ' 2 = x 2 - c 2 t 2 > O , whose signature is opposite of s 2. Now the trajectory of a stationary observer is the ct-axis. Therefore a transportation from the inside of the ct-cone to the outside is necessarily discontinuous (Jacobian-- -1). Indeed the modified LT x' = (x - u t ) / [ ( u 2 / c 2) - 1] 1/2, ct' = (ct - .xlc)l[(u2/c
y' = y,
2) -
1] 1/2,
z' = z,
u > c,
(15)
of Miles is exactly this, reference [5], and it cannot be physically realized in terms of actual observations by using sound signals: i.e., the fixed observer and the moving observer of equations (15) cannot communicate information acoustically. It transforms the wave operator axx+ayy4-azz-(1/c2)O, to -ax,x,+~y,r or, equivalently, the quadratic form x 2 + y2 + z 2 _ c2t 2 into - x 2 + y2 + z 2 + c2t 2. This shows that the rotational symmetry in (x, y, z)-space is lost in supersonic motion. This gives rise to the Mach cone. Actually the physical meaning of the discontinuity of the transformation (15) is that the Mach cone divides the medium into two acoustically disconnected regions. Now in order to retrieve the wave equation in the region of disturbance (viz., inside the Mach cone), at first sight it appears just sufficient to replace x ' and t' in equations (15) by ix' and it' respectively. In that case, these quantities will lose the meaning of physical distance and time for the moving observer. Also the generalized function ~ ( x - u t ) in the source term will then contain an imaginary argument whose mathematical definition disallows its presence. Therefore, instead, one can interchange x' and ct' to retrieve the wave operator and the resulting transformation is
X'=(X--C2I/U)I(1--C2/U2) 112,
y'=y,
Z'=Z,
t'=(t--X/lt)/(1--C2/li2) i/2, (16)
appropriate for the supersonic case [9]. Notice that equations (16) have been obtained by putting c 2 / u in place of v in equations (2). The Mach number o / c = x / c t < 1 is the parameter in equations (2) which connects two points on the hyperboloid x 2 - c 2 t 2 = s 2 inside the "ct-cone" x 2 - c2t 2 = O. In the region outside the "ct-cone", the interchange of space and time co-ordinates has given c / u < 1 as the parameter in the transformation (16). (The reason is that this change has, in effect, put supersonic motion back into the ct-cone.) As the inside and the outside of the ct-cone are two disjoint regions (i.e., there does not exist a LT with Jacobian +1 connecting them), it is natural that the parameters are different for them. However, since both the parameters are <1, in what sense is the LT (16) a transformation for supersonic motion? This is a pertinent question as the
396
T. S, S H A N K A R A A N D S. N. M A J | I I
origin O' of the primed frame in (16) moves with velocity c2/u < c and does not coincide with the source S which is moving with speed u > c. The meaning may be seen in the following way. As shown in Figure 4, where O is the origin and S is the supersonic source, the triangles OPS and O 0 ' P are similar, from which one has x / ( y 2 + Z2) 1/2 = R / ( l t 2 t
2 - - R 2 ) 1/2 =
1/(M 2 -- 1) 1/2.
(17)
S
o
.,
J
Figure 4. Observersin the LT (16).
Further, since P lies on the surface of the Mach cone the two roots of equation (7) coincide at P. Therefore from equation (6) one has 1/(M 2 - 1 ) 1/2 = ( y 2 + z 2 ) l / 2 / ( u t - x ) .
(18)
Solving for x from these two equations gives xut = R 2= c2t 2 whence x = c2t/u. This shows that the velocity of O' is c2/u < c and therefore the LT (16) is the one which connects the two observers at O and O'. Coming back to the question of how equations (16) represent a supersonic LT, one can first note the crucial point of the problem that, after all, an observer moving with S cannot gather any information about the sound field by using sound signals and hence the S frame represented by equations (15) is physically irrelevant; S does not even generate the sound field directly. Any physical measurement incorporating sound signals has to take place only within the Mach cone. To find an "effective observer" O'.in this region who is equivalent to one moving with S, one can note that O 0 ' / O S = ( c 2 t / u ) / u t = M -2 and therefore O' is uniquely obtained from S at any given time by a constant contraction mapping. In other words, a unique LT which is physically realizable is associated with every modified LT which is only a mathematical transformation. The moving frame attached to O' is equivalent to the one attached to S and the corresponding transformation between O and O' is a legitimate LT for supersonic motion, meeting both physical and mathematical requirements. As proof of this it is demonstrated in subsequent sections that the new LT (16) does indeed lead to the same solutions as the conventional ones for linearized supersonic problems [4, 5]. 4. KUSSNER'S METHOD FOR A POINT SOURCE IN SUPERSONIC MOTION With the appropriate LT for supersonic motion in hand, it is now possible to make use of the invariance of the wave operator just as in the subsonic case. Consider the wave equation for the velocity potential of a point source moving at speed U > c along the x-axis. It is 02~b/Ox2+c32~b/c3y2+O2~/Oz2- (1/c2)O2~/Ot 2 = - q ( t ) 8(x - Ut) 8(y) 8(z).
(19)
A LORENTZ-LIKE
TRANSFORMATION
397
In the primed frame fixed to the observer O' of the transformation (16) this becomes a2r
+
a2~/ay,2 + a 2 e , / a z, 2
_
(1/c2)a2~,/at 'z =
(r/U)q
(Fx'/U)
a(t') a(y ')(z '),
(20)
where _ F ' = ( 1 - c 2 1 U 2 ) 1/2. It may be noticed that the source term now has a(t') in the transformed variables in place of ~(x - Ut) in the old variables, yielding an initial value problem in contrast to the origin problem which is a boundary value problem. This feature of the transformation will be very advantageous in the oscillating airfoil case (see section 5). Taking the Laplace transformation [10] with respect to t' and the Fourier transform with respect to the other variables in equation (20) gives O(x') = q ( F x ' / U ) .
(21)
X [ Q [ x ' q- (c2/'2 - y ,2 _ z ,2) 1/2] _1_Q [ x ' - (c2t '2 - y,2 _ z '2)]}/(c2t'2 - y,2 _ z ,2)1/2,
(22)
~**#(t~ 2"t-fl2"b l.t2 q - p 2 / c 2 ) ~- - ( F / 2 1 r U ) Q * ( A ),
On inversion this gives ~(x', y', z ', t') = - (F/4~rU)
(23)
(y,2 d" Z'2) 1/2 ~ ct'.
In terms of the unprimed variables the inequality (23) reduces to 2
which is exactly the Mach cone (equation (18)) deduced by considering the geometry of a point source [4] moving with Mach number M. Also the solution (22) reduces to [(U~/c2)-t-R1/c'~ tp(R, t ) =
qt
~
. [ ( U x / c 2 ) - t + R1/c~
)*qt
M2-1
)
4 lr [(Ut - x ) 2 - (M 2 - 1)(y 2 + z2)] 1/2
-- q (t - R §
1 + q (t - R - / c ) / 4 r r R t,
(25)
which is the same as equation (14). 5. OSCILLATING AIRFOIL IN SUPERSONIC FLOW In this section the linearized potential equation for supersonic flow past a thin oscillating wing is considered. This has been investigated by various authors using different methods. In particular, Miles [5] started with the wave equation for the velocity potential r in dimensionless co-ordinates (non-dimensionalized by the characteristic quantities !, length and l/c, time) Cxx + Cyy + ~p~ - ~ r r = 0, (26) where X is measured downstream from the foremost point of the airfoil at time zero, y and z are positive and T is the dimensionless time. The boundary conditions on equation (26) are ~ ]z=o = - v in S,
~p[==o = 0 outside S,
(27)
where S is the projection of the wing surface on the plane z = 0. The variables in those equations are changed according to the transformation
[t;] = fl-l(D 1M)[XTT]' [ff] =f1-1(M-1 -1]//,j,"<'~r"'l
398
T. S. SHANKARA AND S. N. MAJHI
For obvious reasons this transformation is called the modified LT which is the same as transformation (15) in dimensionless form and fixes the co-ordinate frame in S; however, equation (26) is not invariant under the modified LT. The problem can now be approached in the spirit of the previous section. The GT
[7]--(;
_Mx
in dimensionless form fixes the ca-ordinate in 5'. It leaves the time invariant though not the wave equation (21). On the other hand the L L T (16) in dimensionless form, viz.,
M/It'J'
(32, 32')
leaves equation (26) invariant but not the time. Consequently, t' is not the true time. However from equations (32) and (31'), one has
[/]=f_,(M-~)[t]'
[t]=f-'(?l
M/It'l'f2'~rx']
(33,33')
Now, for the case of an oscillating wing for which the prescribed motion (in Galilean co-ordinates) is given by v = Re [5(x, z) exp (ikMt)],
(34)
the potential may be assumed in the form = Re [~5(x, z) exp (ikMt)],
(35)
where k = to//U is the dimensionless frequency (and o~ is the true radian frequency). Then by equations (33) and (33'),
kMt = kMf1-1 ( - x ' + Mt') = 71(Mfl- ix - x'),
(3 6)
with r / = kMfl -~. Hence one can write
v = V(t', z) e i"X', V = 5 exp (ikM2fl-2x),
~p = @(t', z) e inx',
(37)
q~ = ff exp ( i k M 2 f - 2 x ) ,
(38, 39)
with x = fit'. The boundary value problem of equations (26) and (27) for this oscillating airfoil (see Figure 5) now reduces to the initial value problem, q~,','- ~zz + r/2q~ = 0, q,, I,=o = - V(t'),
t'>O,
,/,1,=o= O,
(40) t'~
~_X t
Figure 5. Two-dimensionalairfoil boundary value problem.
(41)
A LORENTZ-LIKE TRANSFORMATION
399
This feature of the transformation (32) is clearly due to the interchange of the roles of x' and ct' that was effected on the modified LT (15) of Miles. Thus equations (32), or equivalently equations (16) not only ensure the invariance of the wave equation inside the Mach cone but also convert the boundary value problem into an initial value problem. Under the Laplace transform with respect to t' equations (40) and (41) become
~z*~ - A 2 ~ , = 0,
h = (s2+ 2)1/2,
(42)
cO
qb* [ , = o = - V * ( s ) ,
q~*=fo q~(z, t ' ) e - ' " d t ' .
(43,44)
The solution of equations (42) which vanishes at infinity and satisfies condition (43) is cb* = h -l V(s) e - ~ ,
z > 0.
(45)
Applying the inversion theorem corresponding to the transform (44) one obtains q~ = Jo
J~
d~r
(46)
Returning to the original co-ordinates reduces this to the same expression as given by Miles. 6. CONCLUSION In 1952 Rosen [6] gave the following result: for a homogeneous and isotropic medium with which a limiting signal speed c' can be associated, it is possible to set up a c'-relativity, i.e., a theory analogous to Einstein's c-relativity, but with c' replacing c. K/issner's [1] application of LT to iinearized subsonic motion is indeed an example of c'-relativity. But with the LT being defined only for Mach numbers M < I , it might appear that a relativistic approach is not available for supersonic motion. It is possible, however, to construct an appropriate LT for Mach numbers M > 1 as follows. First one can note that the rotational symmetry in the physical (x, y, z) space is lost when a source is moving supersonically. In contradistinction to the rest of the medium, it is only the interior of the Mach cone which is the active zone and here the wave equation continues to be invariant, therefore a physically valid and unique L L T can be obtained from Miles' modified LT by interchanging the roles of space and time variables. This has the further advantage that a boundary value problem is changed into an initial value problem which is simpler to solve in many cases. In this paper, we have first outlined in section 2 the inherent symmetry of the LT in subsonic kinematics. In section 3 the L L T has been constructed from a detailed study of the topology of the medium when a source is in supersonic motion. In section 4 it has been applied to a supersonically moving source. In the moving frame, this becomes an initial value problem and, in addition, the solution at once exhibits the two effective sources and the Mach cone with no extra effort. The potential of the new transformation has been shown further in section 5 dealing with an oscillating airfoil. It appears that sonic relativity can provide a unified foundation for the acoustics of both subsonic and supersonic sources.
ACKNOWLEDGMENT The authors are grateful to the referees for their critical comments and suggestions.
400
T. S. SHANKARA AND S. N. MAJHI REFERENCES
1. H. G. KUSSNER 1940 NACA TM 979. Allgemeine Tragslachentheorie. 2. R. W. TRUITT 1950 North Carolina State Department of Engineering Research Bulletin No. 47. (See also 1949 Bulletin No. 44 (unpublished).) 3. W. R. SEARS 1954 General Theory of High Speed Aerodynamics, Volume VIC (editor W. R. Sears). Princeton, New Jersey: Princeton University Press. 4. P. M. MORSE and K. U. INGARD 1968 TheoreticalAcoustics. New York: McGraw-Hill. 5. J. W. MILES 1959 T/re Potential Theory of Unsteady Supersonic Flow. London: Cambridge University Press. 6. N. ROSEN 1952 American Journal o/Physics 20, 161-164. Special theories of relativity. 7. 1. M. GELFAND and G. E. SHILOV 1964 Generalized Functions Volume I. New York and London: Academic Press. 8. J. A. WHEELER and E. F. TAYLOR 1966 Spacetime PI,ysics. San Francisco: Freeman. See p. 139. 9. T. S. SHANKARA and K. K. NANDI 1978 Journal of Applied Physics 49, 5783-5789. Transformation of coordinates associated with linearized supersonic motions. 10. H. BATEMAN 1954 Tables ofbztegral Transforms. New York: McGraw-Hill.