Applied Mathematics
General solution of the Cauchy problem for
the acoustic equation in the form of dynamic
ray expansion
Valery G. Yakhno and Faruk Uygul
Dokuz Eylul University, Faculty of Arts and Sciences, Buca, 35160, Izmir, Turkey e-mail:valery.yakhno@deu.edu.tr
e-mail:faruk.uygul@deu.edu.tr Received: December 07, 2001
Summary.
Main object of this paper is the acoustic equation. Co-ecients of this equation are smooth functions of the space variables. Existence and uniqueness theorem of the generalized solution of the Cauchy problem in the form of the dynamic and classical ray series is given.Key words:
Cauchy problem, generalized solution, acoustic equa-tion, ray series expansionMathematical Subject Classication (2000): 35D05, 35L15,58J47
1. Introduction
Construction of a solution by classical series expansion for Helmholtz (wave) and hyperbolic equations was studied by V. M. Babich 1]. The fundamental solutions for the second order hyperbolic equations in the form of Hadamard series expansion were studied by V. M. Babich 1,2], V. G. Romanov 3]. We note that dierent forms of the presen-tation for generalized solutions of hyperbolic equations have been applied to study the modern problems of mathematical physics and applied mathematics (see, for instance, works 1,3,4]). The main ob-ject of the present paper is the dynamic ray series expansion for a generalized solution of the Cauchy problem for the acoustic equation.
Let x,tbe variables, x= (x1x2x3) 2R 3t 2Rx 0= (x0 1x 0 2x 0 3) 2R 3
be a parameter. Consider the following acoustic equation (1) utt =v2(x)u ;v 2(x) r(lnm(x))ru+f(xx 0t):
Here u(xx0t) is a preasure of the acoustic medium at the point
x and at the moment t v(x) describes the velocity of the wave,
v(x)>0 m(x) is the density of the acoustic media x, m(x) > 0
f(xx0t) is a given density of externally acting forces is the
Laplace operator with respect to the variable x r is the gradient
operator relative to the variablex. The equation (1) we will consider with the initial condition
(2) u(xx0t)
jt< 0= 0
and suppose thatf(xx0t) =(x ;x
0t) which means a pulse point
source concentrating at the point x =x0 and at the momentt= 0.
From mathematical point of view(x;x
0t) is the Dirac delta
func-tion with the support at the points x = x0, t= 0. A generalized
function u(xx0t) which satises (1), (2) is called a generalized
(fundamental) solution of the Cauchy problem (1),(2).
Let (xx0) be the time required for the signal to get from the
point x0 to the point x . This function (xx0) satises the eikonal
equation jr(xx 0) j 2 = 1 v2(x)
and the condition
j(xx 0)
jx !x
0 = 0:
The equation(xx0) =tdenes the wave front from the point source
at x0 at the time t. The surface (xx0) = t is the characteristic
conoid. The method of constructing characteristic conoid(xx0) =t
consists in constructing separate lines, called bicharacteristics, that lie on the conoid and jointly form it. Projection of the bicharacteristic onto the space of x is called a ray. The rays are orthogonal to the surfaces(xx0) =t. For nding rays we need to solve Euler's system
(see, for instance, 3]).
In this paper x0 is a xed point and the following assumptions
hold.
(A1)
functionsv(x), m(x) have constant values inU" =fx2R 3 x
;x 0
"g
where "is xed positive small number, that is ,v(x) =v0 m(x) =m0
(A2)
functions v(x), m(x) are smooth (innitely dierentiable) with positive values(A3)
the family of the rays fT(x 0x) gx0 2R 3x 2R 3 is regular. Thismeans that for each pair of points x0, x from R3 there exists a ray
T(x0x) which connects these points and this ray is unique.
Remark 1.
The requirement(A3)
may be formulated in the other terms (see, for example 3]). For this we introduce ray coordinates. Letx0 be a xed point in R3, andxbe an arbitrary point inR3. Let= ( 1 2 3) be an arbitrary unit vector. Let us consider a ray
which goes throughx0 and has the tangent vector at the direction of
. Letxbe an arbitrary point of this ray then this point can be given with the Riemann (ray) coordinates x = f(x0), where = ,
( = (123)).
(A3a)
For each pair of points x0, from R3 the inequality@f(x0) @ det @f(x0) @ ! 6 = 0
holds, and the function f(x0) has the inverse one =g(xx0) for
which g(x0x0) = 0, and J(x) @g(xx0) @x = 2 4 @f(x 0) @ =g(xx 0 ) 3 5 ;1 @g(xx0) @x x=x 0 = 1:
Here J(x) is Jacobian of the transformation from Cartesian to ray (Riemann's) coordinate system.
Remark 2.
The requirement(A1)
means that the rays inside the"-neighborhoodU"(x0) are straight lines, and outsideU"(x0) are
suciently arbitrary smooth curves.
2. Existence and uniqueness theorem
Let v0, m0, ", T be xed positive numbers, t0 = "=v0 < T. We
suppose here that x 2 R 3, t
2 0T]. This section deals with the
existence and the uniqueness of the Cauchy problem (1),(2) in the class of functions having form of the dynamic ray series expansion forx2R
3,t
20T].
Theorem 1.
Letx0be a xed point. Under assumptions(A1){(A3)
there exists unique generalized solution having the form
(3) u(xx0t) = 0(t) N X k=;1 k(x)k(t;(xx 0)) +uN(xt)]
where ;1(;) =(;) is the Dirac delta function.
0(;) =
1 ; 0
0 ; <0
is the Heaviside function and
k(;) = (;k)!k0(;) k= 123:::
the functions k(x) k =;10123:::for x 2U"(x 0), t 20t 0] are dened by (4) ;1(x) = 1 4v2 0 jx;x 0 j k(x) = 0 k= 012::: uN(xt)0 for x2U"(x 0) t 20t 0]
the functionsk(x) k =;1012:::forx2R 3 nU"(x 0),t 2t 0T] are dened by (5) ;1(x()) = 1 4v2 0" s v(x)m(x) v0m0J(x) k(x) = s v(x)m(x) J(x) (6) Z T(x 0x ) 1 2 s v()J() m() r(lnm())r k;1() ; k;1()] jdj k= 0123:::N
the function uN(xt) for x 2 R 3
nU"(x 0), t
2 t
0T] has the form
uN(xt) =N
+1(t
;(xx
0))~u(xt), where u~(xt) is a function from
CN;1(R3 t
0T]).
Proof.Substituting (3) into (1) and using tools of generalized func-tions theory and reasoning from the works 3, pp.111{121], 4] and the following properties of the Dirac delta and Heaviside functions
(t)(t;(xx 0)) = 0 0(t) k(t;(xx 0)) = 0 k= 012::: 0(t) (t;(xx 0)) = ;4(xx 0)v3 0(x ;x 0t) @ @tk(;) =k;1(;)k= 012::: @ @t;1(;) = 0(;) ; =t ;(xx 0)
we nd the following recurrence relations fork(x),k=;101:::N (7) 2r ;1(x) r; ; ;1(x) r(lnm(x))r; + ;1(x) ; =0 and 2rk(x)r;;k(x)r(lnm(x))r; + k(x) ; (8) =r(lnm(x))r k;1(x) ; k;1(x) k = 012:::
and equations for the remainder termuN(xt)
(9) uNtt =v2(x)uN ;v 2(x) r(lnm(x))r xu N +f N(xt) (10) uN(xt 0) = 0 uNt(xt0) = 0 where fN(xt) =v2(x) N(x);r(lnm(x))r N(x)]N(;):
Since functions m(x), v(x) are innitely dierentiable then J(x),
(xx0), (x) are innitely dierentiable for
jx;x 0 j ". Hence fN(xt) 2 HN(R 3 (t
0T)). Applying technique of the work 1,
pp.98{105] we obtain formulas (4){(6) from equations (7),(8). Using hyperbolic equations theory to the problem (9),(10) we can nd the existence and uniqueness of the solution of (9),(10)uN(xt) having
the form uN(xt) =N +1(t ;(xx 0))~u(xt)u~(xt) 2C N;1(R3 t 0T]):
3. Formal relation of the dynamic and classical ray series
Let us consider the solutionu(xt) of the problem (1), (2) in the form (3) in whichN =1,t2R,x2R 3, that is (11) u(xt) = 0(t) 1 X k=;1 k(x)k(t;(xx 0)):
A formal application of the Fourier transform with respect to the time variablet2Rto the series (11) leads to classical ray series (see
1]) (12) u~(x!) =ei!(xx 0 ) 1 X k=;1 k(xx0) 1 (;i!)k +1
where ~ u(xt) =Ftu(xt)](!) Ftu(xt)](!) = Z 1 ;1 ei!tu(xt)dt x 2R 3 t 2R:
For proving it we need to use the following properties of the Fourier transform. Ftf](!) = 1( ;i!)m Ftdmdtfm(t)](!) Ftk(t;(xx 0))](!) = 1 (;i!) k +1Ft(t ;(xx 0))](!) = 1 (;i!) k +1e i!(xx 0 ) k= ;1012::::
Remark 3.
Consider the acoustic equation in the frequency domain (13) !2u~+v2(x)u~;v 2(x)
r(lnm(x))r~u+(x;x
0) =0:
If we will seek a solution of this equation in the form of the classical ray series (12) applying the reasoning of the work 1], we get the relations (7), (8) fork(x) . We note also that in this case we nd a
formal solution of the equation (13).
References
1. Babich, V. M. (1999): The higher-dimensional WKB method or ray method. Its analogues and generalizations, in: Partial Dierential Equations (Edited by Fedoruk M. V. ) Springer-Verlag, Berlin, 91{131.
2. Babich, V. M. (1965): On short-wave asymptotics of a solution of the point source problem in an inhomogeneous medium. USSR Comput. Math. Math. Phys. 5,No.5, 247{251.
3. Romanov, V. G. (1986):Inverse Problems of Mathematical Physics,VNU Sci-ence Press, Utrecht.
4. Yakhno, V. G. (2001): Multi-dimensional inverse problem for acoustic equa-tion in ray statement, in: Inverse Problems in Underwater Acoustics, Springer-Verlag, New York, 231{266.
