In the present study we consider the cases of integrability of hyperbolic equation with variable coefficients. For this purpose, a Fourier transform is used in combination with a special representation of the transform in image space. Various versions of closed solutions are formulated with the help of introduced arbitrary functions. The solutions obtained in the work are absent in the known reference manuals on differential equations, and the results obtained for continuously-heterogeneous anisotropic media with cylindrical or spherical symmetry at certain ratios of elastic constants of the material complement the well-known studies of wave processes in similar media [4,5] .
2U (r, t)
2U (r, t) r 2 t 2 A(r) U (r, t)
r where A(r), B(r) C[1,a] . represented by a Fourier integral, recorded as formulas: differential equation in the
area
B(r)U (r, t) 0 ,
(
0
We apply transformation (
2 ~ ~ U (r, p) U (r, p)
[B(r) p 2 ]U~(r, p) 0 .
r 2 r
The solution of the equation is represented in the form: U (r, p) (r) G(s), s p (r) ,
where (r), (r) и G(s) are twice continuously-differentiable functions of their
arguments. As a result of setting (
ds 2
2 ( ' ) 2
1 ( ' ) 2
' ' G s ( ' ) 2 2 'ψ ψ'
A dG
' ds
'' A ' B G 0,
which can be satisfied in various ways. In works [6, 7] we obtained closed solutions
of equation (
Suppose we have the following equations: ' ' (r) (r) (r) ' (r) A(r) (r) 2
(r) ' (r) ' (r)
0; ' ' (r) A(r) ' (r) B(r) (r) 6; [ ' (r)]2
The general solution of the latter equation can be represented as follows [8]: G(s) C1 ( p) 3s cos[s C2 ( p)] (1 s32 ) sin[s C2 ( p) . (10)
Performing convertion equations (
Suppose that condition (
2 1; [ ' (r)]2 ' (r) (r) ' (r)
' ' (r) A(r) ' (r) B(r) (r) 2 ; Here I (s),Y (s) - Bessel function of " " order of I and II kind.
Performing conversion of expression (
Further, from (15) it follows that B(r) 1
2 A'(r) A2 (r) 4 2. Problem statement
The differential equation of motion of continuously-heterogeneous anisotropic elastic medium in the case of its axis-symmetric deformation, as well as the equations of state, connecting components of the stress tensor and the displacement vector, are recorded as follows [9].
r * r*r n( r*r * ) r *
* r*r c11 r * nc1*2 Ur ** , * U * 2U * t*2
0, * U * * *yy c12 r * c2*2 (n 1)c2*3 U * r * .
Here r*r (r * , t * ), *yy (r * , t * ), * (r * , t * ) are the relevant components of the normal stresses; U * (r * , t * ) are the radial component of the displacement vector; ci*k (r* ), * (r* ) are respectively elastic characteristics and density of heterogeneous anisotropic medium; r * , t * are radial coordinate and time; n 1,2 is value, corresponding to the cylindrical and spherical cavities.
After substitution of correlations (21) into (20) and introducing the dimensionless quantities by formulas (19) (20) (21) сik rr с*
i*k , r a33 r*r ,
* a33 r*
,U a
U * a ,
, t * 0 1 a
* a33 t*, 0 * , yy * a33 *yy .
* a33
Here a is the cavity radius; a3*3, * are corresponding stiffness coefficients of density of homogeneous anisotropic medium. Equation (20) and correlation (21) in area :{t 0,1 r a} are defined by the system of equations 2U (r, t)
С(r)
1 . This corresponds to a constant speed of propagation of elastic waves in anisotropic heterogeneous medium. Consequently, the differential equation (22) of hyperbolic type models wave processes propagating at finite speed.
Case 1 * .
From comparing the respective equations (24) and (13), it follows: Theorem III.
Expression (11) defines dynamic displacements, and by formulas (23) and heterogeneous anisotropic medium stress, if its elastic characteristics сik (r) satisfy (22) (23) (24) (25) (26) the functional equation 2 4n2 (n 1)(с12 с23 ) с22 r dс12 .
dr
Let us consider as an example a heterogeneous anisotropic medium, the elastic characteristics of which are periodic functions of the radial coordinate, i.e., с11 (r) a11 sin mr, с12 (r) a12 sin mr, 1 (27) с23 (r) a23 sin mr, с22 (r) a22r 2 (sin mr ). sin mr
After substitution of equations (27) in the criterion formula (26) and simple algebraic transformations, we obtain relations connecting the elastic constants of the material functional relation, connecting characteristics сik (r) 2 4n 2 (n 1)(с12 с23 ) с22 r dс12 .
dr
Suppose that characteristics of material сik (r) are changed by the power law с11(r) a11r m , с12(r) a12r m , с23(r) a23r m , с22(r) a22r m.
Here m is an index of anisotropic medium heterogeneity; aij ; i, j 1,2,3 are dimensionless constants of anisotropic material.
Then relation (29) takes the form 2 a11 n(n 1)a23 a22 n(n 1) ma12 .
From (31) it follows that three of the four parameters aik can be selected arbitrarily. If the elastic characteristics of the material change by the cosine law, с11(r) a11 cos mr, с12 (r) a12 cos mr, с23(r) a23 cos mr, с22 (r) a22r 2 (cos mr the elastic constants aik of the material satisfy relations a12 a11 , a22 2 m 2 4n a11, a23 1
1 4 1 4 2
a11 . n(n 1)
1 cos mr ), (28) (29) (30) (31) (32)
Thus, in this case, only one elastic constant of the anisotropic material is arbitrary.