Investigation of Mathematical Model of Acoustic Wave Propagation Through Relax Environment in Ultrasound Diagnostics Problems

Abstract

A mathematical model of the process of an acoustic wave propagation in a relax environment has been investigated. This mathematical model is widely used to describe and determine the basic parameters of the wave process in the problems of ultrasound diagnostics. The model is formulated in the form of the Cauchy problem for hyperbolic equation of third order with the initial data, which are analytical functions. The class of entire functions, which is the class of existence and uniqueness of the Cauchy problem solution for the partial differential equation, which describes this wave, is established. In the selected class of functions, the Cauchy problem solution is constructed using the differential-symbol method. Examples of solving problems with specific initial data are given. The obtained results and the indicated methodology allow us to determine the basic parameters of the process of acoustic wave propagation in the problems of ultrasound diagnostics.

Keywords

• Mathematical model
• wave process
• ultrasound diagnostics
• initial problem
• differential-symbol method

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Simulation of biomechanical processes in medicine is an extremely important area of scientific researches [1–3]. Such modeling is often based on existing models. For example, models of continuous-environment mechanics (in vibration problems [5–7]) and models of gas-hydrodynamics problems [8, 9] are used particularly in models of biological and medical processes [4]. A characteristic feature of modern models is the using of nonlinear partial differential equations, in addition to the ordinary differential equations. The study of such models is quite complicated (see, in particular, [10–12]). Numerical, qualitative, and asymptotic methods are used to research such models [13–15].

In recent years, in modeling complicated biomedical processes of diverse nature, the interest has been increasing not only in traditional partial differential equations of second order, but also in equations of higher order. The wave processes with dispersion and absorption in water dynamics problems [16, 17], viscosity theory [18], and geophysics [19] are described by partial differential equations of third order with respect to time. In particular, such equations include the equations of fourth order in spatial variables which describe the processes of vibration of mechanical systems [20, 21].

The hyperbolic equations of third order in time, which are intensely studied in ultrasound diagnostics, include the equation of the form

  3tu3  c 2f  ut   2tu2  ce2u  f (t, x), x  (x1, x2 , x3 )  3 , ( 1 ) in which u(t, x) is dynamic pressure,  is relaxation time, constants ce and c f are limiting phase speeds of sound,   2 x12  2 x22

2  x32 is three-dimensional Laplace operator.

In research [22], the solution of the Cauchy problem for equation ( 1 ) is given by the fundamental solution of equation ( 1 ) using modified Bessel functions.

The work is aimed to:  study of the mathematical model of the acoustic wave propagation process in a relax environment with given initial data, which are entire analytical functions;  establishing a class of unique solvability of the corresponding Cauchy problem;  presentation of the analytical method of solving the problem;  study of the process of acoustic wave propagation for the specific initial data of the problem, development of a method of finding the determining parameters of the wave process. 2

Posing of the problem and main results Let us consider the Cauchy problem    2  2       t  t 2  t 2

   u(t, x)  f (t, x), (t, x) 

 u u(0, x)  0 (x),

t where  is the constant which belongs to the interval ( 0,1 ) , 2u (0, x)  1(x), t 2 (0, x)   2 (x), x 

3,   3,   (0, ) .

( 2 ) ( 3 )

Note that equation ( 2 ) is obtained from ( 1 ) by introducing dimensionless variables xi  xi / (cf ) і   cf / ce .

 dt3

In work [23], the solution of problem ( 2 ), ( 3 ) is based on the fundamental solution of equation ( 2 ), but it has a very complicated structure and needs simplification.

In this research, we recommend another approach to solving problem ( 2 ), ( 3 ). We use the differential-symbol method, which was effectively applied to solving the Cauchy problem [24] and two-point in time problems [25–27].

Let us write ordinary differential equation  d 3  ddt22   ddt ( 4 )

  V (t, , )  0 ,  in which 

is the Laplace operator symbol, that is  | |2 12  22  32 ,   (1, 2 , 3 )  3 .

Let 1  1( , ) , 2  2 ( , ) , 3  3 ( , ) is the roots of the algebraic equation

 3  2     0 .

They belong to the set  1    3 3 2(1 3 ) 3A  A ,  1  (1 i 3)(1 3 )  (1 i 3) A  , 33 2 3 33 4 A 6 3 2  ( 5 ) ( 6 ) They are entire functions of variable  12  22  32 and vector-parameter   (1, 2 , 3 ) according to the Poincare’s theorem [28].

Provided 1  2  3 functions ( 6 ) have the form where A  3 B  4(1 3 )3  B2 , B  2  9  27 , i2  1.

Remark 1. If   0 , then B  2 , A   3 2 , roots 1 , 2 and 3 of equation ( 5 ) are independent of the parameter  , in particular, 1( , 0)  1, 2 ( ,0)  0, 3( ,0)  0 .

1 1 1 Remark 2. If    ,   , then 1  2  3   . For the other pairs ( , ) 3 9 3 at least the two roots are different.

In the case 1  2  3  1 , the elements of normal fundamental system of solutions of equation ( 4 ) have the form

V0 (t, , )  32 (3  2 )e1t  31(3  1)e2t  21(2  1)e3t ,

(3  2 )(2  1)(3  1) V1(t, , )  V2 (t, , )  (32  22 )e1t  (32  12 )e2t  (22  12 )e3t

(3  2 )(2  1)(3  1) (3  2 )e1t  (3  1)e2t  (2  1)e3t (3  2 )(2  1)(3  1) .

, ,

, e1t  e2t  (2  1) t e2t .

If 1  2  3 , then the normal fundamental system of solutions of equation ( 4 ) is the following:

V0  1 1t  1 t 212  e1t ,

 2  V1  t 1 1 t  e1t , V2  1 t 2e1t .

2

Remark 3. According to Remark 1, if   0 the condition 1  2  3 is fulfilled and functions ( 6 ) have the following form:

V0 (t, , 0)  1, V1(t, , 0)  t, V2 (t, , 0)  et 1 t .

Remark 4. If    1 and   1 , then according to Remark 2 all roots of equa3 9

1 tion ( 5 ) are identical and equal to  . Functions ( 6 ) have such form

3 V0  1 1 t  1 t 2  e13t ,  3 18 

1  1t V1  t 1 3 t  e 3 , 

1 V2  1 t 2e3t .

2

Let the initial functions 0 (x), 1(x), 2 (x) and right-hand side f (t, x) of equation ( 2 ) are arbitrary entire functions. Then there is only one solution of problem ( 2 ), ( 3 ) in the class of entire functions. This solution can be presented in the form 2    u(t, x)  k    Vk (t, , ) e x  k 0  O

2  et   kVk (t, , )   ,    k 0  f       3   2        e x      0, O 1    ,  2    and f   ,   are obtained from the functions 0 (x) , 1(x) , 2 (x) and f (t, x) with the change x to  

and t to .

 

The differential expressions 0    , 1    ,  2    and f   ,   for entire data of problem can be defined as corresponding Maclaurin series. The actions of these expressions on functions in curly brackets of formula ( 8 ) are correct because the functions in brackets are entire functions of first order with respect to the vector  and the variable  in the last brackets.

Function ( 8 ) satisfies equation ( 2 ). It follows from the commutativity of differentiation operators  ,  ,  and  , and from the fact that functions ( 6 ) satisfy t x   equation ( 4 ). Also, the function u(t, x) of the form ( 8 ) satisfies initial conditions ( 3 ), since these differentiation operators are commutative, and corresponding initial conditions are satisfied for functions ( 6 ).

The fact that the found solution of the Cauchy problem in the class of entire functions is unique, can be proved by method of contradiction (see, for example, [25]).

The main result. The process of an acoustic wave propagation in a relax environment with data at the initial (zero) moment of time is described by the Cauchy problem ( 2 ), ( 3 ) for hyperbolic equation of third order; equation ( 3 ) is important in the problems of ultrasound diagnostics; a class of entire functions as a class of uniqueness solvability of problem is established; formula ( 8 ) for constructing the solution of the problem is proposed. 3

The examples of application of the developed method and constructing the solution of problems with specific initial data Let us investigate the process of acoustic wave propagation for specifically given initial functions and the right-hand side of the equation, which are integer functions. We use the method of constructing the solution of problem from the previous section.

Example 1. Let the initial functions in problem ( 2 ), ( 3 ) and the right-hand side of equation ( 2 ) are polynomials, such as 0 (x)  x1  x2  x3 , 1(x)  2x2 , 2 (x)  x12 x3 , f (t, x)  3 . Then the solution of problem exists in the class of entire functions. It is unique and can be found by formula ( 8 ) due to Remark 1 and 3:

  u(t, x)     1 

  2   

V0 (t, , ) e x  3   O  2  V1(t, , ) e x   V0 (t, , )

 3  O V1(t, , )  O  0  1 t3  (1 2 )t 2  x12 (et 1 t)  3 t 2  2t  2  2et .

3  2

Since the initial data were polynomials, the obtained solution of the problem is also a polynomial. Only operations of differentiation were used to construct the solution.

We note that the solution of problem linearly depends on parameter  .

Example 2. Let us describe the process of an acoustic wave propagation with zero initial conditions for   1 under the influence of external force f (t, x)  et sin x1 .

9 3 Therefore, we find the solution of problem ( 2 ), ( 3 ), for 0 (x) 1(x) 2 (x)  0 x and f (t, x)  et sin 1 . Then according to Remark 4 by formula ( 8 ), we get 3

   1   u(t, x)  e  sin  3  1  H ( , ,t, x) ( 7 ).

We have u(t, x)  e sin  1  

 3  1  H ( , ,t, x)  sin  1    3  1  H ( , , t, x) Therefore, u(t, x)   27  et  1  2 t  2 t2  e13t  sin x1 .

8   3 9   3

The obtained solution of the problem u(t, x) does not depend on the coordinates x2 and x3 . If t   it goes to zero.

The function u(t, x) describes periodic oscillations of an acoustic wave with a period T  2 3 for coordinate x1 (see fig. 1). The amplitude of these oscillations is determined by the formula

A(t)   27  et  1  2 t  2 t2  e13t  .

8   3 9   Graphic time dependence of amplitude 3  is depicted on Fig. 2 with solid and dashed lines. 4

A(t) and u t, x for x1  3  6 and profiles of sound wave for x1  Only values of function H ( , , t, x) in the points (1, i, 0, 0, t, x) and (1, i, 0, 0, t, x) were used to solve the problem. The equalities that are correct for an arbitrary integer function H ( , , t, x) with variables  1 ,  2 ,  3 and  were used. 4

Conclusions The class of existence and uniqueness of the Cauchy problem solution for the hyperbolic equation of third order has been established. The method of constructing the solution of Cauchy problem for arbitrary entire initial functions and an arbitrary entire right-hand side of the equation is given. In the case if the data of problem has a quasipolynomial form, according to the proposed method, the solution of Cauchy problem can be found only with operations of differentiation. In particular, it is illustrated by Examples 1 and 2.

The results of the researches can be used in the problems of ultrasound diagnostics. The obtained results and the developed method constitute an important theoretical basis for the mathematical modeling of the acoustic wave propagation process in a relax environment.

The main conclusion of the application of the presented results in medical practice is the possibility to find exactly the determining parameters of the wave process.

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