Mathematical Model of the Process of Ultrasonic wave Propagation in a Relax Environment with its Given Profiles at three Time Moments

3.1. Posing of the Problem

Eq. (1) of ultrasound oscillations by replacement x =(c₁τ)⁻¹y, α= (с₂)⁻¹c₁ is transformed into the one-parameter (parameter α Î (0, 1)) hyperbolic equation

(2)

The mathematical model of the process of ultrasound wave propagation which contains Eq. (2) and the profile of oscillation is given at three equidistant moments of time t = jh, where 0 (Eq. 3):

(3)

is considered [13]. In the presented paper, we obtain more general results.

Let us consider the problem for the partial differential equation of the third order (Eq. 4).

(4)

Which describes not only ultrasound waves as Eq. (2), but also different waves depending on the operator coefficients and , where

(5)

are complex numbers (Eq. 5).

The profiles of wave are given at time moments t₀, t₁, t₂ by conditions (Eq. 6):

(6)

Let us find a solution to problem (4)-(6) using the differential-symbol method [22].

According to Eq. (4), the ordinary differential equation with parameter μ has the form (Eq. 7):

(7)

where .

To simplify the form of formulas, we will write a = a (μ), b = b (μ), c=c(μ). Let us denote the roots of the characteristic (cubic) equation (Eq. 8):

(8)

by . These roots have the following image (Eq. 9):

(9)

Where and the cubic root of the one belongs to the second quarter.

If the determinant

Of the polynomial does not equal zero, then Eq. (8) has simple roots, moreover, if D=0, then the roots (9) are multiples, in particular

for a² ≠ 3b and λ₁ = λ₂ = λ₃ = −a/3 for a² 3b.

For real numbers a, b, c if D ≥ 0 there are real roots (if D > 0 they are different) and if D < 0 we get real and two complex roots.

The geometric interpretation of the real roots is given in Fig. (1), where

is the set of roots and σ = S+ i Imσ ϵ C, φϵ [0,2π/3), r≥0 (we have a double root for φ = 0, φ = π / 3 and a triple root S for r = 0).

For the equation of the third order (7) with the coefficients, which is integered by the vector-parameter μ, we write the normal fundamental at the point t ₀ system of the solutions which is defined uniquely. In addition, we find the fundamental system of solutions of Eq. (7), which satisfy conditions

(10)

The system can be constructed only for vectors for which the following condition

(11)

is realized.

Therefore,

(12)

Set M, which is defined by formula (12), is divided in three sets , moreover for vectors μ ϵ M₀ roots of Eq (8) are simple, for μ ϵ M₁ there are simple and double roots among the roots of Eq. (8) and a triple root for vectors μ ϵ M₂.

Let us consider the array , which has the form

and array of the form

If μ ϵ M ₀ (simple roots), then

(13)

For the case μ ϵ M₁ (root λ₁ is simple and λ₂ is double), then

(14)

Eqs (13, 14) are correct (determinants are nonzero) because m = 1,2,3 and we have

For the case μ ϵ M₂ (triple root) formulas of the elements of the fundamental system of solutions of Eq. (7), which satisfy conditions (10) are simplified and take the form

(15)

We introduce the quasipolynomial functions of the form

(16)

where Q(x) is nonzero polynomial of variables x₁, x₂, x₃ with complex coefficients, ν₁, ν₂, ν₃ are complex numbers, moreover, the vector ν = (ν₁, ν₂, ν₃) satisfies the condition δ(ν) = 0, i.e. ν ϵ M where M is the set (12).

For the set (12), K_M is a class of functions which consists of the quasipolynomials of variables x₁, x₂, x₃ which can be represented as a finite sum of quasipolynomials of the form (15) with the different vectors. Also, we suppose that the zero quasipolynomial g(x) = 0 belongs to K_M.

For each quasipolynomial of the form (15), the differential expression can be obtained by replacing the vector x with the vector-derivative . The differential expression of the infinite order acts on the analytic function γ(μ) at the neighborhood of m= n by the formula

Similarly, we determine the effect of differential expressions of infinite order on the analytic functions for more general quasipolynomials of the class K_M. We also assume that ≡ 0 for g = 0.

3.2. Main result

For arbitrary functions g ₀(x), g₁(x), g₂(x) from the class K_M there is a unique solution of the problem (4), (6), which is represented by the formula

(17)

moreover, u(t,.) ϵ K_M for all t ϵ (0,∞).

In Eq. (17) we assume O = (0,0,0) .

The fact that function (17) satisfies the partial differential Eq. (4) follows from the commutativity of the differentiation operators , , and that the functions satisfy the ordinary differential Eq. (7).

For the function of the form (17) three-point conditions (6) are carried out, since satisfy conditions (10) and the equalities

are realized.

The solution of three-point problem (4), (6) in the class of quasipolynomials for each fixed t ϵ (0,∞) belongs to K_M, is unique. This is proved by differential-symbol method (see, for example [23],). Choice of the functions g ₀(x), g₁ (x), g₂ (x) from class K_M is significant.

Obviously, if the functions g ₀ (x), g₁ (x), g₂ (x) in conditions (6) belong to the set K_M, then according to Eq. (17), the solutions of problem (4), (6) are quasipolynomials.

Numerous studies [26-30] have been devoted to the construction of solutions of some partial differential equations and boundary value problems for these equations in the form of quasipolynomials.

Thus, the process of oscillation propagation, in particular, ultrasound wave in a relaxed environment with given wave profiles at three points at time is investigated.

The process is modeled by the problem (4), (6) for the partial differential Eq. (4) of the third order in time with three-point time conditions (6). The method of constructing the solution of problem (4), (6) is indicated. The class of quasipolynomials KM is distinguished as the class of existence of the unique solution of problem (4), (6). Eq. (4) belongs to partial differential equations, which are widely used in problems of ultrasound diagnostics.

3.3. Example of Research of the Process of Ultrasound wave Propagation with its given Profiles at three Moments of Time

Let us investigate the process of acoustic oscillations using the proposed method for a specific example.

Example. We study the process of propagation of an ultrasound wave in relax environment, which is described by problem (4), (6) with the following parameters:

The profile of wave at the moment of time t=1 is a sum of two quasipolynomials of the form (16) that is

in which the polynomials are respectively, then the vector-parameters in the exponent are equal to and −ν. These vectors ν and −ν belong to the set M₂ and the corresponding root of Eq. (8) is triple:

Function (14) take the form

Since , then

The solution of problem (4), (6) with given parameters can be found by Eq. (16):

We calculate

Finally, we obtain the solution to the problem

Let us investigate the solution of the problem on parallel planes x₁ + x₂-x₃ - 3h = 0 of the space R³, where . Then the solution of problem has the form (Eq. 18):

(18)

and it is 2p-periodic function by variables h.

The graph of the function (18) of two variables t and h is given in Fig. (2).

Therefore, function (17) describes the periodic oscillations of the ultrasound wave with the period T = 2π by the variable η. For the amplitudes of oscillation are given in the following form:

These amplitudes are shown in Fig. (3) by a solid and dashed line.

As mentioned above, the found solution of problem is unique in the class of quasipolynomials and belongs to K_M each fixed t. Therefore, the ultrasound wave oscillates in a limited range at any time and exponentially goes to zero for on planes which is parallel to the plane x₁ + x₂-x₃ = 0 and acquire zero values on the planes .