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The problem of diffraction of a plane acoustic wave by a finite soft (rigid) cone is investigated. This one is formulated as a mixed boundary value problem for the three-dimensional Helmholtz equation with Dirichlet (Neumann) boundary condition on the cone surface. The diffracted field is sought as expansion of unknown velocity potential in series of eigenfunctions for each region of the existence of sound pressure. The solution of the problem then is reduced to the infinite set of linear algebraic equations (ISLAE) of the first kind by means of mode matching technique and orthogonality properties of the Legendre functions. The main part of asymptotic of ISLAE matrix element determined for large indexes identifies the convolution type operator amenable to explicit inversion. This analytical treatment allows one to transform the initial diffraction problem into the ISLAE of the second kind that can be readily solved by the reduction method with desired accuracy depending on a number of truncation. All these determine the analytical regularization method for solution of wave diffraction problems for conical scatterers. The boundary transition to soft (rigid) disc is considered. The directivity factors, scattering cross sections, and far-field diffraction patterns are investigated in both soft and rigid cases whereas the main attention in the near-field is focused on the rigid case. The numerically obtained results are compared with those known for the disc.

A contemporary nondestructive testing and acoustic diagnostics of materials exploit the modelling simulation. The latter provides for interaction of waves with defects of canonical shapes for which some analytical and semi-analytical solutions of corresponding diffraction problems can be obtained. These solutions play a key role in benchmark data for common numerical methods. On the other hand, it is of importance to take into account physical characteristics of defects and constructions for obtained reliable results of diagnostics in a wide frequency range. It is clear that solutions of diffraction problems on impedance surface very often cannot be obtained in analytical forms. But one can obtain a solution by analytical method for soft and rigid surfaces which are the boundary cases of impedance. So here, we contemplate as a model of construction or defect a finite cone with these surfaces.

In the scientific literature, a significant number of works are devoted to the study of diffraction of acoustic waves in semi-infinite cones with different types of boundary conditions (Dirichlet, Neumann, impedance boundary condition). Infinite circular cones [

The Wiener-Hoрf method in combination with the method of Kontorovich-Lebedev integral transformations is used for the solution of the diffraction problem on finite hollow cones (where discs are considered as particular cases of cones) [

In this article, based on analytical regularization procedure [

Let us consider the perfectly soft (S) rigid (R) hollow finite cone

where

Since the velocity potential

where

The unknown potential

and satisfies the Dirichlet (S) or Neumann (R) boundary conditions on the surface of the cone

Here

In order to obtain the unique solution to the problem (2), (3), the additional conditions must be imposed on the unknown velocity potential

and the condition of the finiteness of energy in any bounded volume (edge condition) given as:

For solution of the diffraction problem let us decompose the space

and determine the total field in the form of

Since the unknown scalar potential

Here

where

For further convenience, in (5) we introduce

The condition (6) guarantees satisfying the boundary condition at the conical surfaces for field presentation (5), as well as the finiteness of energy in the conical vertex. The Equation (5) satisfies the radiation condition at infinity.

We expand the scalar potential of an incident plane acoustic wave (1) in the series of spherical functions. Accounting a definition of indices

with

To find the unknown expansion coefficients in the (5), we use the mode matching technique

Substituting the relationship (5), (7) into Equation (8) leads to the series equations. In order to take into account the singularity of velocity of particles

where the prime indicates the derivation with respect to the argument.

In order to reduce series Equation (9) to the infinite system of linear algebraic equations (ISLAE), we use a property of orthogonality of Legendre functions, which leads to [

Here upper sign (“+”) corresponds to

First, we analyze the series Equation (9) for soft cone (S-case). For this purpose we substitute series (10) into Equation (9). Next, limiting the finite number of unknowns and excluding

where

The main reason of this limitation is to provide the correct transition from Equation (12) to ISLAE (

Next, in Equation (12) we pass to limit

Here

where

Then we turn to analysis of rigid cone (R-case). To obtain the correct solution we take into account the values of pressure independent from

and the others are determined by Equations (12a), (12b) with

According to our previous step for S-case, we introduce a growing sequence

For this case we use the definition of roots of transcendental equations by way of (6b).

Further, passing to limit

Taking into account the asymptotic properties of the modified Bessel and Macdonald functions for large indices, it is found that

which is correct for S and R cones.

Let us introduce the operator formed with the main parts of the asymptotic expression (15) as

and

Here

where

Next, we formulate original diffraction problem (14) via the ISLAE of the second kind as follows:

The technique described above is elaborated in [

ISLAE (22) admits the solution in the class of sequences

We represent the other unknown coefficient in both S and R cases through the solution (22) by way of

where

Let us rewrite the basic ISLAE (22) for both Dirichlet and Neumann cases by way of

where

We take into account the low frequency asymptotic (19) and estimate the terms in expression (16) as:

Neglecting the terms of order

Let us introduce a contour integral

where the circle

Substituting (20b) into (25) and taking into consideration the expression (27) we arrive at

The expression (28) gives the approximate solution of the diffraction problem in low-frequency case as series of

Let us consider the particular case of the problem when cone opening angle

with

Let us present the kernel function (21) in explicit form with split function

Then, the couple of the regularization operators (20) is simplified and looks as:

Summarizing the above results, we prove that the solution of the wave diffraction problem for soft and rigid disc is reduced to the ISLAE, which we obtain from (22), taking into account expressions (29)-(31).

All characteristics of the scattered field are calculated by reduction of ISLAE (22). The order of reduction has been chosen from the condition

Let us express the far-field pattern as

where

With the help of (32), we analyze a diffraction pattern for soft and rigid finite cones when the incident plane wave (7) illuminates the apex (

In order to obtain a profound knowledge of the scattering mechanism, we compare the scattering properties of soft and rigid finite cones.

We verified our results by comparing them with those obtained for circular soft (rigid) disc when

Our further examination aims at studying the energy characteristics of scattering. First of all, we determine the directivity factor [

Applying (32) for (33a) it is found that

In

Let us express the total scattering cross section

The scattering cross section

The curves shown in

with the increase of

Let us derive the total field potential representation at the point

This gives the value of the total field potential at the vertex, if

In

A more complicated diffraction effect

Scattering cross-section | Cone-generating angle ( | ||||||||
---|---|---|---|---|---|---|---|---|---|

10˚ | 20˚ | 30˚ | 40˚ | 50˚ | 60˚ | 70˚ | 80˚ | 90˚ | |

0.108625 | 0.211077 | 0.325297 | 0.443974 | 0.558198 | 0.658833 | 0.737536 | 0.787636 | 0.810569 |

Here the upper sign and

In our investigation, we limit oneself to angles

(

along

(

The mode matching technique together with the analytical regularization procedure is developed for the solution of the canonical diffraction problem of a plane acoustic wave by finite soft and rigid cones in axial irradiation. The diffraction problem has been reduced to ISLAE of the second kind, which satisfies all the necessary conditions. The simple analytical solution in the static case has been derived. In addition, the limit cases of soft and rigid discs are considered, and the inverse operators in explicit form for these cases are obtained.

Numerical solution is used for examination of the finite cone scattering characteristics in a wide frequency range. It is shown that for soft and rigid cases, the main lobe of the far-field pattern is formed in the forward direction for vertex irradiation, while in both the forward and the back directions they are formed for opposite irradiation. The global minima in low-frequency range for scattering cross section in soft case have been obtained, and the feebly resonating character of scattering cross section in

By examination of the near field diffraction effect, the formation of periodical oscillations and good amplification in maxima of these are shown. Distribution of pressures along lateral conical surface indicates the effect of acoustic energy accumulation in rigid conical cavity.

Dozyslav B.Kuryliak,Zinoviy T.Nazarchuk,Victor O.Lysechko, (2015) Diffraction of a Plane Acoustic Wave from a Finite Soft (Rigid) Cone in Axial Irradiation. Open Journal of Acoustics,05,193-206. doi: 10.4236/oja.2015.54015