Diffraction of a Plane Acoustic Wave from a Finite Soft ( Rigid ) Cone in Axial Irradiation

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.


Introduction
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 [1]- [7] are mainly considered.Diffraction elliptic cone is reviewed in [8] and one of limiting cases of the cone such as diffraction of acoustic waves on plane sectors is studied in [9] [10].A semi-transparent cone is investigated in [11].Scattering of electromagnetic wave by infinite cones is also considered (see, for example [12]).It should be noticed that the infinite cone is explored from the mechanical point of view in [13].
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) [14] [15] and on the semi-infinite cone formed by the finite and semi-infinite conical surfaces with different boundary conditions [16].In publication [17], the appropriate problem is solved on a finite cone with internal termination in one of the sectors.Analytical regularization procedure for diffraction problems on fragments of circular conical surfaces is proposed earlier in [18] [19] where an excellent survey of known results for diffraction by finite cone is done.This procedure is used for investigation of the finite cone [20] in the electromagnetic case.Geometrical theory of diffraction is used in [21].
In this article, based on analytical regularization procedure [18], we investigate a scattered field of a plane acoustic wave from the perfectly soft (rigid) finite cone in a different frequency range.

Statement of the Problem
Let us consider the perfectly soft (S) rigid (R) hollow finite cone 1) in a spherical coordinate system ( ) , , r θ ϕ .Cone Q is irradiated by a plane monochromatic acoustic wave that propagates along the symmetry of a cone in the direction z = n i with the velocity potential where n is the vector, that defines a position of any point on the wave front, with the component ( ) , r = r ; n is the normal vector, ( ) ; z i is the unit vector, ( ) is a wave number, ω is the circular velocity, 0 c is the phase of the sound.Time factor ( ) is suppressed throughout this paper.Since the velocity potential ( ) i U is symmetrical and independent of azimuth angle ϕ , than the scattering field is estimated in terms of the scalar (velocity) potential, satisfying the three-dimensional Helmholtz equation The unknown potential Here 0 ρ is mean density; ∇ is the nabla operator.
In order to obtain the unique solution to the problem ( 2), (3), the additional conditions must be imposed on the unknown velocity potential ( ) : the radiation condition written as: and the condition of the finiteness of energy in any bounded volume (edge condition) given as:

Solution of the Diffraction Problem
For solution of the diffraction problem let us decompose the space and determine the total field in the form of Here ( ) x are unknown expansion coefficients; ( ) P η − ⋅ is the Legendre function; ( ) P η − ⋅ is the associated Legendre function, which is defined in [22] by the expression ( ) ( ) For further convenience, in (5) we introduce 1 2 and the unknown 0 x for R cone.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 n z for S and R cones leads to one ( ) ( ) 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 ( ) ( ) ρ → , where ρ is the distance to the edge in local coordinate system, we present these equations by way of 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 [18] ( ) ( ) Here upper sign ("+") corresponds to , cos 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 ( ) we come to finite system of linear algebraic equations as follows: , The main reason of this limitation is to provide the correct transition from Equation ( 12) to ISLAE ( , , P K N → ∞ ), the solution of which satisfies the Meixner condition at the conical edge.For this purpose, we introduce a growing sequence of roots { } 1 Next, in Equation (12) we pass to limit , , P K N → ∞ ( N P K = + ) and arrange the ISLAE according to se- quence (13) as: Here A is the infinite matrix with the elements given as: : where Then we turn to analysis of rigid cone (R-case).To obtain the correct solution we take into account the values , which are determined by the unknowns 0 x , ( ) y , separately.This is done because the minimal (first) positive roots of the Equation (6b) 1 2 η = .Substitut- ing (10) into (9) and limiting the finite number of the unknowns, we exclude 0 x , ( ) and the others are determined by Equations (12a), (12b) with 2, p P = and 2, k K = respectively.Here According to our previous step for S-case, we introduce a growing sequence { } 1 in the following: For this case we use the definition of roots of transcendental equations by way of (6b).
Further, passing to limit , , P K N → ∞ ( 1 N P K = + − ) and arranging ( 12) and ( 17) according to (18), we ar- rive at ISLAE in form of ( 14) with 1 2 . Thus, it is easy to prove that 0 0 x ≡ for the other unknown 0 x .

Regularization of ISLAE
Taking into account the asymptotic properties of the modified Bessel and Macdonald functions for large indices, it is found that , , ; 1 2 , 0, which is correct for S and R cones.
Let us introduce the operator formed with the main parts of the asymptotic expression (15) as , , Here 1 cos cos for S, , , , π cos π cos cos for R, ( ) ( ) region, where upper sign corresponds to S and lower to R cones.Furthermore, the product of operators (20) represent the identity matrix I, [ ] The technique described above is elaborated in [18] [19] and called the analytical regularization procedure.
ISLAE (22) admits the solution in the class of sequences ( ) { } for R cases.This fulfils all the necessary conditions for the existence of a unique solution of ISLAE (22), including the Meixner condition on the edge [18].We represent the other unknown coefficient in both S and R cases through the solution ( 22) by way of

Low-Frequency Solution
Let us rewrite the basic ISLAE (22) for both Dirichlet and Neumann cases by way of ( ) ( ) where 1, 2, 3, k =  .We take into account the low frequency asymptotic (19) and estimate the terms in expression ( 16) as: Neglecting the terms of order ( ) ( ) , we immediately derive the approximate solution (23) as: Let us introduce a contour integral where the circle R C with radius t R = that envelopes the simple poles of the integrand at for R cases.Next, using the residual theorem, it is found that 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 sc .

Transition from Finite Cone to Disc
Let us consider the particular case of the problem when cone opening angle π 2 γ = and becomes the disc.For this case, it is found that indices n z and q ξ are determine as: 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).

Numerical Calculation
All characteristics of the scattered field are calculated by reduction of ISLAE (22).The order of reduction has been chosen from the condition N sc q = + with ( ) Based on the solution we consider a far-field characteristics for soft and rigid cones and near-field characteristics for rigid cone as they are more practical in use.

Far-Field Characteristics of Soft and Rigid Cones
Let us express the far-field pattern as In order to obtain a profound knowledge of the scattering mechanism, we compare the scattering properties of soft and rigid finite cones.Figure 3 shows the far-field patterns scattered by the rigid cone with the same geometrical parameters as in the previous case.Comparison of the curves in Figure 2(a), Figure 2(b) and Figure 3(a), Figure 3(b) visually, we find the similar scattering properties for soft and rigid finite cones illuminated by the plane wave, which propagates along the conical axis.The main difference is the inherent backscatter effect for the sharp rigid cone and its lack for the same soft cone.
We verified our results by comparing them with those obtained for circular soft (rigid) disc when 90 γ =  and 5 kc = .In Figure 4, the magnitudes of the velocity potential is plotted as a function of the polar angle θ .
The solid curve 1 (3) is obtained by us for soft (rigid) disc, while the dashed ones 2 (4) are obtained in [6].There is an excellent agreement for all values of θ .
Our further examination aims at studying the energy characteristics of scattering.First of all, we determine the directivity factor [23] as:   Applying (32) for (33a) it is found that ( ) ( ) In Figure 5(a), we can see the monotonous increase of the directivity factor Ω for soft finite cone with the growth of wave parameter kc and cone-generating angle γ .In the case of a rigid cone (Figure 5(b)), the growth of Ω is similar, however with oscillations.This indicates that the formation of far-field radiation in di- rection 0 θ = is not monotonuous.The maximum value is expected to be observed for wide cones ( π 2

γ →
).For long wave ( 0 kc → ), we have about 1 Ω = ( 3 Ω = ) for a soft (rigid) cone.The latter corresponds to the case of the pulsating (oscillating) disc.So we see that directivity factor can be improved only by increasing the wave parameter kc and cone-generating angle γ .Besides, one can obtain a good concentration of energy in forward direction.
Let us express the total scattering cross section s σ [23] as: The scattering cross section s σ as a function of the parameter kc for soft and rigid cones with different opening angles γ is shown in Figure 6 with the increase of γ to the low-frequency region.This is observed as a result of the well-known piston ac- tion in low-frequency range.Its lowest level is observed for cone-generating angle 20 γ =  and wave parameter about 3 kc ≤ , with the higher peaks for a wide cone, for instance, when 90 γ =  , the wave parameter is 2 kc ≤ .

Some Near-Field Characteristics
Let us derive the total field potential representation at the point 0 r = located on the finite rigid conical surface as a function of the dimensionless parameter kc as follows: , for region ; , sh , for region .In Figure 8(a), the dependences of the normalized sound pressure from kc at the center of rigid disc ( 90 γ =  ) is shown.The curve on this figure shows the tripling of the pressure near π kc = and a minimum at 2π kc = , where the pressure almost equals that of the incidence plane wave ( ) i U .Further increase of the fre- quency leads to the periodical alternation of the maxima and minima.It indicates the formation of Fresnel zones.Comparison of our results with those obtained theoretically and experimentally [24] are also shown in Figure 8(b).The difference of our theoretical results (curve 1) and experimental results (curve 3) is caused by two reasons: the finite thickness of the disc (0.25 in) used in experiment, and the experimental errors specified by external probe microphone.From Figure 8(b), we can observe the excellent agreement our result (see curve 1) and theoretical result of [6] (see curve 2).
A more complicated diffraction effect ( ) ( ) U U can be obtained, if we put down θ γ = and observe it for range 0 r c ≤ ≤ .It give us the cone's surface pressure distribition along the cone-generating lengh c which is shown in Figure 9 and calculated by way of Table 1.First term of scattering cross-section expansion for soft cones (see also Figure 6(a)).Comparison of our data calculation (curve 1) with curve 2 obtained in [6] and experimental results (curve 3) from [24].D is distributed almost symmetrically with two maxima about 0.25c , 0.75c .At the rear side the main maximum ( ( ) ( ) ) is observed at the conical bottom, where pressure .It is because the piston mode gives the main contribution for pressure formation at the bottom of the conical cavity.This shows that the cone cavity is applicable for accumulation of acoustic energy.Another behavior of the pressure we observe for the disc surface (see Figure 9(b)).Here the main maximum is formed near 0.75c with the good amplification ( ( ) ( )

2.6
) for the illuminated side 2 D while with field is related to the sound pressure p and to the velocity of particles υ by way of 0 the Dirichlet (S) or Neumann (R) boundary conditions on the surface of the cone Q as follows:

Figure 1 .
Figure 1.Geometrical scheme of the problem.
) and come to ISLAE, where the first equation for indices 1 p k = = looks as: are located at the real axis out of the Π ; formulate original diffraction problem(14) via the ISLAE of the second kind as follows:

δ
is Kronecker symbol; upper indexes in brackets correspond to region 1 The integrand (26) decays as t ε − , if R → ∞

Figure 2
for its physical matter determines the scattered field in region 3 D .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 ( 30 γ =  ) directly and aperture ( show the far-field patterns for soft cone.From Figure 2(a) we can observe a formation of the main lobe of diffraction pattern in the direction of forward scattering 0 θ =  (curve 1), while backscat- tering ( 180 θ =  ) radiation tends to zero.The side lobes are formed for observation angles 20 80 main lobe essentially grows with increasing kc .We also observe a typical peak about 60 θ =  , which corres- ponds to specular reflection (see curve 2) and the wide deep shadow region for 80 180 θ < ≤   , whereas the contribution of radiation can be neglected.In Figure 2(b), we can observe the far field patterns in the case of plane wave irradiation of the cone aperture ( 150γ =  ).As it is seen from the behavior of curve 1 (compared to curve 1 and 2 in Figure2(a)), the magnitudes of field scattering in direction 0 θ = is almost the same and sig- nificant radiation is observed in the range of

7 .
This gives the value of the total field potential at the vertex, if π 2 γ < and at the deepest point of the bottom of conical cavity, if π 2 γ > .Dependences of the normalized sound pressure from kc at the point 0 r = for these two cases are shown in Figure The curve in Figure 7(a) is plotted for 30 γ =  and shows the in- terference of the periodical oscillations at the vertex for 12 kc > with the period of about 7 kc = .In Figure 7(b), the same dependence at the bottom of the cone with 150 γ =  is shown.The pressure also has almost periodical oscillation with the period 45 kc = .So we see that one can obtained a good amplification of pressure for some frequencies.For example, when 150 γ =  and value of kc is near 25, then the amplification is about ( ) 30 i U in comparison with an incident wave.

Figure 9 .D and 2 DFigure 9 (
Figure 9. Distribution of the pressure on lateral conical surface and disc.(a) Cone-generating angle is equal 30 degree; (b) Cone-generating angle is equal 90 degree.Sign "+" denotes the illuminating side ( 2 D ) and "-" corresponds to the shadow side ( 1 D ); curves 1 are for 1 kc = and curves 2 are for 6 kc = .