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The method suggested by Waterman has been widely used in the last years to solve various light scattering problems. We analyze the mathematical foundations of this method when it is applied to layered nonspherical (axisymmetric) particles in the electrostatic case. We formulate the conditions under which Waterman’s method is applicable, that is, when it gives an infinite system of linear algebraic equations relative to the unknown coefficients of the field expansions which is solvable (i.e., the inverse matrix exists) and solutions of the truncated systems used in calculations converge to the solution of the infinite system. The conditions obtained are shown to agree with results of numerical computations. Keeping in mind the strong similarity of the electrostatic and light scattering cases and the agreement of our conclusions with the numerical calculations available for homogeneous and layered scatterers, we suggest that our results are valid for light scattering as well.

Scattering of light by small particles is an essential part of various applications in physics of atmosphere, ecology, biophysics, astrophysics, and many other fields. The problem of light scattering is frequently treated by a method suggested by Waterman [

Applicability of this method has been considered in the analytical and numerical ways. The results derived were controversial. For instance, calculations demonstrated serious computational problems for spheroids with the aspect ratio

The state-of-the-art analysis has been presented in [

It should be emphasized that Waterman’s method has been thoroughly analyzed just for

In this paper, we extend the electrostatic analysis of applicability of Waterman’s method from [

We consider this problem for a particle with

The boundary conditions at the layer surfaces are

As usual, we separate all the fields in two parts

This method is based on the well-known surface integral equations called the extended boundary condition (see, e.g., [

We expand the Green function

Note that when one considers the light scattering by small particles in comparison with the wavelength of the incident radiation, the problem is reduced to the electrostatic one with a

The main contribution to the scattered field in the far-field zone is then known to be given by a dipole term. So, in the expansions (

In the usual way, we get a system of linear algebraic equations relative to the unknown potential expansion coefficients (see, e.g., [

Using (

Solution of system (

Here we extend our analysis of Waterman’s method, made for homogeneous particles in the electrostatic case in [

For such particles, it was shown in [

The quantities

Note that application of these results to homogeneous spheroids (

Thus, condition (

We apply the approach described in [

Let us start with a core-mantle particle. From (

The asymptotic behavior of the integrals

For

The use of these asymptotic formulae allows us to find the conditions under which system (

To satisfy conditions (

Our accurate consideration of the restrictions (

It is not difficult to prove the obvious general statement that for particles with

Thus, condition (

The spheroidal shape is the most often used one in modeling of nonspherical scatterers. Here we discuss the modification of condition (

Following the consideration of homogeneous spheroids in [

In the particular case of spheroids with the

Thus, while Waterman’s method is applicable to

These particles present a very important particular case when there is a well-known analytical solution to the electrostatic problem (see, e.g., [

Our analysis says that the method is applicable under condition (

This electrostatic problem has always the solution that allows one directly to calculate the polarizability [

It is not difficult to obtain the polarizability from the solution by Waterman’s method; for example, in the case of

Relation (

We have proven that this series converges (to the given value) under condition (

Thus, using the case of confocal core-mantle spheroids, we completely confirmed our condition of applicability of Waterman’s method in electrostatics and saw in detail what happens when the method’s applicability condition is not satisfied: the solution of the problem exists but cannot be found by the method because of the divergence of a series.

The problem is as follows. When in electrostatics we consider a homogeneous spheroid with the semiaxes

Some authors believe that, for a spheroidal shape,

Our analysis of confocal core-mantle spheroids shows that the situation is not as simple. We have firmly established the applicability condition

Some understanding of this problem is given by so-called inverse spheroids: axisymmetric particles obtained from homogeneous spheroids with the profile

The light scattering problem is known to be reduced to the electrostatic one when the size of a scatterer is small in comparison with the wavelength of the incident radiation (see for details [

Despite these differences, the similarity of the electrostatic and light scattering cases is strong enough because of the resemblance of the wave functions, of the wave fields in the presence of a particle, of their expansions, and of singularities of their analytic continuations (see for more details [

This extension is supported by the agreement of our electrostatic conclusions with the theoretical works and numerical calculations made in light scattering case.

Earlier analyses of the method in this case are discussed in detail in [

Concerning the numerical tests, the electrostatic condition (

It is not simple to compare our electrostatic results even for layered spheroids with numerical light scattering calculations based on Waterman’s method as they should not have been published because of their unexpectedly narrow range of parameter values. In [

In the electrostatic case, we have obtained the general condition of Waterman’s method applicability to layered scatterers, which depends only on the shape of their layers. We have considered this condition in the particular case of layered spheroids and noted that the set of such particles that can be treated by Waterman’s method is surprisingly small, which has a large practical value. Using the explicit solution for confocal core-mantle spheroids, we consider what happens when the condition is not satisfied. We have found that the conclusions of our analysis agree with the results of different numerical calculations. A consideration of core-mantle spheroids has revealed another aspect of the “virtual singularities” of the analytic continuation of the field inside a homogeneous spheroid which still requires a better understanding.

The similarity of the electrostatic and light scattering cases, earlier theoretical works on homogeneous scatterers (see for a review [

The authors declare that there are no competing interests regarding the publication of this paper.

The work was partly supported by the Ministry of Education and Science of Russia within the state assignment for SUAI, the research grants of St. Petersburg Univ. (6.38.18.2014), and RFBR (16-02-00194 and 16-52-45005).