Scattering of a spherical wave by a small sphere: an elementary solution

Area of Science:

• Acoustics
• Wave Scattering
• Mathematical Physics

Background:

• Rayleigh scattering typically analyzes plane waves interacting with small objects.
• Near-field interactions and complex geometries necessitate a more comprehensive Green's function approach.
• Previous models often simplify incident wave characteristics.

Purpose of the Study:

• To derive an analytic solution for the diffraction of a spherical sound wave by a small, soft sphere.
• To extend the understanding of wave scattering beyond plane wave assumptions.
• To provide a solution applicable in scenarios with complex scatterer arrangements.

Main Methods:

• Utilizing asymptotic expansions of an exact solution.
• Developing an elementary analytic solution for spherical wave diffraction.
• Validating the solution against established theoretical frameworks.

Main Results:

• An approximate analytic solution for spherical wave scattering by a small, soft sphere was derived.
• The solution is valid for all regions outside the sphere.
• The solution correctly reduces to known results from Kelvin and Rayleigh under specific conditions.

Conclusions:

• The derived solution offers a more generalized approach to scattering problems involving spherical waves.
• This work is crucial for understanding wave interactions in confined or complex environments.
• The findings bridge the gap between simplified models and real-world scattering phenomena.