Elastic Waves in Solids, Volume 2

Appendix 1Developments in Cylindrical and Spherical Harmonics

The use of cylindrical or spherical coordinates simplifies the resolution of equations that govern propagation in structures with axial or spherical symmetry. In this appendix, we establish the solutions to the Helmholtz equations in these two curvilinear coordinate systems.

A1.1. Cylindrical harmonics

Let R_c be the reference frame of cylindrical coordinates (r, θ, z) and center O. The Cartesian coordinates are then defined in terms of the cylindrical coordinates by the relations:

[A1.1]

Considering expression [A1.7] for the Laplacian in cylindrical coordinates given in Appendix 1 of Volume 1, the Helmholtz equation is written with M = L or T, depending on the type of wave:

[A1.2]

The solution to this equation is found using the separation of variables method:

[A1.3]

Figure A1.1. Cylindrical coordinate system

Substituting this trial solution into equation [A1.2] and dividing by RΘZ leads to:

[A1.4]

The third term is a ...