Phase velocity and attenuation of plane waves in dissipative elastic media: Solving complex transcendental equation using functional iteration method

Abstract

An iteration method to find the roots of a complex transcendental equation is under scanner. This method identified as functional iteration method is being used mainly in wave propagation problems to calculate the phase velocity and the attenuation of plane harmonic waves in dissipative elastic plates. Few mathematical concerns are explained in this study, which put a question mark on the definition as well as application of this functional iteration method. A condition specified to ensure the convergence of iterative sequence to a valid solution is found to be mathematically invalid. The criterion to conclude an iteration process based on the change in two successive approximations may not yield a valid solution. Another serious concern is that the existence of root is not ensured before starting the iteration process to find it. It must be a necessary requirement as the secular equation derived to study the wave propagation is always a transcendental equation, which may not have a solution at all. Then the absence of solution implies that the mathematical model used does not represent the propagation of defined wave in the
medium considered.