Curved wave rays in a ripple tank

TL;DR

This paper presents a detailed mathematical analysis of curved wave rays resulting from the interference of two coherent, synchronous waves in a ripple tank, including explicit parametric equations and a focus on simple harmonic transverse waves.

Contribution

It provides explicit parametric equations for bent wave rays and wavefronts in a ripple tank, using confocal conics mathematics, for the first time in this context.

Findings

Derived explicit equations for curved wave rays and wavefronts.

Analyzed interference patterns of coherent waves in a ripple tank.

Focused on simple harmonic transverse waves with equal frequencies and amplitudes.

Abstract

In its broadest sense the term bent wave rays hints at light or electromagnetic waves bending in a strong gravitational field or in their progress through a transparent medium of nonuniform index of refraction. However, there are instances where curved wave rays are observed as a result of the superposition of two or more waves in an elastic medium. Amongst others, this paper aims at presenting a detailed examination of the phenomenon of interference of waves emanating from two synchronous and coherent sources and disturbing the surface of a liquid that is supposed ideally devoid of viscosity. Hence, it provides people with explicit parametric equations of the bent wave rays as well as those of their corresponding curved wavefronts originating from the superposition of two waves in a material medium. To ensure the formality of the equations of the wave rays and those of the corresponding wavefronts, a preliminary introduction to the mathematics of confocal conics is given in a quite intensive fashion. Therefrom, and for the sake of simplicity, and with no loss of generality, the study will be confined to transverse waves of the simple harmonic type having equal frequencies and equal amplitudes, thus, avoiding unnecessary complex situations that transcends the desired scope.