Equivalence of the waveguide invariant and two path ray theory methods for range prediction based on Lloyd's mirror patterns

Previous work discusses the use of normal mode theory as a means of determining the range of a contact from a Lloyd's Mirror interference pattern. This method relies on the long range interference pattern between different modes. For shallow, range-independent environments where the sound is dominated by low-order modes, the constant which characterizes the modal interference pattern, the waveguide invariant, is approximately equal to one. The speed of a contact can be determined from the asymptotic behavior of its tonal frequencies from the Doppler shift. This information can be used along with the change in broadband striation frequencies in a Lloyd's Mirror pattern over time to extract the range of the contact as it transits through CPA. If instead of using normal mode theory, the Lloyd's Mirror Pattern is derived as the coherent interference between a straight-line direct and surface-reflected path, a relationship between the striation frequencies and time of a crossing contact can also be derived. This relationship can be shown to be identical to the result obtained from the normal mode approach when the value of the waveguide invariant is equal to one.