A physical model of the waves' propagation would have to consider that there is more than one relevant wavelength: besides the sound wave that bounces off the structure, there are also the solidborne waves in the structure. It is also known that the longitudinal wave travels along the interface between the elastic solid and the water more quickly than the shear waves and the Lamb waves. In fact, the Lamb waves can be up to two orders of magnitude shorter than the wavelength of the sound in water. And yet, the Lamb wave effects in the bounced sound waves contain a great deal of information. For example, one can understand the physical properties of an elastic shell by examining the Lamb wave's resonance effects.

An advanced SONAR receiver must therefore deal with a complex signal made up of multiple waves that, taken together, determine the echo's resonant structure. Making matters more complex is the fact that there are no analytical models that describe such activity, so it is necessary to unscramble the signal by knowing the underlying physics. In short, one needs to know what to look for in such a signal. Only with a mathematical model can a researcher predict the form and structure of the low-frequency waveform emitting from a submerged object.

The NATO Undersea Research Centre team built such a model using COMSOL Multiphysics. This multiphysics model describes the frequencydomain elastic-displacement wave equation for a submerged object and couples it to an acoustic-wave equation that describes the waves in the fluid domain. Because the Lamb waves are so short, a finite-element model of the object must have many more degrees of freedom than one might expect. According to the researchers, the slow, or short, waves present a challenge in the modeling process, and getting a mesh and convergence is difficult.

For their initial model, that of a cylinder with two end caps representing a scuba tank, the team treated the 3D geometry as an axisymmetric 2D problem (see Figures 2 and 3). They then solved a added these solutions together through an azimuthal Fourier series to reconstruct the 3D field. Although the target must be axisymmetric, the incident SONAR signal need not be so. The Sommerfeld radiation condition is approximated numerically by Bérenger PMLs (Perfectly Matched Layers), which the researchers easily implemented into the multiphysics software. The PML layers absorb outgoing waves, making it possible to simulate efficiently a target immersed in an infinite fluid domain using a finite-sized mesh (see Figure 3).