Professor Gopalakrishnan has made seminal contribution in the areas of wave propagation in complex mediums and Structural Health Monitoring. Wave propagation is a fertile and well-researched area where Prof. Gopalakrishnan has made a mark both nationally and internationally through his seminal contributions. He has practically looked at wave propagation in most of the complex mediums such as metals, anisotropic materials such as laminated composites, functionally graded materials, carbon nanotubes and nanotubes reinforced nano composites structures, periodic structures such as hexagonal lattices, sandwich structures, fluids and electromagnetics.

He has successfully solved many of these wave propagation problems through a unique combination of sound mathematical skills coupled with innovative and original ideas. His notable contribution is the invention of a new numerical method called Spectral Finite Element Method (SFEM), which has already become a tool to solve many wave propagation problems in complex media involving very high frequency input. The reason for SFEM popularity is that it can solve high frequency content problem at a fraction of a cost compared to many established analysis tools such as finite element method. In addition, SFEM makes the solution of inverse problems such as source or system identification simple and straightforward, which is not the case with conventional analysis tools.

This has opened up the possibility of solving many difficult inverse problems such as load monitoring in aircrafts, crack detection and quantification in structures and systems, earthquake/Tsunami modeling, new material design for impact mitigation etc.

His research efforts are primarily responsible for bringing out clarity and complete understanding of many wave based phenomena in complex media. His contributions can be summarized as follows:

• Development of Spectral Element Method as an analysis tool for wave propagation problems. Here, he has developed two variants of the method, one based on Fourier Transform and the other based on Wavelet Transform. The application of these methods to various practical problems involving composites, SHM and active control of structures were published by reputed international publishers as two different monographs. Till date, these two are the only books available on this modeling tool.

• Propagation and scattering of waves in laminated composite 1-D and 2-d waveguides with defects such as notches/debands/delaminations/fiber breakages/degraded regions/corroded/porous regions.

• Propagation and scattering of anisotropic, inhomogeneous (in Functionally Graded structures) and Thermal waves and their interactions with structural discontinuities.

• Propagation and characteristics of Lamb waves/Raleigh waves/Stonely waves in a 2-D composite, anisotropic and inhomogeneous waveguides.

• Propagation and frequency band gap estimation of periodic structures such as hexagonal lattices, sandwich cores, single and multiwall carbon nanotubes and nanotube reinforced nano composite structures. In particular, he studied the effect of scale parameter on the evolution of band gaps in acoustic and optical phonon modes

• Propagation of waves in. Propagation of waves in acoustic fluids and electromagnetic medium

• Development of novel computational models for non-linear wave propagation in hyper elastic solids.