Pulse compression thermography has been recently developed to increase the signal-to-noise ratio (SNR) and enhance depth resolvability. However, a distinctively high SNR and depth resolvability is usually required for the accurate characterization of the (thin) topcoat thickness and defects of multilayer ceramic/metal composite solids (CMCSs). To this end, based on the Green function approach, a photothermal wave model for three-layer CMCSs consisting of a semitransparent ceramic coating and two-layer opaque metal solids is first derived, where a spatially varied volumetric heat source and an interior interface heat source described as the Dirac delta function are considered. The dependence of the thermal wave behaviors on both the thickness and photothermal properties of the semitransparent coating is investigated using the model. Inspired by high-precision radar target detection, two nonlinear frequency modulation (NLFM) waveforms, whose instantaneous frequency curves are respectively convex and concave quadratic function, are explored. Results indicate the NLFM waveform with the instantaneous frequency curve of concave quadratic function outperforms conventional LFM and Barker waveforms in terms of the SNR and depth resolvability. Finally, we reveal that a proper window function can further significantly improve the pulse compression quality. Our investigation can provide direct guidance for the accurate and reliable non-destructive testing and evaluation of multilayer CMCSs.