Geometric methods are useful for understanding the magnitude and timing of high-frequency echoes from elastic objects in water. This presentation emphasizes, but is not limited to, elastic contributions associated with leaky waves on circular cylinders. The regions of strong coupling between the incident acoustic wave and guided leaky waves can be predicted using Fermat’s principle while the size of the associated Fresnel zones are useful for understanding signal magnitudes [P. L. Marston, J. Acoust. Soc. Am. 102, 1628–1638 (1997); F. J. Blonigen and P. L. Marston, 110, 1764–1769 (2001)]. Ray theory predictions are supported by measurements with tilted cylinders [S. F. Morse, P. L. Marston, and G. Kaduchak, J. Acoust. Soc. Am. 103, 785–794 (1998); K. Gipson and P. L. Marston, J. Acoust. Soc. Am. 106, 1673–1680 (1999); F. J. Blonigen and P. L. Marston, J. Acoust. Soc. Am. 112, 528–536 (2002)]. Related applications include signatures and images of tilted cylinders near flat surfaces [D. S. Plotnick and P. L. Marston, J. Acoust. Soc. Am. 140, 1525–1536 (2016)], time-frequency signatures [S. F. Morse and P. L. Marston, J. Acoust. Soc. Am. 111, 1289–1294 (2002)], and signatures for other targets with corners. [Work supported by ONR.]