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This study contains a mathematical analysis of constrained-layer damping (CLD) in plates of infinite extent, with an emphasis on the physical understanding of some special features, including fluid loading. Previous work is expanded to cover extensional waves. Some essential aspects of fluid loading may be understood by applying thin-plate theory. Therefore, thin-plate theory of extensional waves in a fluid-loaded plate was developed as a counterpart to that for flexural waves. The description and examples of CLD follow three models: the first is an extension of Kerwin's 1959 model, the second a hybrid model for which the base plate is treated by exact elasticity theory, and finally a fully exact model for all three layers. Examples and comparisons are given.