Analytic expressions are derived for the variation of modal wave numbers due to variations in five basic properties of a layered environment: the compressional wave speed, shear wave speed, density, and thickness of any layer, and the depth of an interface between layers. Both fluid and solid layers are permitted. The layers may absorb energy, and the modal wave numbers and their variations are generally complex valued. The analytic expressions provide an efficient way to linearize the relation between the modal wave numbers and many geoacoustic parameters, for use in field sensitivity studies, range-dependent propagation models, partially linearized matched-field inversion schemes, and in the study of modes generally.