Sound Propagation in Stratified Fluids Stratified fluids whose densities sound speeds and other parameters are functions of a single depth coordinate occur widely in nature. Indeed the earths gravitational field imposes a stratification on its atmosphere oceans and lakes. It is well known that their stratification has a profound effect on the propagation of sound in these fluids. The most striking effect is probably the occurrence of acoustic ducts due to minima of the sound speed that can trap sound waves and cause them to propagate hori­ zontally. The reflection transmission and distortion of sonar signals by acoustic ducts is important in interpreting sonar echoes. Signal scattering by layers of microscopic marine organisms is important to both sonar engi­ neers and marine biologists. Again reflection of signals from bottom sediment layers overlying a penetrable bottom are of interest both as sources of unwanted echoes and in the acoustic probing of such layers. Many other examples could be given. The purpose of this monograph is to develop from first principles a theory of sound propagation in stratified fluids whose densities and sound speeds are essentially arbitrary functions of the depth. In physical terms the propagation of both time harmonic and transient fields is analyzed. The corresponding mathematical model leads to the study of boundary value problems for a scalar wave equation whose coefficients contain the pre­ scribed density and sound speed functions. 1. Introduction 2. The Propagation Problems and Their Solutions 1. Summary 2. The Acoustic Propagator 3. Solutions with Finite Energy 3. Spectral Analysis of Sound Propagation in Stratified FluidS 1. Summary 2. The Reduced Propagator A 3. Solutions of the Equation A =  4. Spectral Properties of A 5. Generalized Eigenfunctions of A 6. The Spectral Family of A 7. The Dispersion Relations 8. The Spectral Family of A 9. Normal Mode Expansions for A 10. Semi Infinite and Finite Layers 4. Transient Sound Field Structure in Stratified Fluids 1. Summary 2. Normal Mode Expansions of Transient Sound Fields 3. Transient Free Waves 4. Transient Guided Waves 5. Asymptotic Energy Distributions 6. Semi Infinite and Finite Layers 5. Scattering of Signals by Inhomogeneous Layers 1. Summary 2. Signals in Homogeneous Fluids 3. The Reflected and Transmitted Signals 4. Construction of the Scattering Operator 5. The Scattering Operator and Signal Structure Appendix 1. The Weyl Kodaira Titchmarsh Theory Appendix 2. Stationary Phase Estimates of Oscillatory Integrals with Parameters References.