Role of Virtual Modes in Surface‐Duct Leakage Free

A three‐layer surface‐duct model is considered. In the top layer the sound velocity increases linearly (slope g₁) from a value C(0) at the surface to C(b) at the bottom of the duct. In the second layer the sound speed decreases linearly from C(b) at depth b to C₀ at depth d (slope |g₂| > g₁). At depths greater than d the sound speed is constant and equal to C₀. For typical ocean conditions C(0)>C₀. By employing a theorem of Titchmarsh [Eigenfunction Expansions (Oxford University Press, London, 1962), Pt. I] it is shown that only a continuous spectrum of modes exists. This result, so contrary to widely held beliefs, is shown to hold for a broad class of surface duct models including that of M. A. Pedersen and D. F. Gordon [J. Acoust. Soc. Amer. 37, 108–118 (1965)]. Most of the continuum functions correspond to a situation where little energy is contained in the duct. However, under certain conditions, continuum functions exist which represent a strong concentration of energy within the duct. These are called virtual modes in analogy with the virtual states of atomic and nuclear physics. The conditions for the existence of virtual modes and their rate of leakage are derived. [Work supported by the Office of Naval Research under contract.]