How the thermocline affects the value of the waveguide invariant in a shallow-water waveguide

On the basis of the principle of stationary phase, Chuprov [Ocean Acoustics: Current State (Nauka, Moscow, 1982)] proposed a scalar parameter-the waveguide invariant β, to interpret the dispersive properties of underwater acoustic waveguide. It has been found that β may be useful in many applications in underwater acoustics. A reasonable prediction of the value of β is often necessary in such applications. It was known that β has some "canonical" values for waveguides with simple sound speed profiles (SSP). When a thermocline exists, β for refractive modes (with a turning point) is no longer constant and can even change its sign. In this paper, by dividing the SSP into the non-refractive part and the refractive part, a clear explanation of how the value of β is affected by the thermocline is presented. The results show that β can be positive and increase continually from around 1 to +∞; or negative from -∞ to about -3. An analytic criterion of the sign of β is developed. This method can also be used to analyze the value of β for any other kind of SSP.