A new expression for the intensity in ray acoustics is derived, which is valid for a general stationary medium in which the index of refraction may be a continuous function of all three space variables. This expression is derived with the WKB approximation on the wave equation, which results in a line integral along the sound ray. The integrand ∇ ⋅ r′, where r′ = dr/ds is the direction of the ray, is given explicitly in such a way that the integral can be evaluated in closed form for the general case. The result is in terms of functions that can be calculated by integrating simultaneously the vector eikonal equations for the ray paths, together with two other associated vector differential equations.
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Peter Uginčius (1969) studied this question.