The acoustic radiation generated by an impulsive point source in a continuously layered fluid with depth‐varying parameters is investigated theoretically with the aid of the modified Cagniard method. Using a one‐sided time Laplace transformation with a real positive transform parameter, a Fourier transformation with respect to the horizontal space coordinates and appropriate one‐sided Green’s functions, the system of transform‐domain differential equations in the depth coordinate is rewritten as a system of integral equations that can be solved by a Neumann iteration. The modified Cagniard method leads to space‐time expressions for the relevant iterates that physically are representative for the successively reflected waves. This iterative method is shown to be convergent in the time domain for any continuous and piecewise continuously differentiable depth profile in the inertia and compressibility properties of the fluid. To show the generality of the method, the fluid is assumed to show anisotropy in its volume density of mass, which is the kind of anisotropy that shows up in the equivalent medium theory of a finely layered fluid. The continuously refracted waves emitted by the source and the singly, continuously, reflected waves are discussed in detail. With this technique, no difficulties arise with ‘‘turning rays,’’ as is the case in the asymptotic ray theory of the (real) frequency‐domain analysis of the problem.