Helmholtz integral in the acoustics of an inhomogeneous medium

Integral relations that generalize the Helmholtz integral for an inhomogeneous medium with arbitrary gradients of its density and the sound velocity in it are obtained. Expressions that determine the Helmholtz integral for problems related to the diffraction and radiation of sound in an inhomogeneous medium are derived. It is shown that, in the case of an inhomogeneous medium, an additional factor depending on the density distribution in the medium appears in the integrand.