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The propagation of diffuse energy on an unwetted flat plate with attached heterogeneities is examined using a statistical, multiple scattering approach. The statistically homogeneous heterogeneities lightly couple the membrane and flexural waves. The problem is formulated in terms of the Bethe–Salpeter equation, which governs the field covariance. It is reduced to a radiative transfer equation in the limit that the attenuations per wave number are small, i.e., when the heterogeneities are weak. This radiative transfer equation governs the diffuse energy propagation as a function of space, time, and propagation direction. Solutions of the radiative transfer equation are presented for the simple case of attached heterogeneities in the form of delta-correlated springs excited by an extensional point source. The results show the evolution of the extensional, shear, and flexural energy densities across the plate as a function of time. A similar approach is expected to apply to the more complicated case of submerged complex structures.