Using a ’’small roughness’’ boundary condition due to Biot, which incorporates the effects of multiple scatter, one can obtain simple, closed‐form solutions for the coherent scattering of transient spherical waves. Combining this boundary condition with the normal coordinate technique leads to explicit solutions for impulsive or other point sources above or on a rough, rigid plane, i.e., for both finite and grazing angles of incidence. This approach distinguishes naturally between two types of arrival: body (or volume) waves and boundary waves; the latter incorporate the backscatter. The boundary wave dies off rapidly with distance from the surface. But, parallel to the boundary, in the absence of attenuation, its amplitude decreases only as r−1/2 so that for sufficiently large r, and for source and receiver at or near the surface, the boundary‐wave will dominate the direct acoustic pulse arrivals. The methods and results of this paper can be extended to problems of scatter and diffraction of spherical pulses by rigid (or free) plane rough surfaces bounding stratified media or by rough objects of other shapes, e.g., spheres, cylinders, ellipsoids, etc. This theory, valid for wavelengths or pulsewidths long compared to the characteristic roughness dimensions, leads to results that are different from those of the usual perturbation techniques and have been verified experimentally by Medwin et al. in a companion paper.