This paper discusses a numerical algorithm and supporting formulation for evaluating ultrasonic pulse transmission through non-planar component geometries. The algorithm is engineered to model experimental configurations where irregularities in surface geometry preclude the use of less rigorous approaches, such as a field expansion about a single entry point. The algorithm formulation represents the transmitted pulse as a surface integral coinciding with a pulse origin aperture, employing the Green function for the water-component system. The model explicitly considers the component surface geometry over the footprint of the incident pulse, thus allowing consideration of smooth yet non-expandable (i.e. in power series about a single point) geometries, such as adjoining flat and fillet surfaces. A computationally efficient algorithm results from use of asymptotic Green function approximations. Approaches are also discussed under conditions where the asymptotic Green function expressions are singular or invalid, due to focusing by surface concavity or transmission near critical angles. Consideration of pulse time dependence represents an extension of previous work [1], as also does treatment of surface concavity and critical angle transmission. The following sections summarize theoretical formulation and algorithmic implementation, followed by the presentation of illustrative computations.