Calculation of the pressure field from transducers having both a convex
and a concave surface geometry is a complicated assignment that often
is accomplished by subdividing the transducer surface into smaller flat
elements of which the spatial impulse response is known. This method
is often seen applied to curved transducers because an analytical solution
is un-known. In this work a semi-analytical algorithm for the
exact solution to a first order in diffraction effect of the spatial impulse
response of rectangular shaped double curved transducers is presented.
The algorithm and an approximation of it are investigated. The approximation
reformulates the algorithm to an analytically integrable
expression which is computationally efficient to solve. Simulation results
are compared with the simulation software Field II. Calculating
the response from 200 different points yields a mean error for the different
approximations ranging from 0.03 % to 0.8 % relative to a numerical
solution for the spatial impulse response. It is shown that the presented
algorithm gives consistent results with Field II for a linear flat, a linear
focused, and a convex non-focused element. Best solution was found
to be 0.01 % with a three-point Taylor expansion.