Use of complex source points to simplify numerical Gaussian beam synthesis

Abstract

It is often desirable to generate the acoustic field due to a so-called Gaussian beam. One way to do this is to use the free-space Greens function for the acoustic field and to sum small area sources over a circular plate with the appropriate shading for the desired Gaussian beam. For very high frequencies and narrow beams, the computation time to give an accurate sum can be large when calculating the sum for many points in the acoustic field. An alternate approach comes from the use of a single point source with complex coordinates R=[Xr+iXi,Yr+iYi,Zr+iZi]. When this complex source point is used in the free-space Greens function, the formal expressions for pressure and particle velocity can be used if careful attention is paid to the interpretation of the complex distance, r, that arises in the exp(ikr)/r term. The singularity is no longer a single point in the case of a complex source, but a circular disk. The far field of a complex source point is a good approximation to a Gaussian beam. Several computational uses of the technique will be demonstrated. Extension to the shear wave Greens function will be explored.