Four-dimensional directivity pattern for fast calculation of the sound field of a phased array transducer

A new approach is presented for an efficient calculation of the complete four-dimensional wave propagation in a transversely isotropic elastic half-space excited by a normal oriented impulsive point force. The sound field will be represented by the dynamic Green's functions for the particle displacement. The Green's functions will be derived by a solution of the elastodynamic wave equation with integral transform methods applying the Cagniard-de Hoop-method. We transform the Green's functions into a four-dimensional directivity pattern by normalizing the time axis. The formulation of the directivity pattern provides a very short calculation time for the complete four-dimensional sound field of a rectangular transducer element of a phased array probe using a point source synthesis. The calculation results in a complete transient time function for the three spatial directions of the particle movement in each point of the half-space. The complete time function of the three-dimensional sound field of a phased array transducer can be calculated by a convolution with the measured time function of the probe and the superposition of the delayed sound fields of multiple transducer elements. This approach provides an efficient method for optimizing the delay laws in phased array applications with consideration of all excited wave modes. We present the derivation of the four-dimensional directivity pattern and calculate sample simulations for the sound field of a phased array probe. Further we compare the modeling with the wave propagation in solids measured by an electrodynamic probe.