A complex-transducer-point model for finite emitting and receiving ultrasonic transducers

Abstract The complex-source-point (CSP) technique, widely used to generate Gaussian beams out of real point source fields by displacing the source location into complex space, is here considered and extended to model finite flat and focused axisymmetric ultrasonic emitters and receivers with Gaussian profiles. The signal voltage generated by such transducers upon reception of an acoustic pressure field p is shown to be proportional to p sampled at a complex location . We arrive at this result by using a complex continuation of Helmholtz's theorem and the conventional surface integral for the voltage of a reciprocal electro-acoustic transducer. This extension to the CSP technique, called herein the complex-transducer-point (CTP) technique, allows efficient modeling of finite emitting and receiving transducers in the presence of layered fluid-elastic configurations as well as in unbounded fluids. For a pair of CTPs interacting in an unbounded fluid, we derive paraxial expressions and present numerical results for the voltage to help understand its behavior with changing receiver parameters. We also show how transducers with arbitrary axisymmetric profiles can be expanded in terms of coaxial CTPs whose parameters are computed from a minimization scheme. We successfully demonstrate this approach by modeling the single-frequency voltage generated in a pitch-catch experiment with a pair of piston transducers using a collection of three CTPs for each transducer. Furthermore, we extend this latter approach to model time-domain voltages by deriving frequency-scaling rules for the CTP parameters for flat and focused apertures. This is numerically implemented for piston transducers and shown to agree well with experimental transient signals. The well-known singularities of the CSP field are also present for the CTP field. Because of this, the range of transducer beam collimation is limited to cases for which the imaginary part of the complex displacement (known as b ) is larger than the operating wavelength λ f . However, this range remains usefully wide even when the CTP parameter b is varied with frequency to synthesize time-domain signals. Finally, we show how to use the CTP technique for emitting and receiving CTPs interacting with a plane layered fluid-elastic configuration.