Incorporating the effect of frequency-independent attenuation within the on-axis spatial impulse response of a circular piston

The spatial impulse response is important for numerical simulations of diagnostic ultrasound. The spatial impulse response, which describes transient diffraction due to an impulsive input, yields closed form analytical expressions for various transducer geometries when the medium is lossless. These analytical expressions are advantageous for simulations that repeatedly evaluate these expressions at hundreds of thousands of points. However, spatial impulse responses evaluated for lossy materials typically require additional numerical calculations that substantially increase the computation time. Thus, analytical or rapidly converging numerical expressions for the lossy spatial impulse response are expected to greatly enhance present simulation methods. This motivates the derivation of on-axis spatial impulse responses for a circular piston that model frequency-independent attenuation. Closed-form analytical expressions for the on-axis spatial impulse response are introduced for two closely related frequency-independent attenuation models. The results show that, as the attenuation constant increases, the peak amplitudes of these lossy on-axis spatial impulse responses decrease. The lossy on-axis impulse response also decreases slightly as time increases beyond the initial arrival time, whereas the lossless on-axis spatial impulse response for a circular piston maintains a constant value after the initial arrival time.