Generation of Ultrasound Pulses in Water Using Granular Chains with a Finite Matching Layer

Wave propagation in granular chains is subject to dispersive effects as well as nonlinear effects arising from the Hertzian contact law. This enables the formation of wideband pulses, which is a desirable feature in the context of diagnostic and therapeutic ultrasound applications. However, coupling of the ultrasonic energy from a chain of spheres into biological tissue is a big challenge. In order to improve the energy transfer efficiency into biological materials, a matching layer is required. A prototype device is designed to address this by using six aluminum spheres and a vitreous carbon matching layer. The matching layer and the precompression force are selected specifically to maximize the acoustic pressure in water and its bandwidth. The designed device generates a train of wideband ultrasonic pulses from a narrow-band input with a center frequency of 73 kHz. An analytical model is created to simulate the behavior of a matching layer as a flexible thin plate clamped from the edges. This model is then verified using free-field hydrophone measurements in water, which successfully predict the increased bandwidth by generation of harmonics. The shapes of the measured and predicted waveforms are compared by calculating the normalized cross-correlation, which shows 83% similarity between both. Since the generation of harmonics is of interest in this study, the total harmonic distortion (THD) and the $-6$-dB bandwidth of the signals are used to analyze signal fidelity between the hydrophone measurements and the model predictions. The acoustic signals in water have a root-mean-square THD of 73%, and the model predicts a root-mean-square THD of 78%. The $-6$-dB bandwidths of individual pulses measured by a hydrophone and predicted with the model are 280 and 252 kHz, respectively. At these high ultrasonic frequencies, it is an experimental demonstration of resonant chains operating in water with a matching layer.

Figure 1

Illustration of the experimental setup. Transducer, ultrasonic horn, granular chain assembly, and the matching layer together form a nonlinear system that can generate wideband impulses. A membrane hydrophone is used to measure the acoustic pressure waveform generated by this system. A computer numerical controller (CNC) is used to control the alignment of the hydrophone. The precompression force is adjusted with a high-precision micrometer translation stage. The granular chain is exaggerated for easier visualization.

Figure 2

(Top) Displacement profile measured at the tip of the ultrasonic horn with a laser vibrometer for a 25-cycle sine wave input with a center frequency of 73 kHz. (Bottom) Frequency spectrum of the measured displacement signal.

Figure 3

(Top) Estimated velocity profile with the analytical model for a six-sphere chain against an immovable steel wall for precompression force of 1.25 N. (Bottom) Corresponding frequency spectrum.

Figure 4

Estimated velocity profile with the analytical model for a chain of six aluminum spheres terminated with different matching layers with a thickness of 0.5 mm and precompression force of 1.25 N.

Figure 5

(Top) Velocity profile estimated at the surface of the vitreous carbon matching layer for different precompression forces. (Bottom) Regarding frequency spectra.

Figure 6

Relation between the precompression force and maximum velocity profile at the surface of the matching layer that represents the acoustic pressure. The maximum harmonic content is calculated after filtering the fundamental signal at 73 kHz with a high-pass filter.

Figure 7

(Top) Ultrasound pressure waveform measured in water and the velocity profile estimated at the surface of the vitreous carbon matching layer are plotted for comparison. (Bottom) Regarding frequency spectra.

Figure 8

(Top) Comparison of measured ultrasound wave and predicted velocity profile with the model. Figure 7 replotted with different time scale for clear visualization. (Bottom) Frequency spectra of single pulse located around 0.2 ms.

Figure 9

Velocity profiles at the surface of the matching layer and the last sphere are plotted for comparison. Blue arrows point at the highlighted section of the waveform where the contact between the last sphere and the matching layer are lost.