Acoustics mainly studies the emission, propagation, reception of sound, the properties of sound, and the interaction between sound and other substances. Acoustics is one of the oldest disciplines in natural science, with descriptions of acoustics dating back to ancient times. In early acoustic research, the motion equation of acoustics could satisfy people’s basic needs by solving it on a linear basis, thus ignoring the nonlinearity of motion and medium. Therefore, the assumption of linear acoustics is mainly based on small amplitude sound waves. In linear acoustics, two sound waves propagating at the same time will not interact with each other, and the vibration of the medium caused by them is equal to the linear superposition of the vibration of the medium caused by them when they exist separately, satisfying the superposition principle. However, if the acoustic equation retains nonlinear terms, the superposition principle is no longer applicable, and it no longer follows the rules of linear acoustics, thus giving rise to a new branch of discipline—nonlinear acoustics. Nonlinear acoustics is a science that studies intensive sound. In nonlinear acoustics, intensive sound is usually referred to as finite amplitude sound waves or large amplitude sound waves, which is a phenomenon between small amplitude sound waves and weak shock waves. The main research object of nonlinear acoustics is phenomena related to the propagation of finite amplitude sound waves, such as shock formation, harmonic distortion, non-constant propagation speed, etc. These phenomena cannot be explained by general linear acoustics. They are caused by the nonlinear effects of the medium on finite amplitude sound waves. So far, in the most famous practical applications, the most noteworthy contribution is Westervelt’s work on acoustic sound of scattering by sound, which eventually formed the theory of acoustic parametric arrays in the 1960s. In the parametric array, the nonlinear interaction of two high-frequency sound beams produces a narrow low-frequency sound beam with almost no sidelobes. This process allows high directivity sound to be radiated from a relatively small transducer, with the additional benefit of being able to transmit a wide frequency range of sound. The proposal of the acoustic parametric array has brought broad prospects for the application of nonlinear acoustics. Parametric arrays with unique technical advantages such as “wideband, high directivity, small size” have been widely used in the field of underwater acoustics engineering. This chapter first reviews the history and current state of research related to parametric arrays. Starting from the four equations used to describe the general motion of viscous heat-conducting fluids, it presents the propagation formulas for finite amplitude waves in non-attenuating fluids and thermoviscous fluids. Based on the Westervelt model and Berktay model of parametric arrays, the relationship between the directivity of the primary wave and the difference frequency sound is analyzed.

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发表日期：

2025年

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Jun Yang,
Peifeng Ji

摘要

Acoustics mainly studies the emission, propagation, reception of sound, the properties of sound, and the interaction between sound and other substances. Acoustics is one of the oldest disciplines in natural science, with descriptions of acoustics dating back to ancient times. In early acoustic research, the motion equation of acoustics could satisfy people’s basic needs by solving it on a linear basis, thus ignoring the nonlinearity of motion and medium. Therefore, the assumption of linear acoustics is mainly based on small amplitude sound waves. In linear acoustics, two sound waves propagating at the same time will not interact with each other, and the vibration of the medium caused by them is equal to the linear superposition of the vibration of the medium caused by them when they exist separately, satisfying the superposition principle. However, if the acoustic equation retains nonlinear terms, the superposition principle is no longer applicable, and it no longer follows the rules of linear acoustics, thus giving rise to a new branch of discipline—nonlinear acoustics. Nonlinear acoustics is a science that studies intensive sound. In nonlinear acoustics, intensive sound is usually referred to as finite amplitude sound waves or large amplitude sound waves, which is a phenomenon between small amplitude sound waves and weak shock waves. The main research object of nonlinear acoustics is phenomena related to the propagation of finite amplitude sound waves, such as shock formation, harmonic distortion, non-constant propagation speed, etc. These phenomena cannot be explained by general linear acoustics. They are caused by the nonlinear effects of the medium on finite amplitude sound waves. So far, in the most famous practical applications, the most noteworthy contribution is Westervelt’s work on acoustic sound of scattering by sound, which eventually formed the theory of acoustic parametric arrays in the 1960s. In the parametric array, the nonlinear interaction of two high-frequency sound beams produces a narrow low-frequency sound beam with almost no sidelobes. This process allows high directivity sound to be radiated from a relatively small transducer, with the additional benefit of being able to transmit a wide frequency range of sound. The proposal of the acoustic parametric array has brought broad prospects for the application of nonlinear acoustics. Parametric arrays with unique technical advantages such as “wideband, high directivity, small size” have been widely used in the field of underwater acoustics engineering. This chapter first reviews the history and current state of research related to parametric arrays. Starting from the four equations used to describe the general motion of viscous heat-conducting fluids, it presents the propagation formulas for finite amplitude waves in non-attenuating fluids and thermoviscous fluids. Based on the Westervelt model and Berktay model of parametric arrays, the relationship between the directivity of the primary wave and the difference frequency sound is analyzed.