Currently, the numerical calculation of the sound propagation problem of directional beams is generally based on the parabolic approximation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation. Since the KZK equation is derived from the Westervelt equation under the parabolic approximation, it inevitably introduces errors in places far from the axis (>20°) or near the sound source. To overcome this limitation, scholars have proposed various methods, most of which are based on the Westervelt equation. It has been proved that the KZK equation can be used to describe the directional finite amplitude wave emitted by a transducer with a finite aperture more accurately. However, the KZK equation has no analytical solution and can only be solved by numerical calculation methods. At present, it can be divided into two types from the applicable range: one is to directly use finite difference to solve term by term in the time domain or frequency domain, and then superimpose the results. This method is numerical solution, and this result is applicable to any parameters; the other is to use the successive approximation method to obtain the result in the weak nonlinearity situation. This method is used to derive the expression of the nonlinear effect result, which is only applicable to the weak nonlinearity situation. This chapter reviews the current KZK model and its solutions, and on this basis, it presents a simple and high-precision algorithm to calculate the Gaussian beam expansion coefficient. This method has some obvious advantages. It can be used in the spatial domain and can be easily extended to the k space domain. It can provide high accuracy, which is proven by the numerical simulation results of the sound source distribution function and the sound field. It is also stable and highly efficient. This chapter also provides a three-dimensional time domain finite difference algorithm. This algorithm transforms the extended KZK equation into the TBE equation to adapt to the far-field spherical wave propagation of the beam, adopts the operator splitting method to decompose various effects into five equations, and uses the IBFD and CNFD difference methods to solve each equation. The effectiveness of the algorithm has been proved by comparing with those of the two-dimensional time-domain algorithm in existing related research. The empirical formula of the sound absorption coefficient in the air is introduced into the time-domain algorithm to calculate the sound field of the rectangular parametric array in the air.

Directional Beam Calculation

章节
发表日期：

2025年

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Jun Yang,
Peifeng Ji

摘要

Currently, the numerical calculation of the sound propagation problem of directional beams is generally based on the parabolic approximation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation. Since the KZK equation is derived from the Westervelt equation under the parabolic approximation, it inevitably introduces errors in places far from the axis (>20°) or near the sound source. To overcome this limitation, scholars have proposed various methods, most of which are based on the Westervelt equation. It has been proved that the KZK equation can be used to describe the directional finite amplitude wave emitted by a transducer with a finite aperture more accurately. However, the KZK equation has no analytical solution and can only be solved by numerical calculation methods. At present, it can be divided into two types from the applicable range: one is to directly use finite difference to solve term by term in the time domain or frequency domain, and then superimpose the results. This method is numerical solution, and this result is applicable to any parameters; the other is to use the successive approximation method to obtain the result in the weak nonlinearity situation. This method is used to derive the expression of the nonlinear effect result, which is only applicable to the weak nonlinearity situation. This chapter reviews the current KZK model and its solutions, and on this basis, it presents a simple and high-precision algorithm to calculate the Gaussian beam expansion coefficient. This method has some obvious advantages. It can be used in the spatial domain and can be easily extended to the k space domain. It can provide high accuracy, which is proven by the numerical simulation results of the sound source distribution function and the sound field. It is also stable and highly efficient. This chapter also provides a three-dimensional time domain finite difference algorithm. This algorithm transforms the extended KZK equation into the TBE equation to adapt to the far-field spherical wave propagation of the beam, adopts the operator splitting method to decompose various effects into five equations, and uses the IBFD and CNFD difference methods to solve each equation. The effectiveness of the algorithm has been proved by comparing with those of the two-dimensional time-domain algorithm in existing related research. The empirical formula of the sound absorption coefficient in the air is introduced into the time-domain algorithm to calculate the sound field of the rectangular parametric array in the air.