Numerical modeling method for underwater complete axial symmetry space three-wave coupling sound field

Abstract

A numerical modeling method for an underwater complete axial symmetry space three-wave coupling sound field belongs to the field of nonlinear acoustics, and comprises the following steps: carrying out simple harmonic hypothesis on sound waves, and substituting the sound waves meeting difference frequency three-wave frequency matching conditions into a KZK equation to obtain a difference frequency three-wave coupling equation set; decomposing the equation set into a diffraction equation, a dissipation equation and a nonlinear equation by using a second-order Strong operator splitting method; a diffraction equation is solved by using a chasing method, a dissipation equation is solved by using an accurate analytical solution, and a nonlinear equation is solved by using an explicit Euler method; and setting boundary conditions, and calculating an underwater complete axial symmetry space three-wave coupling sound field. According to the method, a three-wave coupling sound field model is established based on a KZK equation, efficient solving of a three-wave coupling equation set is achieved by utilizing a second-order Strong operator splitting method and combining finite difference and an explicit Euler method, and the three-wave coupling sound field in the underwater complete axial symmetry space is accurately calculated. According to the invention, the calculation complexity is reduced, the calculation efficiency is improved, and the calculation result is accurate.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear acoustics technology, specifically relating to a numerical modeling method for a fully axisymmetric spatial three-wave coupled sound field underwater. Background Technology

[0002] In nonlinear acoustics, the energy and momentum exchange process that occurs when three waves meet specific conditions is called three-wave interaction. This process strictly follows the three-wave coupling equations, and the corresponding sound field is usually called the three-wave coupled sound field. In underwater acoustic engineering, underwater acoustic parametric arrays are an important application of three-wave interaction. Existing research has largely relied on quasi-linear approximation methods without rigorously solving the three-wave coupling equations. This limits the application of traditional parametric array theory in handling practical problems with strong nonlinear effects. Therefore, establishing an accurate model of the three-wave coupled sound field is of great significance for promoting the development of parametric simulation technology.

[0003] Modeling of three-wave coupled sound fields can be traced back to the 1970s. Rudenko and his team extended the parametric amplification phenomenon in optics to acoustics using analytical expressions of the three-wave coupling equations based on the Burgers equations (Rudenko OV, Parametric interaction of traveling sound waves [J]. Soviet Physics Acoustics. 1974, 20: 63-64.;2 Novikov B. K, Rudenko O V. Degenerate parametric amplification of sound [J]. Soviet Physics Acoustics. 1976, 22: 258-259.;3 Rudenko OV, Soluyan S I. Theoretical foundations of nonlinearacoustics. [M] New York: Springer, 1977: 86-94, 137-145.). Subsequently, Lan Chaofeng et al. obtained analytical expressions of the three-wave coupling equations under optimal phase-matching conditions using the one-dimensional Burgers equations and the spectral expansion method, and conducted underwater experimental verification. The results show that the nonlinear interaction of sound waves can regulate the weak wave by the pump wave, thereby achieving stable amplification of the weak wave (Lan CF, Yang DS, Lu D, Guo XX, Zhou ZH. The Theory and Experiment of Parametric Amplification of Three-wave Nonlinear Interaction in Water[J]. Chinese Journal of Electronics,2013,22(2):308-312.).

[0004] In summary, existing three-wave coupled acoustic field models largely rely on analytical solutions to one-dimensional equations. While analytical solutions are convenient to calculate, their ideal assumptions severely limit their application in complex real-world scenarios. Numerical solutions have a wider range of applications, but the three-wave coupled equations, which tightly couple three waves through non-homogeneous terms, are complex in form, posing significant challenges for numerical computation. Summary of the Invention

[0005] To address the problems of high computational complexity, low efficiency, and inaccurate results in existing numerical solution methods for three-wave coupled acoustic field models, this invention provides a numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater.

[0006] This invention establishes a three-wave coupled acoustic field model based on the KZK equations in a fully axisymmetric underwater space. It utilizes the second-order Strang operator splitting method, combined with finite difference and explicit Euler method, to achieve efficient solution of the three-wave coupled equations and accurately calculate the three-wave coupled acoustic field in a fully axisymmetric underwater space.

[0007] The technical solution adopted by this invention to solve the technical problem is as follows:

[0008] The present invention provides a numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater, which specifically includes the following steps:

[0009] Step S1: Make a simple harmonic assumption on the sound wave, and substitute the sound wave that satisfies the frequency matching condition of the three-wave difference into the KZK equation to obtain the three-wave difference coupling equation set.

[0010] Step S2: Using the second-order Strang operator splitting method, the difference-frequency three-wave coupling equation set is decomposed into diffraction equation, dissipation equation and nonlinear equation according to the physical meaning;

[0011] Step S3: The diffraction equation is solved using the chasing method, the dissipation equation is solved using the exact analytical solution, and the nonlinear equation is solved using the explicit Euler method.

[0012] Step S4: Set boundary conditions and calculate the underwater fully axisymmetric spatial three-wave coupled acoustic field.

[0013] As a preferred embodiment, the mathematical expression of the KZK equation is as follows:

[0014]

[0015] in, For sound pressure, For the speed of sound, For the density of the medium, Here is the dissipation coefficient. These are nonlinear coefficients. For axial spatial variables, For time variables, For the transverse Laplace operator, For the horizontal spatial variable; the first term of the KZK equation For the propagation of sound waves along the axial direction, the second term The third term is the diffraction term. This is the decay term, the last term. This is a nonlinear term.

[0016] In a preferred embodiment, in step S1, the difference frequency three-wave process is considered, and its difference frequency three-wave frequency matching condition is:

[0017]

[0018] in, and These are the angular frequencies of the two pump waves, respectively. ω is the angular frequency of the difference frequency sound wave.

[0019] As a preferred embodiment, it is assumed that the expression for the sound wave satisfying the difference-frequency three-wave frequency matching condition is as follows: , For sound pressure, The imaginary unit, Let t be the angular frequency and t be the time. Representing pump wave 1, pump wave 2, and difference frequency acoustic wave respectively, substituting them into the KZK equations yields the following set of three-wave coupling equations:

[0020]

[0021] in, and There are two pump waves, It is a difference frequency sound wave. Let the wave number be pump wave 1. Let be the wave number of pump wave 2. The wave number of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the sound pressure of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the pump wave 2 acoustic pressure.

[0022] In a preferred embodiment, step S3 involves setting a difference scheme, the mathematical expression of which is as follows:

[0023]

[0024]

[0025]

[0026] in, For sound pressure, For axial spatial variables, For horizontal spatial variables, For the first The sound pressure at the point, For the first The sound pressure at the point, For the first The sound pressure at the point, For the first The sound pressure at the point, The axial spatial step size, This represents the horizontal spatial step size.

[0027] In a preferred embodiment, the mathematical expression of the diffraction equation is as follows:

[0028]

[0029] in, and There are two pump waves, It is a difference frequency sound wave. Let the wave number be pump wave 1. Let be the wave number of pump wave 2. The wave number of the difference frequency sound wave;

[0030] Substituting the difference scheme into the diffraction equation yields the difference matrix, whose mathematical expression is as follows:

[0031]

[0032] in, , , , For sound field sound pressure, for direction, for direction, For different wave numbers of sound fields, For different sound fields, These represent pump wave 1, pump wave 2, and difference frequency sound wave, respectively; then the Thomas algorithm is used to solve the difference matrix.

[0033] In a preferred embodiment, the mathematical expression of the dissipation equation is as follows:

[0034]

[0035] The dissipation equation is solved using an exact analytical solution, and its mathematical expression is as follows:

[0036]

[0037] in, and There are two pump waves, It is a difference frequency sound wave. and These are the angular frequencies of the two pump waves, respectively. The angular frequency of the difference frequency sound wave. For the speed of sound, Here is the dissipation coefficient. and These are the initial values ​​of the two pump waves, This is the initial value of the difference frequency sound wave.

[0038] In a preferred embodiment, the mathematical expression of the nonlinear equation is as follows:

[0039]

[0040] The nonlinear equations are solved using the explicit Euler method. Substituting the difference scheme into the nonlinear equations yields:

[0041]

[0042] in, , , These are the coefficients of the nonlinear equation for pump wave 1, the nonlinear equation for pump wave 2, and the nonlinear equation for difference frequency acoustic wave, respectively. and There are two pump waves, It is a difference frequency sound wave. The imaginary unit, For the speed of sound, For the density of the medium, It is the conjugate form of the complex amplitude of the sound pressure of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the pump wave 2 acoustic pressure; For the first The sound pressure of pump wave 1 is applied. For the first The sound pressure of pump wave 2 is applied. For the first The sound pressure of the difference frequency sound wave at the point. For the first The complex conjugate sound pressure of the point difference frequency sound wave, For the first The complex conjugate sound pressure of pump wave 2.

[0043] As a preferred implementation, in step S4, the Sommerfeld boundary condition is used to simulate the outward radiation behavior of the wave at infinity, ensuring that the wave has no reflection at the boundary.

[0044] In a preferred embodiment, the mathematical expression of the Sommerfeld boundary condition is as follows:

[0045]

[0046] in, For sound pressure, For the speed of sound, The imaginary unit, For wave number.

[0047] The beneficial effects of this invention are:

[0048] (1) This invention uses the second-order Strang operator splitting method to decompose the three-wave coupling equation system into three sub-equations: diffraction, dissipation and nonlinearity, which reduces the solution complexity and has high computational efficiency.

[0049] (2) The present invention can accurately calculate the results of three-wave coupled sound field, providing strong support for subsequent research on the interaction law of three waves in three-wave coupled sound field.

[0050] (3) Compared with the traditional quasi-linear approximation method, the present invention ensures the integrity of the three-wave coupled equation system, which provides convenience for subsequent research under strong nonlinear conditions.

[0051] (4) The calculation results of the present invention are accurate and the implementation steps are simple. Attached Figure Description

[0052] Figure 1 The flowchart shows a numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater provided by the present invention.

[0053] Figure 2 Example diagram of the calculation region.

[0054] Figure 3 This represents the two-dimensional sound pressure distribution of the pump wave.

[0055] Figure 4 This represents the two-dimensional sound pressure distribution of the difference frequency sound wave.

[0056] Figure 5 This represents the variation of the sound pressure level of the pump wave along the axial direction with distance.

[0057] Figure 6 This represents the variation of the sound pressure level of a difference-frequency sound wave along the axial direction with distance. Detailed Implementation

[0058] The present invention will be further described in detail below with reference to the accompanying drawings.

[0059] This invention provides a numerical modeling method for a three-wave coupled acoustic field in a fully axisymmetric underwater space. By using the second-order Strang operator splitting method, the three-wave coupled equations based on the KZK equations are decomposed into several sub-equations, and the three-wave coupled acoustic field in a fully axisymmetric underwater space is accurately calculated.

[0060] See Figure 1The present invention provides a numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater, the specific implementation process of which is as follows:

[0061] Step S1: Make a simple harmonic assumption on the sound wave, and substitute the sound wave that satisfies the frequency matching condition of the three-wave difference into the KZK equation to obtain the three-wave difference coupling equation set.

[0062] In this invention, the mathematical expression of the KZK equation is as follows:

[0063]

[0064] in, For sound pressure, For the speed of sound, For the density of the medium, Here is the dissipation coefficient. These are nonlinear coefficients. For axial spatial variables, For time variables, For the transverse Laplace operator, It is a horizontal spatial variable.

[0065] The first term of the KZK equation For the propagation of sound waves along the axial direction, the second term The third term is the diffraction term. This is the decay term, the last term. This is a nonlinear term. Considering the difference-frequency three-wave process, its difference-frequency three-wave frequency matching condition is:

[0066]

[0067] in, and These are the angular frequencies of the two pump waves, respectively. ω is the angular frequency of the difference frequency sound wave.

[0068] Assuming the expression for a sound wave that satisfies the above difference-frequency three-wave frequency matching condition is: , For sound pressure, The imaginary unit, Let t be the angular frequency and t be the time. For different sound fields, Let represent pump wave 1, pump wave 2, and the difference frequency acoustic wave, respectively. Substituting these into the KZK equations yields the three-wave coupling equation set, the mathematical expression of which is as follows:

[0069]

[0070] in, and There are two pump waves, It is a difference frequency sound wave. Let the wave number be pump wave 1. Let be the wave number of pump wave 2. The wave number of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the sound pressure of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the pump wave 2 acoustic pressure.

[0071] Step S2: Using the second-order Strang operator splitting method, the difference-frequency three-wave coupling equation system is decomposed into three sub-equations according to the physical meaning;

[0072] In this invention, the three sub-equations are the diffraction equation, the dissipation equation, and the nonlinear equation.

[0073] The mathematical expression for the diffraction equation is as follows:

[0074]

[0075] The mathematical expression for the dissipation equation is as follows:

[0076]

[0077] The mathematical expression for the nonlinear equation is as follows:

[0078]

[0079] Step S3: The diffraction equation is solved using the chasing method, the dissipation equation is solved using the exact analytical solution, and the nonlinear equation is solved using the explicit Euler method.

[0080] (1) Define the difference scheme, the mathematical expression of which is as follows:

[0081]

[0082]

[0083]

[0084] in, For the first The sound pressure at the point, For the first The sound pressure at the point, For the first The sound pressure at the point, For the first The sound pressure at the point, The axial spatial step size, This represents the horizontal spatial step size.

[0085] (2) The diffraction equation is solved using the pursuit method (Thomas algorithm). First, the difference scheme is substituted into the diffraction equation to obtain the difference matrix, the mathematical expression of which is as follows:

[0086]

[0087] in, , , , For sound field sound pressure, for direction, for direction, For different sound fields, the wave number is denoted as .

[0088] Then, the Thomas algorithm is used to solve for the difference matrix obtained above.

[0089] (3) The dissipation equation is solved using an exact analytical solution, and its mathematical expression is as follows:

[0090]

[0091] in, and These are the initial values ​​of the two pump waves, This is the initial value of the difference frequency sound wave.

[0092] (4) The nonlinear equation is solved using the explicit Euler method. Substituting the difference scheme into the nonlinear equation, we can obtain:

[0093]

[0094] in, , , These are the coefficients of the nonlinear equation for pump wave 1, the nonlinear equation for pump wave 2, and the nonlinear equation for difference frequency acoustic wave, respectively. For the first The sound pressure of pump wave 1 is applied. For the first The sound pressure of pump wave 2 is applied. For the first The sound pressure of the difference frequency sound wave at the point. For the first The complex conjugate sound pressure of the point difference frequency sound wave, For the first The complex conjugate sound pressure of pump wave 2.

[0095] Step S4: Set boundary conditions and calculate the underwater fully axisymmetric spatial three-wave coupled acoustic field;

[0096] When performing sound field calculations, especially for difference frequency sound fields with low frequencies and long wavelengths, significant reflections will occur at the boundaries if appropriate lateral boundary conditions are not applied.

[0097] The Sommerfeld boundary condition is a classic open boundary condition, often used to simulate the outward radiation behavior of waves at infinity, thus ensuring that the waves do not reflect at the boundary. Its mathematical expression is as follows:

[0098]

[0099] Using the Sommerfeld boundary condition can effectively reduce reflections at lateral boundaries.

[0100] This invention uses the second-order Strang operator splitting method to decompose the original three-wave coupling equations into three sub-equations, accurately obtaining the acoustic field of three-wave coupling in a fully axisymmetric underwater space.

[0101] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0102] Example 1: Numerical Modeling of a Fully Axisymmetric Spatial Three-Wave Coupled Sound Field Underwater

[0103] Set the speed of sound in water water medium density kinematic viscosity coefficient Transducer radius (like Figure 2 As shown), the sound source radiates a Gaussian beam. Initial sound pressure amplitude radial distribution Pump wave frequency of the sound source radiation The generated difference frequency sound wave frequency Axial calculation step size Radial calculation step size .

[0104] The two-dimensional sound pressure distribution of the pump wave is as follows Figure 3 As shown, the pump wave exhibits a Gaussian distribution in the two-dimensional plane, and the sound pressure gradually decreases with the direction of propagation.

[0105] The two-dimensional sound pressure distribution of difference frequency sound waves is as follows Figure 4 The sound field of the difference frequency sound wave shown is relatively concentrated, and it first increases and then decreases with the direction of propagation.

[0106] The change of the sound pressure level of the pump wave along the axial direction with distance is as follows: Figure 5 As shown, in the axial direction, the pump wave sound field as a whole decreases with the propagation distance.

[0107] The variation of sound pressure level of a difference frequency sound wave with distance along the axial direction is as follows: Figure 6 As shown, in the axial direction, the difference frequency wave sound field exhibits a characteristic of first increasing and then decreasing.

[0108] The computation time for Example 1 was 6.14 seconds, demonstrating that this invention can efficiently calculate the three-wave coupled sound field in a fully axisymmetric underwater space. Furthermore, the calculation results of this invention effectively reflect the spatial distribution of the pump wave sound field and clearly describe the generation process of the difference frequency sound wave and the spatial sound field distribution, providing a scientific basis for accurate modeling of the underwater parametric array sound field.

[0109] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. However, these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater, characterized in that, Includes the following steps: Step S1: Make a simple harmonic assumption on the sound wave, and substitute the sound wave that satisfies the frequency matching condition of the three-wave difference into the KZK equation to obtain the three-wave difference coupling equation set. Step S2: Using the second-order Strang operator splitting method, the difference-frequency three-wave coupling equation set is decomposed into diffraction equation, dissipation equation and nonlinear equation according to the physical meaning; Step S3: The diffraction equation is solved using the chasing method, the dissipation equation is solved using the exact analytical solution, and the nonlinear equation is solved using the explicit Euler method. Step S4: Set boundary conditions and calculate the underwater fully axisymmetric spatial three-wave coupled acoustic field.

2. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 1, characterized in that, The mathematical expression of the KZK equation is as follows: ； in, For sound pressure, For the speed of sound, For the density of the medium, Here is the dissipation coefficient. These are nonlinear coefficients. For axial spatial variables, For time variables, For the transverse Laplace operator, For the horizontal spatial variable; the first term of the KZK equation For the propagation of sound waves along the axial direction, the second term The third term is the diffraction term. This is the decay term, the last term. This is a nonlinear term.

3. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 2, characterized in that, In step S1, considering the difference frequency three-wave process, the frequency matching condition for the difference frequency three-wave process is as follows: ； in, and These are the angular frequencies of the two pump waves, respectively. ω is the angular frequency of the difference frequency sound wave.

4. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 3, characterized in that, Assume the expression for a sound wave that satisfies the difference-frequency three-wave frequency matching condition is: , For sound pressure, The imaginary unit, Let t be the angular frequency and t be the time. Representing pump wave 1, pump wave 2, and difference frequency acoustic wave respectively, substituting them into the KZK equations yields the following set of three-wave coupling equations: ； in, and There are two pump waves, It is a difference frequency sound wave. Let the wave number be pump wave 1. Let be the wave number of pump wave 2. The wave number of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the sound pressure of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the pump wave 2 acoustic pressure.

5. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 1, characterized in that, In step S3, the difference scheme is set, and its mathematical expression is as follows: ； ； ； in, For sound pressure, For axial spatial variables, For horizontal spatial variables, For the first The sound pressure at the point, For the first The sound pressure at the point, For the first The sound pressure at the point, For the first The sound pressure at the point, The axial spatial step size, This represents the horizontal spatial step size.

6. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 5, characterized in that, The mathematical expression for the diffraction equation is as follows: ； in, and There are two pump waves, It is a difference frequency sound wave. Let the wave number be pump wave 1. Let be the wave number of pump wave 2. The wave number of the difference frequency sound wave; Substituting the difference scheme into the diffraction equation yields the difference matrix, whose mathematical expression is as follows: ； in, , , , For sound field sound pressure, for direction, for direction, For different wave numbers of sound fields, For different sound fields, These represent pump wave 1, pump wave 2, and difference frequency sound wave, respectively; then the Thomas algorithm is used to solve the difference matrix.

7. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 5, characterized in that, The mathematical expression of the dissipation equation is as follows: ； The dissipation equation is solved using an exact analytical solution, and its mathematical expression is as follows: ； in, and There are two pump waves, It is a difference frequency sound wave. and These are the angular frequencies of the two pump waves, respectively. The angular frequency of the difference frequency sound wave. For the speed of sound, Here is the dissipation coefficient. and These are the initial values ​​of the two pump waves, This is the initial value of the difference frequency sound wave.

8. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 5, characterized in that, The mathematical expression of the nonlinear equation is as follows: ； The nonlinear equations are solved using the explicit Euler method. Substituting the difference scheme into the nonlinear equations yields: ； in, , , These are the coefficients of the nonlinear equation for pump wave 1, the nonlinear equation for pump wave 2, and the nonlinear equation for difference frequency acoustic wave, respectively. and There are two pump waves, It is a difference frequency sound wave. The imaginary unit, For the speed of sound, For the density of the medium, It is the conjugate form of the complex amplitude of the sound pressure of the difference frequency sound wave. It is the conjugate form of the complex amplitude of the pump wave 2 acoustic pressure; For the first The sound pressure of pump wave 1 is applied. For the first The sound pressure of pump wave 2 is applied. For the first The sound pressure of the difference frequency sound wave at the point. For the first The complex conjugate sound pressure of the point difference frequency sound wave, For the first The complex conjugate sound pressure of pump wave 2.

9. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 1, characterized in that, In step S4, the Sommerfeld boundary condition is used to simulate the outward radiation behavior of the wave at infinity, ensuring that the wave has no reflection at the boundary.

10. The numerical modeling method for a fully axisymmetric spatial three-wave coupled acoustic field underwater according to claim 9, characterized in that, The mathematical expression for the Sommerfeld boundary condition is as follows: ； in, For sound pressure, For the speed of sound, The imaginary unit, For wave number.

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