Ssum frequency wave suppression method under underwater acoustic pulse three-wave mixing

Abstract

A sum frequency wave suppression method under underwater acoustic pulse three-wave mixing belongs to the field of nonlinear acoustics, and comprises the following steps: performing time integration on a KZK equation, and converting the KZK equation into a TBE equation by using generalized traveling wave transformation; dividing the TBE equation into four partial differential equations according to physical meanings by utilizing an operator splitting thought; the X-direction diffraction equation, the Y-direction diffraction equation and the dissipation equation are all solved by adopting a finite difference method, and the nonlinear equation is solved by adopting linear interpolation; setting initial conditions and boundary conditions; setting a sound source boundary condition to send a pulse signal, and adding a phase factor in a pulse signal expression; and changing the phase factor to obtain sum frequency wave suppression and difference frequency wave enhancement under the optimal phase factor. According to the method, the KZK equation is solved through the time domain, the sound field under the first-class pulse three waves is accurately described, and powerful support is provided for follow-up further research on influence factors of the sum frequency sound field and the difference frequency sound field. According to the invention, the difference frequency wave energy can be increased by 5dB at most.

Description

Technical Field

[0001] The present invention belongs to the technical field of nonlinear acoustics, and particularly relates to a method for suppressing the sum-frequency wave under underwater acoustic pulse three-wave mixing. Background Technique

[0002] Nonlinear acoustics refers to a three-wave process in which two pump waves with the same propagation characteristics generate sum and difference frequencies as a type of three-wave mixing. In the three-wave mixing of nonlinear acoustics, the sum-frequency wave and the difference-frequency wave have a relatively complex coupling relationship, and there is a certain energy competition effect. Since both the sum-frequency wave and the difference-frequency wave originate from the nonlinear interaction of the pump waves, when the energy of the pump wave is certain, the energy of the pump wave used to generate the sum-frequency wave is greater than that used to generate the difference-frequency wave, resulting in the energy of the difference-frequency wave being less than that of the sum-frequency wave. However, since the difference-frequency wave has a wider application range underwater, people hope to increase the energy of the difference-frequency wave. Therefore, suppressing the energy of the sum-frequency wave is of great significance for increasing the energy of the difference-frequency wave.

[0003] In 1995, Lee and Hamilton proposed a time-domain method for calculating the KZK equation to simulate the propagation of pulsed finite-amplitude sound beams. In the cylindrical coordinate system, this method considered the beam expansion effect, used the generalized traveling-wave transformation to transform the KZK equation into the TBE equation, and decomposed the equation using the operator splitting algorithm, reducing the difficulty of time-domain solution (Y.S. Lee, M.F. Hamilton. Time-domain mode ling of pulsed finite-amplitude soundbeams[J]. The Journal of the Acoustical Society of America, 1995, 97(2):906–917). In 2006, A.M. Gavrilov used the KZK equation to study the process of sum-frequency wave generation by the nonlinear interaction of three-frequency pump waves. By considering the amplitude-phase relationship of the pump waves, an analytical expression for the sum-frequency wave was obtained. The results showed that the sum-frequency wave could be amplified and suppressed by adjusting the amplitude and phase of the pump waves, and the accuracy of the results was verified by experiments (A.M. Gavrilov. Theoreti cal model of phase exclusion conditions for the sum-frequency wave produced by a nonlinear acoustic source. Acoust. Phys. 2007, 53:572–583). In 2013, Lan Chaofeng et al. obtained the analytical expression of the three-wave nonlinear interaction equation under the optimal phase-matching condition through the one-dimensional Burgers equation using the spectral expansion method and verified it experimentally. The results showed that the nonlinear interaction of sound waves could enable the pump wave to adjust the weak wave for a long time and achieve the stable amplification of the weak wave (C. Lan, D. Yang, D. Lu, X. Guo and Z. Zhou. The Theory and Experiment of Par ametric Amplification of Three-wave Nonlinear Interaction inWater[J]. Chinese Jou rnalof Electronics, 2013, 22(2):308-312.).

[0004] In the prior art, there is still a blank in the suppression of sum-frequency waves generated by two pulsed pump waves with the same propagation characteristics in a water medium. Summary of the Invention

[0005] The object of the present invention is to provide a method for suppressing the sum-frequency wave under underwater acoustic pulse three-wave mixing. The present invention can suppress the sum-frequency wave sound field under a class of nonlinear acoustic pulse three-wave mixing and improve the energy of the difference-frequency sound field.

[0006] The present invention proposes a method for suppressing the sum-frequency wave generated by two pulse pump waves under weak nonlinearity. Considering the high directivity of the high-frequency pump wave, starting from the numerical solution of the KZK equation in the time domain, the initial phases of the two pump waves are modified. The present invention finds a method for suppressing the sum-frequency wave, thereby increasing the energy of the difference-frequency wave to a certain extent.

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

[0008] A method for suppressing the sum-frequency wave under underwater acoustic pulse three-wave mixing provided by the present invention is characterized by comprising the following steps:

[0009] (a) Perform a first-order time integration on the KZK equation, and use the generalized traveling wave transformation to transform the KZK equation into the TBE equation;

[0010] (b) Using the idea of operator splitting, divide the TBE equation into an X-direction diffraction equation, a Y-direction diffraction equation, a dissipation equation, and a nonlinear equation according to physical meanings;

[0011] (c) The X-direction diffraction equation, the Y-direction diffraction equation, and the dissipation equation are all solved by the finite difference method, and the nonlinear equation is solved by linear interpolation;

[0012] (d) Set the initial conditions and boundary conditions;

[0013] (e) Set the sound source boundary conditions to send a pulse signal, and add a phase factor to the pulse signal expression

[0014] (f) Change the phase factor Obtain the suppression of the sum-frequency wave and the enhancement of the difference-frequency wave under the optimal phase factor.

[0015] Further, the mathematical expression for performing a first-order time integration on the KZK equation is as follows:

[0016]

[0017] Among them, p is the sound pressure, z is the axis direction of sound wave propagation, x and y are the transverse coordinate and the longitudinal coordinate respectively, δ is the dissipation coefficient, β is the nonlinear coefficient, ρ0 and c0 are the medium density and the sound speed respectively, t' is the delay time coordinate, and t” is the time integration variable.

[0018] Further, the mathematical expression of the generalized traveling wave transformation is as follows:

[0019]

[0020] Among them, σ, X, and Y are all dimensionless coordinates, x and y are the horizontal coordinate and the vertical coordinate respectively, a and b are the horizontal characteristic length and the vertical characteristic length of the sound source respectively, d is the Rayleigh distance, τ is the dimensionless delay time, ω0 = 2πf0 is the angular frequency, f0 is the sound wave frequency, p0 is the initial sound pressure, c0 is the sound speed in the medium, t′ is the delay time coordinate, and P is the dimensionless sound pressure.

[0021] Furthermore, the mathematical expression for converting the KZK equation into the TBE equation is as follows:

[0022]

[0023] Among them, and are the diffraction lengths in the horizontal and vertical directions respectively, A = α0(f0)d is the absorption term coefficient, α0(f0) is the absorption coefficient, which is related to the frequency, f0 is the sound wave frequency, is the nonlinear term coefficient, is the discontinuous distance, τ is the dimensionless delay time, τ' is the dimensionless time integration variable, P is the dimensionless sound pressure, σ, X, and Y are all dimensionless coordinates, ω0 is the angular frequency, p0 is the initial sound pressure, β is the nonlinear coefficient, ρ0 and c0 are the medium density and the sound speed respectively, a and b are the horizontal characteristic length and the vertical characteristic length of the sound source respectively, and d is the Rayleigh distance.

[0024] Furthermore, the calculation formulas for the X-direction diffraction equation, the Y-direction diffraction equation, the dissipation equation, and the nonlinear equation are respectively:

[0025]

[0026] Among them, P is the dimensionless sound pressure, σ, X, and Y are all dimensionless coordinates, and are the diffraction lengths in the horizontal and vertical directions respectively, ω0 is the angular frequency, c0 is the sound speed in the medium, a and b are the horizontal characteristic length and the vertical characteristic length of the sound source respectively, d is the Rayleigh distance, τ is the dimensionless delay time, τ' is the dimensionless time integration variable, N is the nonlinear term coefficient, and A is the absorption term coefficient.

[0027] Furthermore, the mathematical expression for the finite difference method is as follows:

[0028]

[0029] Among them, is the coefficient of the horizontal diffraction equation, is the coefficient of the vertical diffraction equation, is the coefficient of the dissipation equation, Δτ and Δσ k are the time calculation step and the calculation step in the axis direction respectively, A is the coefficient of the diffraction term, j represents the transverse spatial node, h represents the longitudinal spatial node, k represents the axial spatial node, and both i and m represent the time nodes. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node i. is the sound pressure value at the transverse spatial node j + 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node m. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node m. is the sound pressure value at the transverse spatial node j - 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node m. is the sound pressure value at the transverse spatial node j + 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. ) is the sound pressure value at the transverse spatial node j - 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h + 1, the axial spatial node k + 1, and the time node m. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h - 1, the axial spatial node k + 1, and the time node m. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h + 1, the axial spatial node k + 1, and the time node i. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h - 1, the axial spatial node k + 1, and the time node i. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i + 1. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i - 1.

[0030] Furthermore, the mathematical expression of the linear interpolation method is as follows:

[0031]

[0032] where Δτ and Δσ k are the time calculation step and the calculation step in the axis direction respectively. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node i - 1. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node i + 1, and N is the coefficient of the nonlinear term.

[0033] Furthermore, the mathematical expressions of the initial conditions and the boundary conditions are as follows:

[0034]

[0035] Among them, X min and X max are the lateral boundaries, Y min and Y max are the longitudinal boundaries, τ min and τ max are the time boundaries, σ, X, and Y are all dimensionless coordinates, and τ is the dimensionless delay time.

[0036] Furthermore, the mathematical expression of the pulse signal expression is as follows:

[0037]

[0038] Among them, is the symmetric pulse envelope function, nπ is the pulse width, A1 and A2 are the amplitudes of the two pump waves respectively, n1 and n2 are both frequency multiples, σ, X, and Y are all dimensionless coordinates, and τ is the dimensionless delay time.

[0039] Furthermore, in step (f), by solving the KZK equation in the time domain, the phase factor controlling the sum-frequency sound field under a class of three-wave mixing under the control pulse signal is accurately obtained.

[0040] The present invention accurately obtains the phase factor controlling the sum-frequency sound field under a class of three-wave mixing under the control pulse signal by solving the KZK equation in the time domain.

[0041] The beneficial effects of the present invention are:

[0042] (1) By solving the KZK equation in the time domain, the sound field under a class of pulsed three-waves is accurately described, providing strong support for further studying the influencing factors of the sum- and difference-frequency sound fields.

[0043] (2) Since there is an energy competition relationship between the sum-frequency wave and the difference-frequency wave in a three-wave structure, compared with the previous studies that only separately studied the sum-frequency wave or the difference-frequency wave, the process of three-wave interaction can be understood more effectively.

[0044] (3) The present invention can increase the energy of the difference-frequency wave by up to 5 dB at most, and has a relatively effective improvement on the energy of the difference-frequency wave. Description of the Drawings

[0045] Figure 1 It is a flowchart of the implementation of the sum - frequency wave suppression method under underwater acoustic pulse three - wave mixing provided by the present invention.

[0046] Figure 2 It is an example diagram of the calculation area.

[0047] Figure 3 It is the spectrogram at the receiving point when the phase difference of the pump wave is 0.

[0048] Figure 4 It is when the phase difference of the pump wave is It is the spectrogram at the receiving point.

[0049] Figure 5 It is when the phase difference of the pump wave is It is the spectrogram at the receiving point.

[0050] Figure 6 It is when the phase difference of the pump wave is It is the spectrogram at the receiving point.

[0051] Figure 7 It is the spectrogram at the receiving point when the phase difference of the pump wave is π. Specific implementation manner

[0052] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific implementation manners.

[0053] As Figure 1 and Figure 2 shown, a sum - frequency wave suppression method under underwater acoustic pulse three - wave mixing provided by the present invention mainly includes two parts: physical modeling and control method, and its specific implementation process is as follows:

[0054] (a) Perform a first - order time integration on the KZK equation, and the mathematical expression of this process is as follows:

[0055]

[0056] Among them, p is the sound pressure, z is the axis direction of sound wave propagation, x and y are the transverse coordinate and longitudinal coordinate respectively, δ is the dissipation coefficient, β is the nonlinear coefficient, ρ0 and c0 are the medium density and sound speed respectively, t' is the delay time coordinate, and t” is the time integration variable.

[0057] The mathematical expression of the generalized traveling - wave transformation is as follows:

[0058]

[0059] Among them, σ, X, and Y are all dimensionless coordinates, a and b are the transverse characteristic length and longitudinal characteristic length of the sound source respectively, d is the Rayleigh distance, τ is the dimensionless delay time, ω0 = 2πf0 is the angular frequency, f0 is the acoustic wave frequency, p0 is the initial sound pressure, and P is the dimensionless sound pressure.

[0060] Substitute the mathematical expressions of the generalized traveling wave transformation into the KZK equation respectively, and transform it into the TBE equation. The mathematical expressions of this process are as follows:

[0061]

[0062] Among them, and are the diffraction lengths in the transverse and longitudinal directions respectively. A = α0(f0)d is the diffraction term coefficient, and α0(f0) is the absorption coefficient, which is related to the frequency. is the nonlinear term coefficient. is the discontinuous distance, and τ' is the dimensionless time integration variable;

[0063] (b) Using the idea of operator splitting, the TBE equation is divided into four partial differential equations according to the physical meaning. These four partial differential equations are specifically as follows:

[0064]

[0065]

[0066] (c) The X-direction diffraction equation (Equation (3)), Y-direction diffraction equation (Equation (4)), and dissipation equation (Equation (5)) are all solved using the finite difference method, and the nonlinear equation (Equation (6)) is solved using the linear interpolation method.

[0067] The mathematical expressions of the so-called finite difference method are as follows:

[0068]

[0069] Among them, is the coefficient of the transverse diffraction equation. is the coefficient of the longitudinal diffraction equation. is the coefficient of the dissipation equation. j represents the transverse spatial node, h represents the longitudinal spatial node, k represents the axial spatial node, and i and m both represent the time nodes. is the sound pressure value at the transverse spatial node j, longitudinal spatial node h, axial spatial node k + 1, and time node i. is the sound pressure value at the transverse spatial node j, longitudinal spatial node h, axial spatial node k, and time node i. is the sound pressure value at the transverse spatial node j + 1, longitudinal spatial node h, axial spatial node k + 1, and time node m. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node m. is the sound pressure value at the lateral spatial node j - 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node m. is the sound pressure value at the lateral spatial node j + 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. ) is the sound pressure value at the lateral spatial node j - 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h + 1, the axial spatial node k + 1, and the time node m. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h - 1, the axial spatial node k + 1, and the time node m. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h + 1, the axial spatial node k + 1, and the time node i. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h - 1, the axial spatial node k + 1, and the time node i. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i + 1. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i - 1.

[0070] The mathematical expression of the said linear interpolation method is as follows:

[0071]

[0072] where, Δτ and Δσ k are the time calculation step length and the axis direction calculation step length respectively, is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node i - 1. is the sound pressure value at the lateral spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node i + 1.

[0073] (d) Set the initial conditions and boundary conditions, and their mathematical expressions are as follows:

[0074]

[0075] where, X min and X max are the lateral boundaries, Y min and Y max are the longitudinal boundaries, τ min and τ max are the time boundaries.

[0076] (e) Set the sound source boundary condition to send a pulse signal, and add a phase factor to the pulse signal expression. The mathematical expression of the pulse signal expression is as follows:

[0077]

[0078] Where, is the symmetric pulse envelope function, nπ is the pulse width, A1 and A2 are the amplitudes of the two pump waves respectively, and n1 and n2 are both frequency multiples.

[0079] (f) Change the phase factor to obtain the sum-frequency wave suppression and difference-frequency wave enhancement under the optimal phase factor.

[0080] The present invention accurately obtains the phase factor by solving the KZK equation in the time domain to control the sum-frequency sound field under a class of three-wave mixing under a pulsed signal.

[0081] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without making creative efforts belong to the scope of protection of the present invention.

[0082] The example parameters are set as follows: the sound speed in water c0 = 1500 m / s, the density of the water medium ρ0 = 1000 kg / m 3 , the nonlinear coefficient β = 3.5, the high-frequency pump waves f1 = 65 kHz, f2 = 60 kHz, the difference frequency f d = 5 kHz, the sum frequency f s = 125 kHz, the sound source level of the high-frequency pump wave SPL = 200 dB, select the typical pump wave phase difference The transmitting transducer is a circular transducer with a radius of 0.1 m, and the distance between the receiving point and the sound source is 120 m.

[0083] The pump wave phase difference The spectrum at the receiving point when is as Figure 3 shown. Through Figure 3 it can be seen that the sum-frequency sound pressure level is 186.858 dB, the difference-frequency sound pressure level is 159.498 dB, and most of the energy is concentrated in the sum-frequency three-wave, while the energy of the difference-frequency three-wave is small.

[0084] The pump wave phase difference The spectrum at the receiving point when is as Figure 4 shown. Through Figure 4 it can be seen that the phase difference is changed to After that, the energy of the sum-frequency wave decreases, and the energy of the difference-frequency wave increases. Compared with when the phase difference is not changed, the sound pressure level of the difference-frequency wave is increased by about 3 dB.

[0085] Pump wave phase difference The spectrum at the receiving point when Figure 5 is as shown in Figure 5 It can be seen that at this time, the energy of the sum-frequency wave further decreases, and the energy of the difference-frequency wave further increases. At this time, compared with when the phase difference is not changed, the sound pressure level of the difference-frequency wave is increased by about 5 dB.

[0086] When the pump wave phase difference is the spectrum at the receiving point is as shown in Figure 6 It can be seen that at this time, the sound pressure level of the difference-frequency wave decreases by 2 dB, which is equivalent to when the phase difference is Figure 6 The sound pressure level of the sum-frequency wave is equivalent to that when and is the sound pressure level.

[0087] When the pump wave phase difference is the spectrum at the receiving point is as shown in Figure 7 It can be seen that the sound pressure level of the difference-frequency wave is equivalent to that when the phase difference is not changed, and the difference is 5 dB compared with when the phase difference is Figure 7 Comparing with the previous results, it can be found that when the phase difference is the sound pressure level of the difference-frequency wave increases greatly, and the sound pressure level of the sum-frequency wave decreases significantly. When the phase difference is 0 or π, the sound pressure level of the difference-frequency wave is relatively low. Therefore, when the phase difference is the effect is more obvious.

[0088] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and modifications can be made, and these improvements and modifications should also be regarded as the protection scope of the present invention.​

Claims

1. A method for suppressing sum-frequency waves under underwater acoustic pulse three-wave mixing, characterized in that, It includes the following steps: (a) Perform a first-order time integration on the KZK equation and use the generalized traveling-wave transformation to transform the KZK equation into the TBE equation; (b) Using the idea of operator splitting, divide the TBE equation into an X-direction diffraction equation, a Y-direction diffraction equation, a dissipation equation, and a nonlinear equation according to the physical meaning; (c) The X-direction diffraction equation, the Y-direction diffraction equation, and the dissipation equation are all solved by the finite difference method, and the nonlinear equation is solved by linear interpolation; (d) Set the initial conditions and boundary conditions; (e) Set the sound source boundary condition to send a pulse signal, and add a phase factor to the pulse signal expression (f) Changing the phase factor Sum-frequency wave suppression and difference-frequency wave enhancement under the optimal phase factor are obtained.

2. A method for suppressing sum-frequency waves under underwater acoustic pulse three-wave mixing according to claim 1, characterized in that, The mathematical expression for performing a first-order time integration on the KZK equation is as follows: where p is the sound pressure, z is the axis direction of sound wave propagation, x and y are the transverse coordinate and the longitudinal coordinate respectively, δ is the dissipation coefficient, β is the nonlinear coefficient, ρ0 and c0 are the medium density and the sound speed respectively, t' is the delay time coordinate, and t” is the time integration variable.

3. A method for suppressing sum-frequency waves under underwater acoustic pulse three-wave mixing according to claim 1, characterized in that, The mathematical expression for the generalized traveling-wave transformation is as follows: where σ, X, and Y are all dimensionless coordinates, x and y are the transverse coordinate and the longitudinal coordinate respectively, a and b are the transverse characteristic length and the longitudinal characteristic length of the sound source respectively, d is the Rayleigh distance, τ is the dimensionless delay time, ω0 = 2πf0 is the angular frequency, f0 is the sound wave frequency, p0 is the initial sound pressure, c0 is the medium sound speed, t′ is the delay time coordinate, and P is the dimensionless sound pressure.

4. A method for suppressing sum-frequency waves in underwater acoustic pulse three-wave mixing according to claim 1, characterized in that, The mathematical expression for transforming the KZK equation into the TBE equation is as follows: Among them, and are the diffraction lengths in the horizontal and vertical directions respectively. A = α0(f0)d is the diffraction term coefficient, α0(f0) is the absorption coefficient, which is related to the frequency, f0 is the acoustic wave frequency, is the nonlinear term coefficient, is the discontinuity distance, τ is the dimensionless delay time, τ' is the dimensionless time integration variable, P is the dimensionless sound pressure, σ, X and Y are all dimensionless coordinates, ω0 is the angular frequency, p0 is the initial sound pressure, β is the nonlinear coefficient, ρ0 and c0 are the medium density and sound speed respectively, a and b are the horizontal and vertical characteristic lengths of the sound source respectively, and d is the Rayleigh distance.

5. A method for suppressing sum-frequency waves under underwater acoustic pulse three-wave mixing according to claim 1, characterized in that, The calculation formulas for the X-direction diffraction equation, the Y-direction diffraction equation, the dissipation equation, and the nonlinear equation are respectively: where P is the dimensionless sound pressure, and σ, X, and Y are all dimensionless coordinates, and are the diffraction lengths in the transverse and longitudinal directions respectively, ω0 is the angular frequency, c0 is the sound speed in the medium, a and b are the transverse and longitudinal characteristic lengths of the sound source respectively, d is the Rayleigh distance, τ is the dimensionless delay time, τ' is the dimensionless time integration variable, N is the coefficient of the nonlinear term, and A is the coefficient of the diffraction term.

6. The sum-frequency wave suppression method under underwater acoustic pulse three-wave mixing according to claim 1, characterized in that The mathematical expression for the finite difference method is as follows: Among them, is the coefficient of the transverse diffraction equation, is the coefficient of the longitudinal diffraction equation, is the coefficient of the dissipation equation, Δτ and Δσ k are the time calculation step and the calculation step in the axis direction respectively, A is the coefficient of the diffraction term, j represents the transverse spatial node, h represents the longitudinal spatial node, k represents the axial spatial node, and both i and m represent the time nodes. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node i. is the sound pressure value at the transverse spatial node j + 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node m is the transverse between node k + 1 and the time node is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node m. is the transverse spatial node j - 1, the longitudinal spatial node h, the axial space is the sound pressure value at node m is the sound pressure value at the node m. ， is the sound pressure value at the transverse spatial node j + 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. is the sound pressure value at the transverse spatial node j - 1, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i. is the transverse spatial node j, the longitudinal spatial node h + 1, the axial spatial node k + 1, and the time node k + 1 and the time node is the transverse spatial node j, the longitudinal spatial node h - 1, the axial node axial is the sound pressure value at the node m. is the transverse spatial node j, the longitudinal spatial node h + 1, between the spatial node k + 1 and node h - 1, the axial space is the sound pressure value at the time node i. is the transverse spatial node j, the longitudinal spatial is the sound pressure value is the sound at the node k + 1 and the time node i is the transverse space at + 1 is the sound pressure value is the time node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i The mathematical expression for the linear interpolation method is as follows: is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k + 1, and the time node i - 1.

7. A method for suppressing sum-frequency waves in underwater acoustic pulse three-wave mixing according to claim 1, characterized in that, length Among them, Δτ and Δσ k are the time calculation step size and the calculation step in the axis direction respectively is the transverse is the sound pressure value at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node i - 1 at point i + 1 is the sound at the transverse spatial node j, the longitudinal spatial node h, the axial spatial node k, and the time node The mathematical expression for the initial conditions and boundary conditions is as follows: is the sound pressure value N is the non - linear term coefficient 8. A method for suppressing the sum-frequency wave under underwater acoustic pulse three-wave mixing according to claim 1, characterized in that, The mathematical expression for the pulse signal expression is as follows: Among them, X min and X max are the horizontal boundaries, Y min and Y max are the vertical boundaries, τ min and τ max are the time boundaries, σ, X, and Y are all dimensionless coordinates, and τ is the dimensionless delay time.

9. A method for suppressing the sum-frequency wave under underwater acoustic pulse three-wave mixing according to claim 1, characterized in that ​ wherein, is a symmetric pulse envelope function, nπ is the pulse width, A1 and A2 are the amplitudes of two pump waves respectively, n1 and n2 are both frequency multiples, σ, X, and Y are all dimensionless coordinates, and τ is a dimensionless delay time.

10. A method for suppressing sum-frequency waves under underwater acoustic pulse three-wave mixing according to claim 1, characterized in that In step (f), the phase factor was accurately obtained by solving the KZK equation in the time domain. The sum-frequency sound field under a type of three-wave mixing under a control pulse signal.

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