A three-dimensional low-frequency sound field calculation method and calculation system under an island topography environment

Abstract

The present application belongs to the field of underwater acoustic physics technology, and particularly relates to a three-dimensional low-frequency sound field calculation method and system under an island and reef topography environment, comprising: establishing a sound field differential equation under a three-dimensional cylindrical coordinate system; based on the operator theory and the forward field assumption, a parabolic differential equation is established under the three-dimensional cylindrical coordinate system based on the parabolic equation theory; the approximate form of the root operator is determined, and the parabolic differential equation is converted into a root form horizontal recursive form equation; according to the high-order Padé rational approximation, it is converted into a basic algebra form horizontal recursive equation; according to the split-step theory, the basic algebra form horizontal recursive equation is converted into an N Zp and N Yp step cycle horizontal recursive equation; according to the boundary conditions of the water body-reef and sea surface and the Galerkin discrete method, combined with the spatial discrete division of the waveguide, the effective representation of the operator action in different directions is obtained; according to the cycle horizontal recursive equation and the representation of the operator action in different directions, the three-dimensional low-frequency sound field under the island and reef topography environment is calculated.

Description

Technical Field

[0001] This invention belongs to the field of underwater acoustic field calculation technology in underwater acoustic physics. Specifically, it relates to a three-dimensional low-frequency sound field calculation method and system in island and reef terrain environments. Background Technology

[0002] As a typical underwater acoustic countermeasure and anti-countermeasure environment, the significant topographical changes in island and reef waters severely limit the use of existing sound field calculation models, leading to the generation of three-dimensional sound field propagation characteristics such as horizontal refraction and horizontal diffraction, as well as the time-frequency characteristics and spatial distribution of modulated signals. To achieve the assessment of low-frequency sonar detection capabilities in island and reef environments, analyzing the three-dimensional low-frequency sound propagation characteristics and establishing a three-dimensional low-frequency sound field calculation method adapted to island and reef topographic environments is of great significance. In existing three-dimensional sound field simulations, the commonly used three-dimensional ray model can describe the horizontal refraction process under island and reef topography, but it cannot effectively realize the propagation process of low-frequency sound field diffraction phenomena. Existing three-dimensional sound field simulation methods adapted to the low-frequency band cannot be applied to multiphase media environments of water-island-air, i.e., situations where the reef is exposed above the sea surface. The N×2D method combined with low-frequency sound field calculation theory can overcome the limitations of multiphase media or island and reef topography, but it cannot characterize three-dimensional phenomena such as horizontal refraction and horizontal diffraction. Summary of the Invention

[0003] To address the problems existing in the prior art, this invention proposes a method for calculating three-dimensional low-frequency sound fields in island and reef terrain environments. This method comprehensively considers the island and reef terrain features and the three-dimensional low-frequency sound propagation characteristics to simulate the three-dimensional low-frequency sound field in island and reef terrain environments.

[0004] This invention provides a method for calculating three-dimensional low-frequency sound fields in island and reef terrain environments, the method specifically including:

[0005] For typical island and reef topographic environments at sea, a differential equation for the acoustic field is established in a three-dimensional cylindrical coordinate system.

[0006] Based on operator theory and forward field assumption, and on the theory of parabolic equations, a parabolic differential equation is established in three-dimensional cylindrical coordinates.

[0007] Determine the approximate form of the radical operator, and transform the parabolic differential equation into a horizontal recursive form equation in radical form;

[0008] Based on the higher-order Padé rational approximation, the horizontal recurrence equation in radical form is transformed into a horizontal recurrence equation in basic algebraic form.

[0009] Based on the split-step theory, under waveguide spatial discretization, the basic algebraic form of the horizontal recurrence equation is transformed into N. Zp and N Yp Step-by-step horizontal recursive equation;

[0010] Based on the boundary conditions of the water-reef and sea surface and the Galerkin discretization method, combined with the spatial discretization of the waveguide, an effective characterization of the action of operators in different directions is obtained.

[0011] Based on the cyclic horizontal recursive equation and the characterization of the action of operators in different directions, the three-dimensional low-frequency sound field under the island and reef terrain environment is calculated.

[0012] As an improvement to the above technical solution, the sound field differential equation is established in a three-dimensional cylindrical coordinate system for typical island and reef topographic environments on the sea surface; the specific process includes:

[0013] Assuming a high-resistivity fluid approximation of the island / reef body, and assuming that the water body and the island / reef-seabed parameters are respectively (ρ... w ,c w ) and (ρ b ,c b ,α b );

[0014] Where, ρ w c is the density of the water. w The speed of sound in water; ρ b c is the density of the seabed-island reef body; b The sound speed of the seabed-island reef; α b For absorption by the seabed and reef bodies; for the sound pressure field excited by a single-frequency harmonic point source with a time factor of exp(-iωt); where i is an imaginary number; ω is the angular frequency; t is time; in the three-dimensional cylindrical coordinate system (r,θ,z), the elliptic Helmholtz equation satisfied by the sound pressure field p(r,θ,z) is:

[0015]

[0016] Where p is the sound pressure of the medium, denoted as p = p(r,θ,z); k 1,2 The medium wavenumber is k1 or k2; when k 1,2 =k1=ω / c w When k represents the wave number of the medium in the water; when k 1,2 =k2=ω / (c b +iα b When ), it is represented as the medium wavenumber of the seabed;

[0017] ρ is the density of the medium, and its value is ρ w or ρ b When ρ = ρ w When ρ = ρ_t, it represents the density of the water; when ρ = ρ_t b At that time, it indicates the density of the seabed;

[0018] The boundary conditions are defined by the sea surface; where the sea surface has a mean value of And the standard deviation is σ srf Gaussian distributed random vector z srf (r) is used to characterize; The vertical translation amount is used to avoid negative coordinates; and All are boundary conditions at the water-reef-seabed interface z = H(r,θ); where H - Characterized as the water body at depth H on one side; H + Characterized as the reef-seabed side at depth H; p(r,θ,z)| z＝D =0 is the cutoff boundary condition introduced at the cutoff boundary z=D;

[0019] The elliptic Helmholtz equation satisfied by the above sound pressure field is used as the sound field differential equation to complete the establishment of the sound field differential equation.

[0020] As an improvement to the above technical solution, the parabolic differential equation is established in three-dimensional cylindrical coordinates based on operator theory and the forward field assumption, and on the theory of parabolic equations; the specific process includes:

[0021] Based on the step approximation method, the phase factor exp(ik0r) is extracted under the reference sound velocity c0 and reference wavenumber k0 = ω / c0, and let:

[0022]

[0023] By employing factorization and operator theory, and neglecting the errors generated during operator commutation, and taking only the forward field, the acoustic field differential equation is transformed into a parabolic differential equation:

[0024]

[0025] Here, Y and Z are operators in two different directions:

[0026]

[0027]

[0028] Where Y is the medium operator in the direction of travel; it takes the value Y1 or Y2; Z is the medium operator in the vertical direction; it takes the value Z1 or Z2.

[0029] When ρ = ρ w c = c w k 1,2 When θ = k1, Y = Y1 represents the operator of the water body in the θ direction; Z = Z1 represents the operator of the water body in the vertical direction.

[0030] When ρ = ρ bc = c b k 1,2 When = k2, Y = Y2, which represents the operator of the seabed-island reef body in the θ direction; Z = Z2, which represents the operator of the seabed-island reef body in the vertical direction.

[0031] As an improvement to the above technical solution, the process of determining the approximate form of the radical operator, converting the parabolic differential equation into a horizontal recursive form equation in radical form, includes:

[0032] Based on the actual topographic features of islands and reefs, the approximate form of the radical operator is established:

[0033]

[0034] Discretize the radial coordinate r into r l =lΔr(l=1,2,...,N) l The radial discrete length is Δr. The parabolic differential equation is transformed into a horizontal recursive form in radical form:

[0035]

[0036] Where u(r) l (θ, z) represents the horizontal distance r l =Calculation of the sound field at lΔr; u(r) l+1 (θ, z) represents the horizontal distance r l+1 The calculation of the sound field at (l+1)Δr.

[0037] As an improvement to the above technical solution, the horizontal recursive equation in radical form is transformed into a horizontal recursive equation in basic algebraic form based on the higher-order Padé rational approximation; the specific process includes:

[0038] Based on the higher-order Padé rational approximation, the horizontal recurrence equation in radical form is transformed into a horizontal recurrence equation in basic algebraic form:

[0039]

[0040] Where u(r) l (θ, z) represents the horizontal distance r l =Calculation of the sound field at lΔr; u(r) l+1 (θ, z) represents the horizontal distance r l+1 =Calculation of the sound field at (l+1)Δr;

[0041] and The j-th term characterizing the action of operator Z Zp =1,2,...,N Zp Padé rational approximation coefficients of order 1 and To characterize the j-th action of operator Y Yp =1,2,...,N Yp Padé rational approximation coefficients of order N Zp With N Yp radicals and The order of the higher-order Padé rational approximation.

[0042] As an improvement to the above technical solution, based on the split-step theory, under waveguide spatial discretization, the basic algebraic form of the horizontal recursive equation is transformed into N... Zp and N Yp The step-by-step recursive equation; its specific process includes:

[0043] When based on the horizontal distance r l = sound field u(r) at lΔr l Calculate r (θ, z) l+1 The sound field u(r) at (l+1)Δr l+1 When , θ, z), the azimuth coordinate θ is discretized into θ m =mΔθ(m=1,2,..N). θ ,) The vertical coordinate z is discretized into z j =jΔz(j=1,2,...,N) z ), where Δθ and Δz are the azimuth and vertical distances from the walk, respectively;

[0044] At a horizontal distance r = r l For each discrete value θ of the azimuth angle m and all vertical discrete values ​​z j According to the split-step theory, N is obtained. Zp Step-by-step horizontal recurrence equation:

[0045]

[0046] Where, j Zp =1,2,...,N Zp Let N be the order of each of the higher-order Padé rational approximations of the Z-radical operator. Zp The input quantity of the step-by-step horizontal recursion is u(r) l ,θ m ,z j Given, let Based on the first step of the horizontal recursion form, from Obtain the first step output Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the Nth...Zp Step output

[0047] In N Zp Based on the calculation of the step-by-step recursive equation, for each vertical discrete value z j and all discrete values ​​of azimuth angle θ m According to the split-step theory, N is obtained. Yp Step-by-step horizontal recurrence equation:

[0048]

[0049] Where, j Yp =1,2,...,N Yp Let N be the order of each of the higher-order Padé rational approximations of the Y radical operator. Yp The input quantity for the step-by-step horizontal recursion is from Give, let Based on the first step of the horizontal recursion form, from Obtain the first step output Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the Nth... Yp Step output Nth Yp Step output equal to radial r l+1 The sound field u(r) at the discrete point = (l+1)Δr l+1 ,θ m ,z j ).

[0050] As an improvement to the above technical solution, the method of obtaining an effective characterization of the action of operators in different directions by combining the boundary conditions of the water-reef and sea surface with the Galerkin discretization method and the spatial discretization of the waveguide; the specific process includes:

[0051] For the vertical operator Z, for the horizontal distance r = r l j Zp =1,2,...,N Zp The iterative equation is used to iterate through the discrete azimuth values ​​θ. m (m=1,2,...,N θ According to z = z srf (r l ,θ m The boundary conditions representing the interface between the water body or reef and the sea surface, and the truncation boundary conditions introduced at z = D, are as follows: and N z ; where Nzsrf For the sea surface z = z srf (r l ,θ m At discrete points in the vertical direction; when the horizontal recursive equation is a computational quantity on the left side, the superscript... When the horizontal recursive equation is a computational quantity on the right side, the superscript is used. For other discrete values ​​z in the vertical direction j Based on the Galerkin discretization method, we can derive j = N zsrf +1,...,N z The effect of -1 at the vertical discrete point can be characterized as:

[0052]

[0053]

[0054]

[0055] Where Δz is the vertical distance from the walk, ρ j =ρ(r) l ,θ m ,z j ) and c j =c(r l ,θ m ,z j ) respectively represent (r l ,θ m ,z j The density of the medium and the speed of sound at that location. and They represent (r) l ,θ m ,z j Correction amount and corrected relative density at ) For (r) l ,θ m ,z j Medium wavenumber at ) The squared difference with reference wavenumber k0;

[0056] For the operator Y in the θ direction, for the horizontal distance r = r l j Yp =1,2,...,N Yp The step-by-step horizontal recursive equation, by reasonably setting the azimuth values, ensures that the initial azimuth θ1 and the ending azimuth are consistent. The environmental parameters change gradually, and the effect of the azimuth operator is negligible; for each discrete value z of the vertical coordinate... j (j = 1, 2, ..., N) z Take the discrete azimuth value θ m (m=1,N θThe operator action of ) is characterized as follows:

[0057]

[0058]

[0059] For the remaining discrete values ​​of θ m (m=2,3,...,N θ -1), according to the Galerkin discretization method, we can derive:

[0060]

[0061]

[0062] Where the superscript is used when the horizontal recursive equation is a computational quantity on the left side. When the horizontal recursive equation is a computational quantity on the right side, the superscript is used. Δθ is the azimuth distance from the walk; ρ m =ρ(r) l ,θ m ,z j ) and c m =c(r l ,θ m ,z j ) respectively represent (r l ,θ m ,z j The density of the medium and the speed of sound at that location. and They represent (r) l ,θ m ,z j Correction amount and corrected relative density at ().

[0063] As an improvement to the above technical solution, the three-dimensional low-frequency sound field under island and reef terrain environment is calculated based on the cyclic horizontal recursive equation and the characterization of the action of operators in different directions; the specific process includes:

[0064] Taking the point where the sound source is mapped onto the horizontal plane as the origin, and according to the parabolic equation, the initial field theory, or normal wave theory, the horizontal distance between the first horizontal discrete point and the origin is r1 = Δr, and the spatial discrete point θ... m (m=1,2,...,N θ ) and z j (j = 1, 2, ..., N) z The sound pressure p(r1,θ) on the surface m ,z j The initial sound field computational cost is obtained as follows:

[0065]

[0066] As N Zp With N Yp Input of the step-by-step horizontal recursive equation Perform N Zp With N Yp The solution is obtained by iteratively solving the loop, resulting in the sound field computational cost at a horizontal distance r2 between the second horizontal discrete point and the origin. With u(r2,θ) m ,z j The sound field computational quantity u(r) at each discrete point of the level is obtained by iteratively solving the problem using the recursive input value of the next level. l ,θ m ,z j (l=3,4,...,N) l (), until the set maximum horizontal distance is calculated;

[0067] According to u(r) l ,θ m ,z j (l=1,2,...,N) l ), (r l ,θ m ,z j The sound pressure at point (r) is used as the three-dimensional low-frequency sound field p(r) in the island and reef topographic environment. l ,θ m ,z j )for:

[0068]

[0069] This invention also provides a three-dimensional low-frequency sound field calculation system for island and reef terrain environments, the system comprising:

[0070] The sound field equation establishment module is used to establish sound field differential equations in a three-dimensional cylindrical coordinate system for typical island and reef topographic environments on the sea surface.

[0071] The parabolic equation establishment module is used to establish parabolic differential equations in three-dimensional cylindrical coordinates based on the acoustic field differential equations, according to operator theory and forward field assumptions and parabolic equation theory.

[0072] The equation transformation module is used to determine the approximate form of the radical operator and convert parabolic differential equations into horizontal recursive form equations in radical form.

[0073] The recursive transformation module is used to transform horizontal recursive equations in radical form into horizontal recursive equations in basic algebraic form based on the higher-order Padé rational approximation.

[0074] The cyclic recursion module is used to transform the basic algebraic form of the horizontal recursive equation into N, based on the split-step theory and with waveguide spatial discretization. Zp and N Yp Step-by-step horizontal recursive equation;

[0075] The operator characterization module is used to obtain effective characterizations of the operator action in different directions based on the boundary conditions of the water-reef and sea surface and the Galerkin discretization method, combined with the spatial discretization of the waveguide.

[0076] The sound field solution module is used to calculate the three-dimensional low-frequency sound field in the island and reef terrain environment based on the cyclic horizontal recursive equation and the characterization of the action of operators in different directions.

[0077] The advantages of this invention compared to the prior art are:

[0078] The method of this invention employs robust operator approximation and water-island-sea surface boundary processing to establish a three-dimensional low-frequency sound field adapted to the island and reef terrain environment. It effectively characterizes the three-dimensional low-frequency sound propagation process such as horizontal diffraction and horizontal refraction, ensuring the stability and computational efficiency of the three-dimensional sound field calculation in the island and reef terrain environment. It overcomes the limitations of the multiphase medium environment of water-island-air and the extreme terrain of islands and reefs on the simulation of three-dimensional low-frequency sound fields. Attached Figure Description

[0079] Figure 1 This is a schematic diagram of a marine island and reef terrain environment model in Embodiment 1 of the present invention, which is a method for calculating three-dimensional low-frequency sound field in an island and reef terrain environment.

[0080] Figure 2 This is a schematic diagram of the discrete division of the horizontal plane space in cylindrical coordinate system in Embodiment 1 of the method of the present invention;

[0081] Figure 3 The result of sound field loss distribution calculation in Embodiment 1 of the method of the present invention;

[0082] Figure 4 This is a comparison chart of the propagation loss curves at a receiving depth of 30m and a 0-degree azimuth in Embodiment 1 of the method of the present invention. Detailed Implementation

[0083] The present invention will now be further described with reference to the accompanying drawings.

[0084] This invention provides a three-dimensional low-frequency sound field calculation method for island and reef terrain environments. The method mainly establishes a three-dimensional underwater low-frequency sound field calculation method adapted to island and reef terrain environments based on parabolic equation theory, robust and efficient radical operator approximation method and reliable interface processing method.

[0085] The method includes:

[0086] Starting from the premise of revealing the three-dimensional low-frequency sound propagation process in the island and reef environment, a high-impedance fluid is used to approximate the island and reef body, and a parabolic equation model satisfied by the sound field in the island and reef environment is established.

[0087] Starting from the actual island and reef terrain features that combine extreme and gradual characteristics, an appropriate radical operator approximation form is established, operators in different directions are separated, and a horizontal recursive form satisfied by the forward sound field is established to achieve effective compatibility between the three-dimensional sound propagation process and the efficient calculation of the underwater long-range sound field.

[0088] Based on higher-order rational approximation and the split-step method, a finite-step horizontal recursive equation is established for island and reef topographic environments. At the same time, under the assumption of fluid in the island and reef body, a reasonable boundary treatment method is established to achieve effective solution of the radial and azimuth recursive equations in the presence of water, reef and air, thus achieving the purpose of low-frequency long-range sound field calculation in island and reef environments.

[0089] Simulation results under typical island and reef terrain show that this method can effectively achieve robust and rapid calculation of three-dimensional low-frequency sound fields in island and reef sea areas, and can be effectively used for the study of three-dimensional sound propagation phenomena such as horizontal refraction and horizontal sound diffraction under island and reef terrain.

[0090] The method specifically includes:

[0091] Based on the need for simulating the three-dimensional sound propagation process in an island and reef environment, we reasonably assume that the high-impedance fluid approximates the island and reef body. For typical island and reef topographic environments on the sea surface, i.e., when the island and reef body may be exposed above the sea surface, we establish the sound field differential equation in a three-dimensional cylindrical coordinate system.

[0092] Specifically, to achieve three-dimensional sound field calculation and three-dimensional sound propagation process analysis in an island and reef environment, a high-impedance fluid is assumed to approximate the island and reef body. The water body and the island / reef-seabed parameters are assumed to be (ρ...). w ,c w ) and (ρ b ,c b ,α b ); where ρ w c is the density of the water. w The speed of sound in water; ρ b c is the density of the seabed-island reef body; b The sound speed of the seabed-island reef; α b For absorption by the seabed and reef bodies; for the sound pressure field excited by a single-frequency harmonic point source with a time factor of exp(-iωt); where i is an imaginary number; ω is the angular frequency; t is time; in the three-dimensional cylindrical coordinate system (r,θ,z), the elliptic Helmholtz equation satisfied by the sound pressure field p(r,θ,z) is:

[0093]

[0094] Where p is the sound pressure of the medium, denoted as p = p(r,θ,z); k 1,2 The medium wavenumber is k1 or k2; when k 1,2 =k1=ω / c w When k represents the wave number of the medium in the water; when k 1,2 =k2=ω / (c b +iα b When ), it is represented as the medium wavenumber of the seabed;

[0095] ρ is the density of the medium, and its value is ρ w or ρ b When ρ = ρ w When ρ = ρ_t, it represents the density of the water; when ρ = ρ_t b At that time, it indicates the density of the seabed;

[0096] The boundary conditions are defined by the sea surface; where the sea surface has a mean value of And the standard deviation is σ srf Gaussian distributed random vector z srf (r) is used to characterize; The vertical translation amount is used to avoid negative coordinates; and All are boundary conditions at the water-reef-seabed interface z = H(r,θ); where H - Characterized as the water body at depth H on one side; H + Characterized as the reef-seabed side at depth H; p(r,θ,z)| z＝D =0 is the cutoff boundary condition introduced at the cutoff boundary z=D;

[0097] The elliptic Helmholtz equation satisfied by the above sound pressure field is used as the sound field differential equation to complete the establishment of the sound field differential equation.

[0098] Based on operator theory and forward field assumption, for island and reef topographic environments where the reef body may be exposed above the sea surface, parabolic differential equations adapted to the acoustic low-frequency band characteristics and island and reef topographic characteristics are established in three-dimensional cylindrical coordinates based on parabolic equation theory.

[0099] Specifically, based on the step approximation method, and considering energy conservation and columnar geometric expansion at the vertical interface, the phase factor exp(ik0r) is extracted under the reference sound velocity c0 and reference wavenumber k0=ω / c0, let:

[0100]

[0101] By employing factorization and operator theory, and neglecting the errors generated during operator commutation, and considering only the forward field, the acoustic field differential equation is transformed into a parabolic differential equation, namely:

[0102]

[0103] Here, Y and Z are operators in two different directions:

[0104]

[0105]

[0106] Where Y is the medium operator in the direction of travel; it takes the value Y1 or Y2; Z is the medium operator in the vertical direction; it takes the value Z1 or Z2.

[0107] When ρ = ρ w c = c w k 1,2 When θ = k1, Y = Y1 represents the operator of the water body in the θ direction; Z = Z1 represents the operator of the water body in the vertical direction.

[0108] When ρ = ρ b c = c b k 1,2 When = k2, Y = Y2, which represents the operator of the seabed-island reef body in the θ direction; Z = Z2, which represents the operator of the seabed-island reef body in the vertical direction.

[0109] Based on the needs of computational stability and deep-sea long-range computing, and based on actual computational requirements, the approximate form of the radical operator is reasonably determined to separate the effects of the azimuth and vertical operators, and to convert the parabolic differential equation into a horizontal recursive form equation in radical form.

[0110] Specifically, taking into account the actual topographic features of islands and reefs, and considering computational stability and the computational complexity of deep-sea long-range operations, an approximate form of the radical operator is established:

[0111]

[0112] Discretize the radial coordinate r into r l =lΔr(l=1,2,...,N) l If the radial discrete length is Δr, then the parabolic differential equation is transformed into a horizontal recursive form equation in radical form:

[0113]

[0114] Where u(r) l (θ, z) represents the horizontal distance r l =Calculation of the sound field at lΔr; u(r) l+1 (θ, z) represents the horizontal distance rl+1 The calculation of the sound field at (l+1)Δr.

[0115] Based on the higher-order Padé rational approximation, the horizontal recurrence equation in radical form is transformed into a horizontal recurrence equation in basic algebraic form.

[0116] Specifically, based on the higher-order Padé rational approximation, the horizontal recurrence equation in radical form is transformed into a horizontal recurrence equation in basic algebraic form.

[0117]

[0118] Among them, u(r) l (θ, z) represents the horizontal distance r l =Calculation of the sound field at lΔr; u(r) l+1 (θ, z) represents the horizontal distance r l+1 =Calculation of the sound field at (l+1)Δr;

[0119] and The j-th term characterizing the action of operator Z Zp =1,2,...,N Zp Padé rational approximation coefficients of order 1 and To characterize the j-th action of operator Y Yp =1,2,...,N Yp Padé rational approximation coefficients of order N Zp With N Yp radicals and The order of the higher-order Padé rational approximation corresponds to the calculation range of the grazing angle and azimuth angle.

[0120] Based on the split-step theory, under waveguide spatial discretization, the basic algebraic form of the horizontal recurrence equation is transformed into N. Zp and N Yp The horizontal recursive equation is used to solve the sound field horizontally;

[0121] Specifically, when based on the horizontal distance r l = sound field u(r) at lΔr l Calculate r (θ, z) l+1 The sound field u(r) at (l+1)Δr l+1 When considering the changes in the acoustic field calculation frequency and the horizontal variation of marine waveguide environmental parameters (θ, z), the azimuth coordinate θ is discretized into θz. m =mΔθ(m=1,2,...,N) θ The vertical coordinate z is discretized into z j =jΔz(j=1,2,...,N)z ), where Δθ and Δz are the azimuth and vertical distances from the walk, respectively;

[0122] At a horizontal distance r = r l For each discrete value θ of the azimuth angle m and all vertical discrete values ​​z j According to the split-step theory, N is obtained. Zp Step-by-step horizontal recurrence equation:

[0123]

[0124] Where, j Zp =1,2,...,N Zp Let N be the order of each of the higher-order Padé rational approximations of the Z-radical operator. Zp The input quantity of the step-by-step horizontal recursion is u(r) l ,θ m ,z j Given, let Based on the first step of the horizontal recursion form, from Obtain the first step output Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the Nth... Zp Step output

[0125] In N Zp Based on the calculation of the step-by-step recursive equation, for each vertical discrete value z j and all discrete values ​​of azimuth angle θ m According to the split-step theory, N is obtained. Yp Step-by-step horizontal recurrence equation:

[0126]

[0127] Where, j Yp =1,2,...,N Yp Let N be the order of each of the higher-order Padé rational approximations of the Y radical operator. Yp The input quantity for the step-by-step horizontal recursion is from Give, let Based on the first step of the horizontal recursion form, from Obtain the first step output Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the Nth... Yp Step output Nth Yp Step output equal to radial r l+1 The sound field u(r) at the discrete point = (l+1)Δr l+1 ,θ m ,z j ).

[0128] Based on the boundary conditions of water-reef and sea surface (water-island-air interface) and Galerkin discretization method, combined with the spatial discretization of waveguide, an effective characterization of the action of operators in different directions is obtained.

[0129] Specifically, regarding the effects of operators Y and Z in the horizontal recursive equation, based on the Galerkin (one-dimensional finite element) discretization method and the boundary conditions of the water-island-air interface, the effects of the Z-radical and Y-radical operators at each Padé step under spatial discretization are processed as follows:

[0130] For the vertical operator Z, for the horizontal distance r = r l j Zp =1,2,...,N Zp The iterative equation is used to iterate through the discrete azimuth values ​​θ. m (m=1,2,...,N θ According to z = z srf (r l ,θ m The boundary conditions representing the interface between the water body or reef and the sea surface, and the truncation boundary conditions introduced at z = D, are as follows: and N z ; where N zsrf For the sea surface z = z srf (r l ,θ m At discrete points in the vertical direction; when the horizontal recursive equation is a computational quantity on the left side, the superscript... When the horizontal recursive equation is a computational quantity on the right side, the superscript is used. For other discrete values ​​z in the vertical direction j Based on the Galerkin discretization method, we can derive j = N zsrf +1,...,N z The effect of -1 at the vertical discrete point can be characterized as:

[0131]

[0132]

[0133]

[0134] Where Δz is the vertical distance from the walk, ρ j =ρ(r)l ,θ m ,z j ) and c j =c(r l ,θ m ,z j ) respectively represent (r l ,θ m ,z j The density of the medium and the speed of sound at that location. and They represent (r) l ,θ m ,z j Correction amount and corrected relative density at ) For (r) l ,θ m ,z j Medium wavenumber at ) The squared difference with reference wavenumber k0;

[0135] For the operator Y in the θ direction, for the horizontal distance r = r l j Yp =1,2,...,N Yp The step-by-step horizontal recursive equation, by reasonably setting the azimuth values, ensures that the initial azimuth θ1 and the ending azimuth are consistent. The environmental parameters change gradually, and the effect of the azimuth operator is negligible; for each discrete value z of the vertical coordinate... j (j = 1, 2, ..., N) z Take the discrete azimuth value θ m (m=1,N θ The operator action of ) is characterized as follows:

[0136]

[0137]

[0138] For the remaining discrete values ​​of θ m (m=2,3,...,N θ -1), according to the Galerkin discretization method, we can derive:

[0139]

[0140]

[0141] Where the superscript is used when the horizontal recursive equation is a computational quantity on the left side. When the horizontal recursive equation is a computational quantity on the right side, the superscript is used. Δθ is the azimuth distance from the walk; ρ m =ρ(r) l ,θm ,z j ) and c m =c(r l ,θ m ,z j ) respectively represent (r l ,θ m ,z j The density of the medium and the speed of sound at that location. and They represent (r) l ,θ m ,z j Correction amount and corrected relative density at )

[0142] Based on the recursive equations of the cyclic level and the characterization of the operator's action, the three-dimensional low-frequency sound field under the island and reef terrain environment is calculated.

[0143] Specifically, taking the mapping point of the sound source on the horizontal plane as the origin, and according to the parabolic equation, the initial field theory, or normal wave theory, the horizontal distance between the first horizontal discrete point and the origin is r1 = Δr, and the spatial discrete point θ... m (m=1,2,...,N θ ) and z j (j = 1, 2, ..., N) z The sound pressure p(r1,θ) on the surface m ,z j The initial sound field computational cost is obtained as follows:

[0144]

[0145] As N Zp With N Yp Input of the step-by-step horizontal recursive equation Perform N Zp With N Yp The solution is obtained by iteratively solving the loop, resulting in the sound field computational cost at a horizontal distance r2 between the second horizontal discrete point and the origin. With u(r2,θ) m ,z j The sound field computational quantity u(r) at each discrete point of the level is obtained by iteratively solving the problem using the recursive input value of the next level. l ,θ m ,z j (l=3,4,...,N) l (), until the set maximum horizontal distance is calculated;

[0146] According to u(r) l ,θ m ,z j (l=1,2,...,N) l ), (rl ,θ m ,z j The sound pressure at point (r) is used as the three-dimensional low-frequency sound field p(r) in the island and reef topographic environment. l ,θ m ,z j )for:

[0147]

[0148] Example 1.

[0149] This invention discloses a three-dimensional low-frequency sound field calculation method for island and reef terrain environments. This method, based on parabolic equation theory, robust and efficient radical operator approximation methods, and reliable interface processing techniques, establishes a three-dimensional underwater low-frequency sound field calculation method adapted to island and reef terrain environments. The method mainly includes: starting from revealing the three-dimensional low-frequency sound propagation process in island and reef environments, approximating the island and reef body using high-impedance fluids, and establishing a parabolic equation model satisfied by the sound field in this environment; and taking the actual island and reef terrain characteristics, which combine extreme and gradually varying features, as the starting point... Starting from this point, an appropriate radical operator approximation form is established, operators in different directions are separated, and a horizontal recursive form satisfied by the forward sound field is established to achieve effective compatibility between the three-dimensional sound propagation process and the efficient calculation of underwater long-range sound fields. Based on higher-order rational approximations and the split-step method, a finite-step horizontal recursive equation is established for island and reef topographic environments. At the same time, under the assumption of fluid in the island and reef body, a reasonable boundary treatment method is established to achieve effective solution of the radial and azimuth recursive equations under the presence of water-reef-air, thus achieving the goal of low-frequency long-range sound field calculation in island and reef environments.

[0150] This invention discloses a three-dimensional low-frequency sound field calculation method adapted to island and reef terrain environments, wherein the waveguide environment model of the island and reef terrain is as follows: Figure 1 As shown: Establish a three-dimensional cylindrical coordinate system (r, θ, z), taking the z-axis downwards as positive, and the water density and sound velocity as ρ. w and c w The reef body adopts the high-resistivity fluid approximation, and the density, sound velocity, and absorption of the seabed-reef body are ρ. b c b and α b The water depth is H, and the shape of the island / reef is approximated by an ideal cone, with the apex of the cone located at an azimuth θ = θ s And r = r s At this location, the radius of the base of the conical reef is R. s The height of the island / reef is H s And the height above sea level is H u For a time factor of exp(-iωt) and a depth of z in the water sThe sound pressure field excited by a single-frequency harmonic point source, the Helmholtz equation satisfied by the sound pressure field p(r,θ,z) is:

[0151]

[0152] Where, k1=ω / c w The wave number in water; k2=ω / (c b +iα b ) represents the seafloor wave number; For sea surface boundary conditions; and The boundary conditions at the water-reef-seabed interface z = H(r,θ); p(r,θ,z)| z＝D =0 is a cutoff boundary condition introduced artificially at the cutoff boundary z=D. To avoid the influence of the reflection effect of the cutoff boundary, a matching layer is needed for additional processing.

[0153] To solve the three-dimensional sound field, based on the step approximation method, considering energy conservation and columnar geometric expansion at the vertical interface, the phase factor exp(ik0r) is extracted under the reference sound velocity c0 and reference wavenumber k0=ω / c0, taking:

[0154]

[0155] By employing factorization and operator theory, and neglecting the errors arising from operator commutation, and considering only the forward field, the elliptic Helmholtz differential equation can be transformed into a parabolic differential equation, i.e.:

[0156]

[0157] Where the operators Y and Z are:

[0158]

[0159] In the solution process, considering both computational stability and the computational complexity of deep-sea remote operations, an approximate form of the radical operator is established:

[0160]

[0161] At the same time, such as Figure 2 As shown, the radial coordinate r on the horizontal plane is discretized into r0. l =lΔr(l=1,2,...,N) l The azimuth coordinates θ are discretized into θ0. m =mΔθ(m=1,2,...,N) θ The radial distance step length is always Δr, and the azimuth distance step length is always Δθ. Generally, the radial distance step length and the azimuth distance step length satisfy Δr≤5λ and Δθ≤5λ, respectively. λ is the sound wavelength; the vertical coordinate z is discretized into z0.j =jΔz(j=1,2,...,N) z The vertical step size is always Δz, and generally satisfies Δz ≤ λ / 10; according to the split-step theory and the higher-order Padé rational approximation, the parabolic differential equation can be transformed into N Zp and N Yp The step-by-step horizontal recursive equation is based on the horizontal distance r. l =Calculation of sound field quantity u(r) at lΔr l Calculate r (θ, z) l+1 The calculated quantity of the sound field at (l+1)Δr is u(r). l+1 When ,θ,z), first perform N Zp Solve the step-by-step horizontal recursive equation, whose equation form is:

[0162]

[0163] Where, j Zp =1,2,...,N Zp Let N be the order of each of the higher-order Padé rational approximations of the Z-radical operator. Zp The input quantity of the step-by-step horizontal recursion is u(r) l ,θ m ,z j Given, let Based on the first step of the horizontal recursion form, from Obtain the first step output Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the Nth... Zp Step output In solving Then, for each vertical discrete value z j and all discrete values ​​of azimuth angle θ m Then proceed with N Yp Solve the iterative recurrence equation step by step; the equation has the following form:

[0164]

[0165] Where, j Yp =1,2,...,N Yp Let N be the order of each of the higher-order Padé rational approximations of the Y radical operator. Yp The input quantity for the step-by-step horizontal recursion is from Give, let Based on the first step of the horizontal recursion form, from Obtain the first step output Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the Nth... Yp Step output Nth Yp Step output equal to radial r l+1 = (l+1)Δr sound field calculation quantity u(r) at discrete points l+1 ,θ m ,z j ).

[0166] The effects of operators Y and Z in the horizontal recursive equation are numerically discretized based on the boundary conditions of the water-island-sea surface and the Galerkin (one-dimensional finite element) method. For operator Z, the discretized value θ is obtained for each azimuth angle. m (m=1,2,...,N θ (where j = 1, 2, ..., N) zsrf and N z The effect at vertical discrete points can be characterized as:

[0167]

[0168] j = N zsrf +1,...,N z The effect of -1 at the vertical discrete point can be characterized as:

[0169]

[0170]

[0171]

[0172] Where, N zsrf For the sea surface z = z srf (r l ,θ m At discrete points in the vertical direction; when the horizontal recursive equation is a computational quantity on the left side, the superscript... When the horizontal recursive equation is a computational quantity on the right side, the superscript is used. Δz is the vertical distance from the walk, ρ j =ρ(r) l ,θ m ,z j ) and c j =c(r l ,θ m ,z j ) respectively represent (r l ,θ m ,z j The density of the medium and the speed of sound at that location. and They represent (r) l,θ m ,z j Correction amount and corrected relative density at ) For (r) l ,θ m ,z j Medium wavenumber at ) The squared difference with reference wavenumber k0;

[0173] For operator Y, at each discrete value z in the vertical coordinate... j (j = 1, 2, ..., N) z ), m=1 and N θ The effect at discrete points of orientation can be characterized as follows:

[0174]

[0175]

[0176] For the remaining discrete values ​​of θ m (m=2,3,...,N θ -1), the operator's action can be characterized as:

[0177]

[0178]

[0179] Where the superscript is used when the horizontal recursive equation is a computational quantity on the left side. When the horizontal recursive equation is a computational quantity on the right side, the superscript is used. Δθ is the azimuth distance from the walk; ρ m =ρ(r) l ,θ m ,z j ) and c m =c(r l ,θ m ,z j ) respectively represent (r l ,θ m ,z j The density of the medium and the speed of sound at that location. and They represent (r) l ,θ m ,z j Correction amount and corrected relative density at )

[0180] For the discrete representation of the cyclic recursive equation and the operator action, based on the parabolic self-initial field theory or other methods, due to the introduction of robust radical operator theory, the three-dimensional sound field solution of the island and reef topographic sea area can be effectively realized by basic methods such as the chasing method, and the three-dimensional low-frequency sound field propagation process under the island and reef topography is given.

[0181] Simulation example:

[0182] Considering the ocean waveguide with an ideal conical reef, the sound velocity in the water is c. w =1500m / s, density ρ w =1.0 g / cm3, the speed of sound on the seabed and on islands and reefs is c b =1650m / s, density is ρ b =1.5g / cm3, absorption as α b =0.25dB / λ, vertices of conical reefs r s =20km and θ s =0 degrees, the height of the island / reef is H s =400m and the height above sea level is H u =200m, the water depth at other locations is 200m, the sound field calculation frequency is 25Hz, and the sound source is located directly below the origin z. s At a distance of 112m, with a receiving depth of 30m, the propagation loss is calculated with an azimuth of 0°, a radial distance step length Δr = 10m, an azimuth distance step length Δθ = 0.05°, a vertical discrete interval Δz = 1m, and a maximum horizontal distance r max =40km, the azimuth calculation range is [-60°, 60°] ° ] Calculate the maximum depth z max =600m, sound field propagation loss distribution as follows Figure 3 As shown, the propagation loss at the receiving depth in the receiving azimuth is compared between the conventional N×2D calculation method and the method of the present invention. Figure 4 As shown.

[0183] like Figure 4 As shown in the comparison results, the sound field calculation method established in this invention can effectively realize the calculation of three-dimensional low-frequency sound field in island and reef terrain environments. The radical operator used meets the needs of solving three-dimensional low-frequency sound field and calculating stability of complex island and reef terrain. It can effectively give the three-dimensional sound field propagation characteristics such as horizontal sound refraction and horizontal sound diffraction in island and reef terrain environments, and has high accuracy.

[0184] Example 2.

[0185] This invention also provides a three-dimensional low-frequency sound field calculation system for island and reef terrain environments, the system comprising:

[0186] The sound field equation establishment module is used to establish sound field differential equations in a three-dimensional cylindrical coordinate system for typical island and reef topographic environments on the sea surface.

[0187] The parabolic equation establishment module is used to establish parabolic differential equations in three-dimensional cylindrical coordinates based on the acoustic field differential equations, according to operator theory and forward field assumptions and parabolic equation theory.

[0188] The equation transformation module is used to determine the approximate form of the radical operator and convert parabolic differential equations into horizontal recursive form equations in radical form.

[0189] The recursive transformation module is used to transform horizontal recursive equations in radical form into horizontal recursive equations in basic algebraic form based on the higher-order Padé rational approximation.

[0190] The cyclic recursion module is used to transform the basic algebraic form of the horizontal recursive equation into N, based on the split-step theory and with waveguide spatial discretization. Zp and N Yp Step-by-step horizontal recursive equation;

[0191] The operator characterization module is used to obtain effective characterizations of the operator action in different directions based on the boundary conditions of the water-reef and sea surface and the Galerkin discretization method, combined with the spatial discretization of the waveguide.

[0192] The sound field solution module is used to calculate the three-dimensional low-frequency sound field in the island and reef terrain environment based on the cyclic horizontal recursive equation and the characterization of the action of operators in different directions.

[0193] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for calculating a three-dimensional low-frequency sound field in an island and reef terrain environment, the method specifically includes: For typical island and reef topographic environments at sea, a differential equation for the acoustic field is established in a three-dimensional cylindrical coordinate system. Based on operator theory and forward field assumption, and on the theory of parabolic equations, a parabolic differential equation is established in three-dimensional cylindrical coordinates. Determine the approximate form of the radical operator, and transform the parabolic differential equation into a horizontal recursive form equation in radical form; Based on the higher-order Padé rational approximation, the horizontal recurrence equation in radical form is transformed into a horizontal recurrence equation in basic algebraic form. Based on the split-step theory, under waveguide spatial discretization, the basic algebraic form of the horizontal recurrence equation is transformed into... and The step-by-step horizontal recursive equation, the for Z The order of the higher-order Padé rational approximation of radical operators, the stated for Y The order of the higher-order Padé rational approximation of radical operators; Based on the boundary conditions of the water-reef and sea surface and the Galerkin discretization method, combined with the spatial discretization of the waveguide, an effective characterization of the action of operators in different directions is obtained. Based on the cyclic horizontal recursive equation and the characterization of the action of operators in different directions, the three-dimensional low-frequency sound field under the island and reef terrain environment is calculated.

2. The method for calculating three-dimensional low-frequency sound field in island and reef terrain environments according to claim 1, characterized in that, For typical island and reef topographic environments on the sea surface, a sound field differential equation is established in a three-dimensional cylindrical coordinate system; Its specific process includes: Assuming a high-resistivity fluid approximation of the island / reef body, and assuming the water body and island / reef-seabed parameters are as follows: and ; in, Density of water; Speed ​​of sound in water; Density of the seabed-island reef body; The speed of sound on the seabed and reefs; Absorption by the seabed-island reef body; for a time factor of The sound pressure field excited by a single-frequency harmonic point source; among which, It is an imaginary number; Angular frequency; For time; in three-dimensional cylindrical coordinates Below, sound pressure field The satisfied elliptic Helmholtz equation is: ； in, The sound pressure of the medium is denoted as . ; The medium wavenumber has a value of [value missing]. or ;when When, it represents the wave number of the medium in the water; when When, it is expressed as the medium wavenumber of the seabed; For the density of the medium, take a value of or ;when When, it indicates the density of the water; when At that time, it indicates the density of the seabed; The boundary conditions are defined by the sea surface; where the sea surface has a mean value of And the standard deviation is Gaussian distributed random vector To characterize; The vertical translation amount is used to avoid negative coordinates; and All are water bodies and reef-seabed interfaces Boundary conditions at; where, Characterized as depth H Located on one side of the water body; Characterized as depth H Located on the reef-seabed side; To cut off the boundary The truncation boundary condition introduced at the point; The elliptic Helmholtz equation satisfied by the above sound pressure field is used as the sound field differential equation to complete the establishment of the sound field differential equation.

3. The method for calculating three-dimensional low-frequency sound field in island and reef terrain environments according to claim 1, characterized in that, Based on operator theory and forward field assumption, and on the theory of parabolic equations, a parabolic differential equation is established in three-dimensional cylindrical coordinates. Its specific process includes: Based on the step approximation method, at the reference sound speed and reference wavenumber Phase factor extraction ,make: ； in, For reference wavenumber, Angular frequency, It is an imaginary number. For sound pressure field, For the density of the medium, For the calculation of the sound field in parabolic equation theory, Radial coordinates, These are the azimuth coordinates. Vertical coordinates; By employing factorization and operator theory, and neglecting the errors generated during operator commutation, and taking only the forward field, the acoustic field differential equation is transformed into a parabolic differential equation: ； in, and For two operators in different directions, It is an imaginary number. Reference wavenumber: in, for Medium operator in direction; value or ; For the vertical medium operator; the value is... or , For the density of the medium, Radial coordinates, These are the azimuth coordinates. For vertical coordinates, The medium wavenumber has a value of [value missing]. or , The wave number of the medium in water; The medium wavenumber at the seabed; when , , hour, This indicates that the water body is in Operators in direction; , represents the vertical direction of the water body; when , , hour, This indicates that the seabed-island reef body is in Operators in direction; , represents the vertical operator of the seabed-island reef body; in, Density of water; Speed ​​of sound in water; Density of the seabed-island reef body; The speed of sound on the seabed and reefs.

4. The method for calculating three-dimensional low-frequency sound field in island and reef terrain environments according to claim 1, characterized in that, The determination of the approximate form of the radical operator transforms the parabolic differential equation into a horizontal recursive form equation in radical form. Its specific process includes: Based on the actual topographic features of islands and reefs, the approximate form of the radical operator is established: ； in, for Medium operator in direction; value or ; For vertical medium operators; Take radial coordinates Discretized The radial discrete length is Transform the parabolic differential equation into a horizontal recursive form in radical form: ； in, Horizontal distance The computational cost of the sound field at that location; Horizontal distance The computational cost of the sound field at that location. These are the azimuth coordinates. For vertical coordinates, It is an imaginary number. For reference wavenumber, for Medium operator in direction; value or ; For vertical medium operators.

5. The method for calculating three-dimensional low-frequency sound field in island and reef terrain environments according to claim 1, characterized in that, The process of transforming the horizontal recurrence equation in radical form into a horizontal recurrence equation in basic algebraic form based on the higher-order Padé rational approximation includes: Based on the higher-order Padé rational approximation, the horizontal recurrence equation in radical form is transformed into a horizontal recurrence equation in basic algebraic form. ； in, Horizontal distance The computational cost of the sound field at that location; Horizontal distance The computational cost of the sound field at that location. These are the azimuth coordinates. For vertical coordinates, It is a radial discrete length; and To characterize the operator The first function Padé rational approximation coefficients of order 1 and To characterize the operator The first function Padé rational approximation coefficients of order 1 and radicals and The order of the higher-order Padé rational approximation.

6. The method for calculating three-dimensional low-frequency sound field in island and reef terrain environments according to claim 1, characterized in that, Based on the split-step theory, under waveguide spatial discretization, the basic algebraic form of the horizontal recursive equation is transformed into... and Step-by-step horizontal recursive equation; Its specific process includes: When based on horizontal distance Sound field calculate Sound field At that time, the orientation is directed to the coordinates. Discretized Vertical coordinates Discretized , and These are the azimuth and vertical distances from the walking distance, respectively. in, It is a radial discrete length; In horizontal distance For each azimuth discrete value and all vertical discrete values According to the split-step theory, we obtain Step-by-step horizontal recurrence equation: ； in, for Z The orders of the higher-order Padé rational approximations of radical operators. radical The order of higher-order Padé rational approximations and To characterize the operator The first function Padé rational approximation coefficients of order 1; The input quantity for the step-by-step horizontal recursion is from Give, let According to the first step of the horizontal recursive form, from Obtain the output of the first step Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the first one is obtained. Step output ; exist Based on the calculation of the step-by-step recursive equation, for each vertical discrete value and all azimuth discrete values According to the split-step theory, we obtain Step-by-step horizontal recurrence equation: ； in, for Y The orders of the higher-order Padé rational approximations of radical operators. radical The order of higher-order Padé rational approximations and To characterize the operator The first function Padé rational approximation coefficients of order 1; The input quantity for the step-by-step horizontal recursion is from Give, let According to the first step of the horizontal recursive form, from Obtain the output of the first step Then As input to the next step of horizontal recursion, the output of the second step is obtained. And so on until the first one is obtained. Step output , No. Step output equal to radial Sound field at discrete points ; in, It is a radial discrete length.

7. The method for calculating three-dimensional low-frequency sound field in island and reef terrain environments according to claim 1, characterized in that, Based on the boundary conditions of the water-reef and sea surface and the Galerkin discretization method, combined with the spatial discretization of the waveguide, an effective characterization of the action of operators in different directions is obtained. Its specific process includes: For vertical operators Z For horizontal distance Place The iterative equation is used to iterate through each discrete azimuth angle value. ,according to The boundary conditions of the interface between the water body or reef and the sea surface, and The truncation boundary conditions introduced at this point are: , and ;in, For the sea surface At discrete points in the vertical direction; when the horizontal recursive equation is a computational quantity on the left side, the superscript... When the horizontal recursive equation is a right-hand side computation, the superscript is used. For other discrete values ​​in the vertical direction Based on the Galerkin discretization method, the following can be derived: The effect at vertical discrete points can be characterized as follows: ； ； ； in, , The medium wavenumber has a value of [value missing]. or , For reference wavenumber, for Z The order of the higher-order Padé rational approximation of radical operators. For vertical coordinates, For vertical distance from walking length, and They represent The density of the medium and the speed of sound at that location, and They represent Correction amount and corrected relative density, for Medium wavenumber With reference wavenumber The difference of squares; for Operators in direction Y For horizontal distance Place The step-by-step horizontal recursive equation, by reasonably setting the azimuth values, ensures the initial azimuth... and termination direction The environmental parameters change gradually, and the effect of the azimuth operator is negligible; for each discrete value of the vertical coordinate... Take the discrete values ​​of the azimuth The operator action is characterized as follows: ； ； For the remaining azimuth discrete values Based on the Galerkin discretization method, the following can be derived: ； ； Where the superscript is used when the horizontal recursive equation is a computational quantity on the left side. When the horizontal recursive equation is a right-hand side computation, the superscript is used. , for Y The order of the higher-order Padé rational approximation of radical operators. For reference wavenumber, and They represent The density of the medium and the speed of sound at that location, For reference sound speed, when the horizontal recursive equation is a calculation quantity on the left side, the superscript is used. When the horizontal recursive equation is a right-hand side computation, the superscript is used. ; The distance from the direction of walking; and They represent The density of the medium and the speed of sound at that location, and They represent Correction amount and corrected relative density.

8. The method for calculating three-dimensional low-frequency sound field in island and reef terrain environments according to claim 1, characterized in that, The three-dimensional low-frequency sound field under the island and reef terrain environment is calculated based on the cyclic horizontal recursive equation and the characterization of the action of operators in different directions. Its specific process includes: Taking the point where the sound source is mapped onto the horizontal plane as the origin, and according to the parabolic equation, the initial field theory, or normal wave theory, the horizontal distance between the first horizontal discrete point and the origin is... Spatial discrete points and sound pressure The initial sound field computational cost is obtained as follows: ； As and Input of the step-by-step horizontal recursive equation ,conduct and By iteratively solving the problem step by step, the horizontal distance between the second horizontal discrete point and the origin is obtained. sound field computation and with The sound field computation at each discrete point is calculated by iteratively solving the problem using the recursive input value for the next level. This continues until the set maximum horizontal distance is reached; according to , The sound pressure at a certain location serves as a three-dimensional low-frequency sound field in an island and reef topographical environment. for: in, For radial discrete length, It is an imaginary number. For reference wavenumber, for Z The order of the higher-order Padé rational approximation of radical operators. for Y The order of the higher-order Padé rational approximation of radical operators. and They represent The density of the medium and the speed of sound at that location, For reference speed of sound.

9. A three-dimensional low-frequency sound field calculation system for island and reef terrain environments, characterized in that, The system includes: The sound field equation establishment module is used to establish sound field differential equations in a three-dimensional cylindrical coordinate system for typical island and reef topographic environments on the sea surface. The parabolic equation establishment module is used to establish parabolic differential equations in three-dimensional cylindrical coordinates based on the acoustic field differential equations, according to operator theory and forward field assumptions and parabolic equation theory. The equation transformation module is used to determine the approximate form of the radical operator and convert parabolic differential equations into horizontal recursive form equations in radical form. The recursive transformation module is used to transform horizontal recursive equations in radical form into horizontal recursive equations in basic algebraic form based on the higher-order Padé rational approximation. The cyclic recursion module is used to transform the basic algebraic form of the horizontal recursive equations into, based on the split-step theory and with waveguide spatial discretization, the equations. and Step-by-step horizontal recursive equation, for Z The order of the higher-order Padé rational approximation of radical operators. for Y The order of the higher-order Padé rational approximation of radical operators; The operator characterization module is used to obtain effective characterizations of the operator action in different directions based on the boundary conditions of the water-reef and sea surface and the Galerkin discretization method, combined with the spatial discretization of the waveguide. The sound field solution module is used to calculate the three-dimensional low-frequency sound field in the island and reef terrain environment based on the cyclic horizontal recursive equation and the characterization of the action of operators in different directions.

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