Method for acquiring mode wave frequency dispersion curve in cylindrical surface radial layered medium

Abstract

The invention discloses a method for obtaining a mode wave frequency dispersion curve in a cylindrical surface radial layered medium, and the method comprises the steps: finding out the wave number initial values of mode waves at the last four frequency points based on a matrix condition number, and calculating the wave number precise values at the last four frequency points based on a matrix determinant Newton iteration method; according to the obtained wave numbers at the last four frequency points, extrapolating to obtain a wave number initial value of the previous frequency point, and calculating a wave number accurate value at the frequency point based on a matrix determinant Newton iteration method; and finally, drawing a mode wave frequency dispersion curve according to all the calculated frequency wave numbers. The method adopts a frequency dispersion characteristic matrix condition number peak searching mode to replace a matrix determinant zero point solving mode to determine the wave number initial value, avoids the problem of unstable root searching caused by matrix morbidity, accelerates the calculation process by applying a numerical extrapolation method, greatly reduces the number of searching times, has a complete result, and has the advantages of higher speed, higher accuracy and the like. And the efficiency is higher and the stability is higher.

Description

Technical Field

[0001] The present invention relates to the field of acoustic detection technology, and in particular to a method for obtaining a mode wave dispersion curve in a cylindrical radially layered medium. Background Art

[0002] Acoustic guided wave testing technology has been widely used in fields such as materials science, geophysical exploration, aerospace, and structural engineering. Understanding the dispersion characteristics of acoustic field modes within target waveguide structures—that is, the variation in wave velocity at different frequencies—is key to effective and accurate acoustic guided wave testing. Cylindrical structures such as pipelines, oil well casings, and submarine cables are common acoustic testing targets and are often composed of radially layered layers of different materials. Obtaining and analyzing the dispersion characteristics of modes within cylindrical radially layered media has important theoretical and practical applications for predicting material behavior, monitoring structural integrity, and optimizing engineering design. For example, in the oil industry, acoustic testing technology is often used to monitor the bonding state of cement in cased wells. The wellbore structure of a cased well is a typical cylindrical layered structure. When the cement is well bonded, the layers of the wellbore, from the inside out, are fluid-casing-cement-formation. When the cement is partially or completely missing, it is replaced by fluid, altering the wellbore structure. Obtaining and analyzing the mode wave dispersion characteristics in the acoustic field under different cement bonding states can help guide the design of acoustic detection instruments and parameter selection, data interpretation and analysis, and propose reliable cementing quality evaluation algorithms.

[0003] Mathematically speaking, obtaining the mode wave dispersion curve lies in finding the roots that satisfy the dispersion equation, namely the real frequency and complex wavenumber, or in other words, finding the zeros of the dispersion characteristic function. Currently, related methods mainly focus on the bisection method and the two-dimensional search method. The bisection method can only obtain the dispersion calculation of the propagation mode wave, but cannot obtain the leakage mode wave dispersion curve. The two-dimensional search method searches for the complex wavenumber point on the two-dimensional grid of real and imaginary wavenumbers at a fixed frequency that makes the dispersion characteristic function reach a minimum. Although it can calculate the propagation mode and leakage mode dispersion simultaneously, the frequency characteristic function itself is very sensitive to changes in the frequency and wavenumber values, requiring a very small search step size to obtain a complete result. Therefore, this method is inefficient and time-consuming. Summary of the Invention

[0004] The present invention aims to solve the problems of incomplete results and low efficiency in existing methods for obtaining mode wave dispersion curves. It provides a method for obtaining mode wave dispersion curves in cylindrical radially layered media, which can quickly and accurately calculate all mode wave dispersion curves in a cylindrical layered structure, thereby guiding the smooth development of related acoustic detection technologies.

[0005] To solve the above problems, the present invention is achieved through the following technical solutions:

[0006] A method for obtaining a mode wave dispersion curve in a cylindrical radially stratified medium comprises the following steps:

[0007] Step 1: Obtain the target cylindrical radially layered medium model parameters;

[0008] Step 2: Based on the parameters of the target cylindrical radially layered medium model, the dispersion characteristic function matrix and its determinant are derived based on the elastic wave theory in solids and the acoustic wave theory in fluids, while utilizing the boundary conditions of each layer; the dispersion characteristic function matrix and its determinant are both binary functions of frequency and wave number;

[0009] Step 3: Set the frequency sequence according to the frequency range to be solved, and set the corresponding wave number sequence according to the frequency sequence;

[0010] Step 4: First, select the first or last P consecutive frequencies in the frequency sequence; then, based on the matrix condition number calculation formula of the dispersion characteristic function matrix, calculate the matrix condition number corresponding to each wavenumber of the wavenumber sequence at each frequency; then, construct a wavenumber-matrix condition number curve at each frequency, and regard each peak of the curve as a mode wave, and the wavenumber corresponding to each peak as the initial wavenumber value of the mode wave; where P is a set positive integer greater than 1;

[0011] Step 5: For each mode wave of each frequency, the initial value of the wave number of the current mode wave of the current frequency is used as the initial iterative wave number, and the final iterative wave number is obtained by the Newton iteration method based on the iterative formula of the dispersion characteristic function matrix determinant, and the final iterative wave number is used as the precise value of the wave number of the current mode wave of the current frequency;

[0012] Step 6: Based on the exact wave number values ​​of all frequencies currently obtained under the same mode wave, a polynomial extrapolation method is used to obtain the initial wave number value of the current mode wave at a frequency adjacent to all the frequencies currently obtained in the frequency sequence;

[0013] Step 7: For each mode wave of an adjacent frequency, the initial value of the wave number of the current mode wave of the current frequency is used as the initial iterative wave number, and the final iterative wave number is obtained by the Newton iteration method based on the iterative formula of the dispersion characteristic function matrix determinant, and the final iterative wave number is used as the precise value of the wave number of the current mode wave of the current frequency;

[0014] Step 8: Determine whether the adjacent frequency is the last or the first frequency in the frequency sequence: if so, use the precise wavenumber values ​​of all frequencies currently obtained in different modes to draw the mode dispersion curve; otherwise, return to step 6.

[0015] In the above step 1, the parameters of the target cylindrical radial layered medium model include the density of each layer of medium, the longitudinal wave velocity of each layer of medium, the shear wave velocity of each layer of medium, the interlayer radius and the sound source type.

[0016] In the above step 3, a frequency sequence is established by setting the minimum frequency, maximum frequency and frequency interval; a wave number sequence is established by setting the minimum wave number, maximum wave number and number of wave numbers.

[0017] In the above step 4, the value range of P is between 2 and 5.

[0018] In the above step 4, regarding the frequency ω i and wave number k j The matrix condition number Q ij The calculation formula is:

[0019] Q ij =||D ij ||1×||D ij -1 ||1

[0020] Where D ij Represents the frequency ω i and wave number k j The dispersion characteristic function matrix, D ij -1 Represents the frequency ω i and wave number k j The inverse matrix of the dispersion characteristic function matrix; i = 1, 2, …, N, N is the number of frequencies of the frequency sequence, j = 1, 2, …, M, M is the number of wave numbers of the wave number sequence; ||*||1 represents the 1 norm.

[0021] In the above steps 5 and 7, regarding the frequency ω i and wave number k j The iterative formula of the determinant of the dispersion characteristic function matrix is:

[0022] k t+1 =k t –|D ij-t | / |D ij-t |'

[0023] Where k t+1 represents the t+1th wave number, k t represents the tth wave number; |D ij-t | denotes the frequency ω i and wave number k j-t The determinant of the dispersion characteristic function matrix, |D ij-t |' represents the frequency ω i and wave number k j-tThe derivative of the determinant of the dispersion characteristic function matrix; i = 1, 2, ..., N, N is the number of frequencies in the frequency sequence, j = 1, 2, ..., M, M is the number of waves in the wave number sequence.

[0024] Compared with the existing technology, the present invention adopts the method of finding the peak of the dispersion characteristic matrix condition number instead of finding the zero point of the matrix determinant to determine the initial value of the wavenumber, avoiding the problem of unstable root finding caused by matrix pathology. At the same time, the numerical extrapolation method is applied to accelerate the calculation process, greatly reducing the number of searches, while achieving complete results, and having the characteristics of faster speed, higher efficiency and greater stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 is a flow chart of a method for obtaining a dispersion curve of a mode wave in a cylindrical radially layered medium;

[0026] Figure 2 Schematic diagram of a cylindrical radial multi-layer acoustic waveguide structure;

[0027] Figure 3 is the wave number-matrix condition number curve at a certain frequency.

[0028] Figure 4 FIG. 1 is a phase velocity dispersion curve of guided waves in a cylindrical structure in an embodiment obtained by using the present invention. DETAILED DESCRIPTION

[0029] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to specific examples and the accompanying drawings.

[0030] A method for obtaining the dispersion curve of mode waves in cylindrical radially layered media, such as Figure 1 As shown, the steps include:

[0031] Step 1: Obtain the target cylindrical radially stratified medium model parameters.

[0032] The parameters of the target cylindrical radial layered medium model include the density ρ of each layer of medium, the longitudinal wave velocity V of each layer of medium p , the shear wave velocity V of each layer of medium s , interlayer radius r and sound source type n. Wherein sound source type n = 0, 1, 2 represent monopole, dipole and quadrupole sound sources respectively.

[0033] like Figure 2 The cylindrical radial multilayer acoustic waveguide structure shown in the figure has the following media: fluid, steel pipe, fluid, steel pipe, fluid and stratum from the inner to the outer layer. The density of each layer is ρ (kg / m 3 )=[1000,7800,1000,7800,1000,2300], longitudinal wave velocity V p(m / s) = [1500, 5860, 1500, 5860, 1500, 4500], shear wave velocity V s (m / s) = [none, 3130, none, 3130, none, 2500], interlayer radius r (cm) = [6, 7, 10, 11, 14, ∞] and sound source type n = 0.

[0034] Step 2: Based on the input target cylindrical radial layered medium model parameters, the dispersion characteristic function matrix D and its determinant |D| are derived based on the elastic wave theory in solids and the acoustic wave theory in fluids while utilizing the boundary conditions of each layer.

[0035] The dispersion characteristic function matrix D and its determinant |D| are both binary functions of the real frequency ω and the complex wavenumber k. The derivation process has been detailed in numerous monographs and references, such as "Quantitative Well Logging Acoustics" and "Acoustic Fields and Waves in Boreholes." Here, we provide only a brief overview.

[0036] According to the foundation of elastic wave dynamics, the displacement potentials Φ, X, and Γ of longitudinal waves, SV shear waves, and SH shear waves in solid dielectric layers satisfy the equations:

[0037]

[0038] Among them, k p =ω / V p and k s =ω / V s Respectively represent the longitudinal and transverse wave numbers of the solid medium, V p and V s are the longitudinal and transverse wave velocities of the medium, respectively. According to the separation of variables method in the cylindrical coordinate system, the general solution form of the above displacement potential functions in the frequency-wavenumber domain satisfies the equation:

[0039]

[0040] Where n represents the type of sound source, n = 0, 1, 2 are monopole, dipole, and quadrupole sound sources respectively. l represents the number of dielectric layers. and are the radial wave numbers of longitudinal and transverse waves in the lth layer of medium, θ is the azimuth angle, and φ is the reference angle. n and K n are the first and second modified Bessel functions, respectively. AF are the reflection and transmission coefficients to be solved. The displacement vectors (u, v, w) in the wavefield represent the radial, circumferential, and axial displacements, respectively, and can be derived from the displacement potential function using the following equations:

[0041]

[0042] Stress vector in the wave field (σ rr ,σrθ ,σ rz ) represent radial, circumferential and axial stresses respectively, which can be derived from the displacement potential function by the following equation.

[0043]

[0044] in, and Represents the Lame coefficient of the solid medium.

[0045] Similarly, based on the theory of acoustic wave equations in fluids, the acoustic pressure and displacement vector in the fluid medium can also be derived.

[0046] Since both ends of the radial interface between different dielectric layers must meet corresponding boundary conditions, such as radial displacement continuity and stress continuity on both sides of the fluid-solid interface, and continuity of displacement vectors and stress vectors on both sides of the solid-solid interface, relevant equations can be constructed based on the continuity conditions of the acoustic field characteristics of the interface at different radial positions. By combining the continuity equations at all interfaces in the entire model, the following equations can be obtained:

[0047] D(k,w)X=0

[0048] Where D is the dispersion characteristic function matrix, whose dimension is related to the number of radial layers of the model; |D| is the determinant of D, and the result is a complex value.

[0049] Step 3: Set the frequency sequence according to the frequency range to be solved, and set the corresponding wave number sequence according to the frequency sequence.

[0050] Frequency sequence {ω i}Through artificial setting,ω i Indicates the i-th frequency of the frequency sequence, i = 1, 2, ..., N, N represents the number of frequencies in the frequency sequence. In the present invention, a frequency sequence is established by setting the minimum frequency, the maximum frequency and the frequency interval, such as by setting the minimum frequency ω1 = 50 Hz, the maximum frequency ω N =20000 Hz and frequency interval Δω=50 Hz, the set frequency sequence ω1=50 Hz, ω2=100 Hz, ..., ω N-1 =19950Hz,ω N =20000Hz.

[0051] Wave number sequence {k j}Through artificial setting, k j Indicates the jth wave number of the wave number sequence, j = 1, 2, ..., M, M represents the number of wave numbers in the wave number sequence. In the present invention, the wave number sequence is established by setting the minimum wave number, the maximum wave number and the number of wave numbers, such as setting the minimum wave number k min =0.1ks , maximum wave number k max =3k f The sum of the wave numbers M = 5120. Among them, the formation shear wave number k s =ω i / v s , fluid wave number k f =ω i / v f ,ω i Equal to the i-th element in the frequency sequence, v s and v f represent the formation shear wave velocity and fluid velocity, respectively.

[0052] Step 4: Find the initial value of the wave number of the mode wave of the selected frequency based on the matrix condition number.

[0053] Step 4.1: Select the first or last P consecutive frequencies in the frequency sequence, and define these P frequencies as selected frequencies.

[0054] The selected frequency number P is a positive integer greater than 1 that is artificially set. Since the frequency number P is related to the precise value and convergence speed of the wave number calculated using the Newton iteration algorithm, the smaller the value of the frequency number P, the fewer the number of matrix inversion operations in the subsequent matrix condition number calculation, but the initial wave number value obtained by interpolation is far from the accurate value, and the convergence speed and accuracy of the subsequent Newton iteration method are lower; conversely, the larger the value of the frequency number P, the more the number of subsequent inverse matrix operations, but the initial wave number value obtained by interpolation is closer to the accurate value, and the convergence speed and accuracy of the subsequent Newton iteration method are higher. In order to balance the amount of calculation and accuracy, the value range of P is preferably between 2 and 5. In the embodiment, the last 4 frequencies (P=4) of the frequency sequence are selected as the selected frequencies.

[0055] Step 4.2: Based on the matrix condition number calculation formula of the dispersion characteristic function matrix, calculate the matrix condition number corresponding to each wavenumber of the wavenumber sequence at each selected frequency.

[0056] Since the dispersion characteristic function matrix D is a binary function of real frequency ω and complex wave number k, when calculating the frequency ω i and wave number k j The matrix condition number Q ij When the frequency ω i and wave number k j The dispersion characteristic function matrix D ij to achieve.

[0057] About frequency ω i and wave number k j The matrix condition number Q ij The calculation formula is:

[0058] Q ij =||D ij ||1×||D ij -1 ||1

[0059] Where D ij Represents the frequency ω i and wave number k j The dispersion characteristic function matrix, D ij -1 Represents the frequency ω i and wave number k j The inverse matrix of the dispersion characteristic function matrix; i = 1, 2, …, N, N is the number of frequencies of the frequency sequence, j = 1, 2, …, M, M is the number of wave numbers of the wave number sequence; ||*||1 represents the 1 norm.

[0060] Step 4.3: Construct a wavenumber-matrix condition number curve at each frequency, and regard each peak of the curve as a mode wave, and the wavenumber corresponding to each peak as the initial wavenumber value of the mode wave.

[0061] In the actual execution of the algorithm, Figure 3 The wave number-matrix condition number curve at each frequency shown does not need to be actually constructed (drawn), but is realized by directly finding the peak.

[0062] Since the selected P frequencies are P consecutive frequencies in the frequency sequence, the frequency values ​​of these P frequencies are not much different, that is, the wave number-matrix condition number curves are not much different. Therefore, the number of peaks in the wave number-matrix condition number curves of these P frequencies is the same, that is, the number of mode waves is the same. At this time, the first peak in the wave number-matrix condition number curve of these P frequencies is called the first mode wave, the second peak in the wave number-matrix condition number curve of these P frequencies is called the second mode wave, the third peak in the wave number-matrix condition number curve of these P frequencies is called the third mode wave, and so on.

[0063] Step 5. For each mode wave of each frequency, the initial value of the wave number of the current mode wave at the current frequency is used as the initial iterative wave number, and the final iterative wave number is obtained by the Newton iteration method based on the iterative formula of the dispersion characteristic function matrix determinant, and the final iterative wave number is used as the exact value of the wave number of the current mode wave at the current frequency.

[0064] The Newton iteration method is a method for approximately solving equations in the real and complex domains. It is a numerical method based on the Taylor series expansion and is used to solve the roots of the nonlinear equation f(x) = 0. Its core idea is to gradually approach the true root by continuously obtaining the tangent line of the function and using the intersection of the tangent line and the x-axis as the new approximate solution.

[0065] For frequency ω i , based on the initial value k of the wave number of each mode wave j-0 The wave number k of each mode wave is calculated using the Newton iteration method. j-ex The process is briefly described as follows:

[0066] Step 5.1, set the error range eps, in this embodiment eps = 0.01; set the initial iteration number t = 0, the initial wave number k0 = k j-0 .

[0067] Step 5.2: Calculate the t+1th wave number k t+1 :

[0068] k t+1 =k t -|D ij-t | / |D ij-t |'

[0069] Where k t represents the tth wave number; |D ij-t | denotes the frequency ω i and wave number k j-t The determinant of the dispersion characteristic function matrix, |D ij-t |' represents the frequency ω i and wave number k j-t The derivative of the determinant of the dispersion characteristic function matrix; i = 1, 2, ..., N, N is the number of frequencies in the frequency sequence, j = 1, 2, ..., M, M is the number of waves in the wave number sequence.

[0070] Step 5.3, when |k t+1 -k t When |≤eps, stop the iteration, and it is considered that k t+1 is the final wave number, that is, the exact value of the wave number k j-ex .

[0071] Step 6: Based on the exact wavenumber values ​​of all currently obtained frequencies under the same mode wave, a polynomial extrapolation method is used to obtain the initial wavenumber value of the current mode wave at an adjacent frequency of all currently obtained frequencies in the frequency sequence.

[0072] Polynomial extrapolation is a numerical method that constructs a polynomial function based on known data points and uses this function to make predictions outside the data range. The core idea is to approximate the data trend through polynomial fitting and extend it into unknown regions. This implementation uses the built-in interp1 function in Matlab software to implement polynomial extrapolation.

[0073] The adjacent frequency of all currently obtained frequencies is the frequency in the frequency sequence that is closest to all currently obtained frequencies. For example, during the first cycle: if P frequencies are selected from the front of the frequency sequence, then the adjacent frequency is the P+1th frequency in the frequency sequence; if P frequencies are selected from the back of the frequency sequence, then the adjacent frequency is the NP-1th frequency in the frequency sequence; during the first cycle: if P frequencies are selected from the front of the frequency sequence, then the adjacent frequency is the P+2th frequency in the frequency sequence; if P frequencies are selected from the back of the frequency sequence, then the adjacent frequency is the NP-2th frequency in the frequency sequence; and so on.

[0074] Step 7. For each mode wave of an adjacent frequency, the initial value of the wave number of the current mode wave of the current frequency is used as the initial iterative wave number, and the final iterative wave number is obtained by the Newton iteration method based on the iterative formula of the dispersion characteristic function matrix determinant, and the final iterative wave number is used as the exact value of the wave number of the current mode wave of the current frequency.

[0075] After obtaining the initial wavenumber values ​​of each mode wave at an adjacent frequency of all the currently obtained frequencies in the frequency sequence, the same method as step 5 can be used to obtain the precise wavenumber values ​​of the adjacent frequency under different mode waves.

[0076] Step 8: Determine whether the adjacent frequency is the last or the first frequency in the frequency sequence: if so, use the precise wavenumber values ​​of all frequencies currently obtained in different modes to draw the mode dispersion curve; otherwise, return to step 6.

[0077] When using the precise wavenumber values ​​of all frequencies currently obtained in different modes to draw the mode dispersion curve, the frequency-wavenumber dispersion curve of each mode can be directly drawn by sorting out the frequency-wavenumber points corresponding to each mode wave, or the frequency-phase velocity dispersion curve can be drawn by the calculation formula of the phase velocity, that is, v = ω / real(k), where v represents the phase velocity, ω represents the frequency, k represents the wavenumber, and real(k) represents the real part of the wavenumber, as shown in Figure 4 shown.

[0078] It should be noted that although the embodiments of the present invention described above are illustrative, they are not intended to limit the present invention. Therefore, the present invention is not limited to the above-mentioned specific embodiments. Without departing from the principles of the present invention, any other embodiments obtained by those skilled in the art under the guidance of the present invention are deemed to be within the protection of the present invention.

Claims

1. A method for obtaining a mode wave dispersion curve in a cylindrical radially layered medium, characterized by: The steps are as follows: Step 1: Obtain the target cylindrical radially layered medium model parameters; Step 2: Based on the parameters of the target cylindrical radially layered medium model, the dispersion characteristic function matrix and its determinant are derived based on the elastic wave theory in solids and the acoustic wave theory in fluids, while utilizing the boundary conditions of each layer; the dispersion characteristic function matrix and its determinant are both binary functions of frequency and wave number; Step 3: Set the frequency sequence according to the frequency range to be solved, and set the corresponding wave number sequence according to the frequency sequence; Step 4: First, select the first or last P consecutive frequencies in the frequency sequence; then, based on the matrix condition number calculation formula of the dispersion characteristic function matrix, calculate the matrix condition number corresponding to each wavenumber of the wavenumber sequence at each frequency; then, construct a wavenumber-matrix condition number curve at each frequency, and regard each peak of the curve as a mode wave, and the wavenumber corresponding to each peak as the initial wavenumber value of the mode wave; where P is a set positive integer greater than 1; Step 5: For each mode wave of each frequency, the initial value of the wave number of the current mode wave of the current frequency is used as the initial iterative wave number, and the final iterative wave number is obtained by the Newton iteration method based on the iterative formula of the dispersion characteristic function matrix determinant, and the final iterative wave number is used as the precise value of the wave number of the current mode wave of the current frequency; Step 6: Based on the exact wave number values ​​of all frequencies currently obtained under the same mode wave, a polynomial extrapolation method is used to obtain the initial wave number value of the current mode wave at a frequency adjacent to all the frequencies currently obtained in the frequency sequence; Step 7: For each mode wave of an adjacent frequency, the initial value of the wave number of the current mode wave of the current frequency is used as the initial iterative wave number, and the final iterative wave number is obtained by the Newton iteration method based on the iterative formula of the dispersion characteristic function matrix determinant, and the final iterative wave number is used as the precise value of the wave number of the current mode wave of the current frequency; Step 8: Determine whether the adjacent frequency is the last or the first frequency in the frequency sequence: if so, use the precise wavenumber values ​​of all frequencies currently obtained in different modes to draw the mode dispersion curve; otherwise, return to step 6.

2. The method for obtaining a mode wave dispersion curve in a cylindrical radially stratified medium according to claim 1, wherein: In step 1, the parameters of the target cylindrical radial layered medium model include the density of each layer of medium, the longitudinal wave velocity of each layer of medium, the shear wave velocity of each layer of medium, the interlayer radius and the sound source type.

3. The method for obtaining a mode wave dispersion curve in a cylindrical radially stratified medium according to claim 1, wherein: In step 3, a frequency sequence is established by setting the minimum frequency, maximum frequency, and frequency interval; a wave number sequence is established by setting the minimum wave number, maximum wave number, and number of wave numbers.

4. The method for obtaining a mode wave dispersion curve in a cylindrical radially stratified medium according to claim 1, wherein: In step 4, the value of P ranges from 2 to 5.

5. The method for obtaining a mode wave dispersion curve in a cylindrical radially layered medium according to claim 1, wherein the step 4, about the frequency ω i and wave number k j The matrix condition number Q ij The calculation formula is: Q ij =||D ij ||1×||D ij -1 ||1 Where D ij Represents the frequency ω i and wave number k j The dispersion characteristic function matrix, D ij -1 Represents the frequency ω i and wave number k j The inverse matrix of the dispersion characteristic function matrix; ||*||1 represents the 1 norm.

6. The method for obtaining a mode wave dispersion curve in a cylindrical radially stratified medium according to claim 1, wherein in steps 5 and 7, the frequency ω i and wave number k j The iterative formula of the determinant of the dispersion characteristic function matrix is: k t+1 =k t –|D ij-t | / |D ij-t |’ Where k t+1 represents the t+1th wave number, k t represents the tth wave number; |D ij-t | denotes the frequency ω i and wave number k j-t The determinant of the dispersion characteristic function matrix, |D ij-t |' represents the frequency ω i and wave number k j-t The derivative of the determinant of the dispersion characteristic function matrix.

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