CN111353251A - Fundamental frequency and higher harmonic frequency domain finite difference calculation method of nonlinear sound field - Google Patents
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Abstract
The invention provides a finite difference calculation method for fundamental frequency and higher harmonic frequency domains of a nonlinear sound field, which comprises the following steps: obtaining a nonlinear wave equation on a frequency domain; calculating a general solution of the frequency domain nonlinear wave equation; solving the fundamental frequency by adopting a linear approximate expression to obtain a corresponding relation between the second harmonic and the fundamental frequency; replacing the second harmonic of the convolution item in the fundamental frequency expression by using the obtained corresponding relation between the second harmonic and the fundamental frequency, and correcting the fundamental frequency expression; the integral term of the second harmonic general solution is written into a form convenient for numerical solution by Riemann and approximation; determining a space step length based on the wavelength of the sound wave, substituting the fundamental frequency total solution into the Riemann sum, and calculating the Riemann sum; and substituting the Riemann sum into a second harmonic general solution to solve the sound field of the second harmonic in the space. The invention considers the difference of the sound velocity and the sound attenuation of the fundamental frequency and the second harmonic in the medium, accurately describes the nonlinear propagation of each order of harmonic in the dispersion and attenuation medium, and has small calculation amount.
Description
Technical Field
The invention relates to a finite difference calculation method for fundamental frequency and higher harmonic frequency domains of a nonlinear sound field, which realizes the conversion from a time domain to a frequency domain by utilizing Fourier transform, can fully utilize the physical conditions of unequal sound velocity and sound attenuation of each order of harmonic, and truly simulates nonlinear sound propagation in a medium with strong attenuation and strong frequency dispersion.
Background
Since the 80 s of the 20 th century, the influence of the nonlinear effect is more and more emphasized by people, and especially in more than twenty years, nonlinear acoustics has penetrated into various occasions and is more and more widely applied, such as the fields of physics, biomedicine, hydroacoustics and the like. Since most of the actual propagation media are not uniform media, such as bubble-containing water media in the underwater acoustic field, contrast agents used in medicine, and the like, all have the characteristics of strong attenuation and strong dispersion, that is, the phase velocity and the attenuation coefficient of the sound wave change along with the frequency change of the sound wave, so that the harmonic phase velocity and the attenuation coefficient generated by the nonlinear effect are unequal when a line of sound waves are propagated by the medium. Typical nonlinear wave equations such as Westervelt and KZK treat the phase velocity and attenuation coefficient of sound waves with different frequencies as equal, which has a large deviation from the actual problem. How to simulate and calculate a nonlinear sound field under the conditions of unequal sound velocity and sound attenuation becomes a problem which needs to be solved urgently. At present, the mainstream numerical calculation methods mainly include the following two methods: finite difference in time domain, finite difference in frequency domain. Among them, the finite difference in time domain is to simulate the propagation of sound field by approximating the space-time partial derivative (Ibrahim M. Hallaj, FDTD simulation of fine-amplitude and temporal fields for biological ultrasound, J.Acoust. Soc.am.105(5), May 1999, L7-L12) with discrete difference, but this method cannot directly see the propagation of sound fieldIn the case of frequency domain harmonic propagation, fourier transform is required to be performed on the sound pressure value at each point in space, which greatly consumes calculation space and storage space, and cannot overcome the inequality of harmonic sound velocity and sound attenuation. Finite difference in frequency domain (s.i.aannsen, t.barkve, j.n.
The discrimination and Harmonic Generation in the Near-Field of a fine amplitude modulated beam, J.Acoust.Soc.Am,1984,75(3):749-768) uses frequency domain Fourier decomposition method to substitute the sound pressure expanded into Fourier series into KZK equation, eliminates time variable, obtains linear partial differential dual equation set satisfied by Harmonic sound pressure, and then uses Finite difference to carry out gridding dispersion solution to the dual equation set in the calculation region. While frequency-domain finite differences can conveniently handle frequency-dependent absorption effects, acoustic propagation when harmonic sound velocities are different is not given.
The invention discloses a novel frequency domain finite difference calculation method by fully considering unequal sound velocity and sound attenuation of harmonic waves. The method can accurately simulate the nonlinear sound transmission of harmonic waves in a medium with strong attenuation and strong dispersion.
Disclosure of Invention
The invention aims to provide a finite difference calculation method for fundamental frequency and higher harmonic frequency domains of a nonlinear sound field, aiming at a frequency dispersion medium of a frequency-dependent absorption effect, realizing the conversion from a time domain to a frequency domain, and conveniently processing the relationship that the attenuation and phase velocity of sound waves depend on the frequency.
The purpose of the invention is realized as follows: the method comprises the following steps:
(a) taking the space z direction as a main sound propagation direction, and carrying out time dimension and space x and y dimension three-dimensional Fourier transform on a Westervelt equation to obtain a nonlinear wave equation on a frequency domain;
(b) obtaining a general solution of the frequency domain nonlinear wave equation by using a one-dimensional Green function;
(c) weak nonlinear approximation is considered, the fundamental frequency adopts a linear approximation expression, and the corresponding relation between the second harmonic and the fundamental frequency is obtained through solving;
(d) replacing the second harmonic of the convolution item in the fundamental frequency expression by using the obtained corresponding relation between the second harmonic and the fundamental frequency, and correcting the fundamental frequency expression;
(e) the integral term of the second harmonic general solution is written into a form convenient for numerical solution by Riemann and approximation;
(f) determining a space step length based on the wavelength of the sound wave, substituting the fundamental frequency total solution into the Riemann sum, and calculating the Riemann sum;
(g) and substituting the Riemann sum into a second harmonic general solution to solve the sound field of the second harmonic in the space.
The invention also includes such structural features:
the frequency domain nonlinear wave equation in (a) is:
in the formula:
w is the angular frequency, ρ is the density of the medium, β is the nonlinear coefficient of the medium, P (r, t) is the time-domain sound pressure, P (k)x,kyZ, w) is the frequency domain complex sound pressure after three-dimensional Fourier transform of the time domain sound pressure, the convolution term and kx,kyW correlation, kx,kyWave numbers in x and y directions, w < 0
And w > 0
c (w) is the speed of sound in relation to angular frequency, α (w) is the acoustic attenuation in relation to angular frequency, in Neper/m.
The general solution in (b) is:
wherein M is β w2/ρc4(w)，
P (0) is the complex sound pressure at z ═ 0.
The corresponding relationship between the second harmonic and the fundamental frequency in (c) is:
in the formula (I), the compound is shown in the specification,
w1、w2negative angular frequencies, c, of fundamental and second harmonics, respectively1、c2The speed of sound of the fundamental and second harmonic, respectively.
The modified fundamental frequency total solution in (d) is:
in the formula:
(g) the second harmonic is given by:
and obtaining second harmonic sound pressure at the point by transmitting N fundamental frequency sound pressures before the distance z, and further obtaining a second harmonic sound field in the space.
Compared with the prior art, the invention has the beneficial effects that: 1) the model fully considers a frequency dispersion medium depending on the absorption effect of frequency, converts a time domain nonlinear wave equation into a frequency domain nonlinear wave equation, and adds sound velocity and sound attenuation corresponding to each subharmonic into an expression to obtain a sound field on each subharmonic space. 2) The model is suitable for various sound attenuation and frequency-dependent media with sound velocity, and has wide application range. 3) And each point in the space does not need to be subjected to Fourier transform, and only a small computing space and a small storage space are consumed.
Drawings
FIG. 1 is a graph of the variation of the fundamental frequency solution before and after approximation with weak non-linearity with propagation distance;
FIG. 2 is a comparison of the modified fundamental frequency solution with a linear approximation, Bessel-Fubini solution;
FIG. 3 is a comparison of the second harmonic with the solution of the Burgers equation under weak non-linear conditions;
FIG. 4 is a comparison of the second harmonic with the solution of the Burgers equation under strongly nonlinear conditions;
FIG. 5 is a plot of second harmonic versus distance in an aqueous medium containing bubbles;
fig. 6 is a flow chart of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
With reference to fig. 6, the steps of the present invention are as follows:
(a) firstly, taking the space z direction as the main sound propagation direction, and carrying out time dimension and space x and y dimension three-dimensional Fourier transform on a Westervelt equation to obtain a frequency domain nonlinear wave equation.
In the formula
In the above formula w is an angleFrequency, ρ is density of the medium, β is nonlinear coefficient of the medium, P (r, t) is time-domain sound pressure, P (k)x,kyZ, w) is the frequency domain complex sound pressure after the time domain sound pressure is subjected to three-dimensional Fourier transform, and the convolution term and k in the formula (3)x,kyW correlation, kx,kyWave numbers in x and y directions, w < 0
And w > 0
c (w) is the speed of sound in relation to angular frequency, α (w) is the acoustic attenuation in relation to angular frequency, in Neper/m.
(b) And calculating to obtain a general solution of the nonlinear wave equation of the frequency domain by using the one-dimensional Green function.
One-dimensional Green's function is expressed as
The final general solution is obtained by calculation
Wherein M is β w2/ρc4(w)，
P (0) is the complex sound pressure at z ═ 0.
(c) And (4) taking weak nonlinear approximation into consideration, solving the fundamental frequency by adopting a linear approximation expression to obtain the corresponding relation between the second harmonic and the fundamental frequency.
For weak non-linear conditions, the following approximation exists
Therefore, the formula (6) can be simplified to
Under the condition of weak non-linear approximation, the fundamental frequency adopts a linear expression
For the second harmonic, substituting the linear expression of fundamental frequency into the second, third and fifth terms of formula (6)
In the formula (I), the compound is shown in the specification,
w1、w2negative angular frequencies, c, of fundamental and second harmonics, respectively1、c2The speed of sound of the fundamental and second harmonic, respectively.
Similarly, the fourth term of the formula (6) has
The total solution of the second harmonic ((8) + (9)) is
In the case of a weakly non-linear approximation, the following equation (12) can be written again
In the formula (I), the compound is shown in the specification,
(d) and replacing the second harmonic of the convolution item in the fundamental frequency expression by using the obtained corresponding relation between the second harmonic and the fundamental frequency, and correcting the fundamental frequency expression.
For the fundamental frequency, the right side of equation (7)
In the formula (I), the compound is shown in the specification,
formula (13) is substituted for formula (14) having
In the formula (I), the compound is shown in the specification,
similarly, formula (12) is substituted for the left side of formula (7) with
Due to the establishment of equation (7), the corrected fundamental frequency is totally solved as
(e) And writing the second harmonic general solution integral term into a form convenient for numerical solution by Riemann and approximation.
The second harmonic general solution integral term is approximated by Riemann's sum
In the formula, Δ z is a space step, and N is the number of Δ z at the space propagation distance z. P (n.DELTA.z, w)1) Complex sound pressure at fundamental frequency z ═ n Δ z, F (P (n Δ z, w)1))＝P2(nΔz,w1)。
(f) And determining a space step length based on the wavelength of the sound wave, substituting the fundamental frequency total solution into the Riemann sum, and calculating the Riemann sum.
And selecting a proper space step length by taking the wavelength of the sound wave as a reference, substituting the fundamental frequency sound pressure at the corresponding distance in the formula (17) into the Riemann sum, and calculating the previous N-term Riemann sum.
(g) And substituting the Riemann sum into a second harmonic general solution to solve the sound field of the second harmonic in the space.
The second harmonic is calculated by the following equation
And solving second harmonic sound pressure at the point by N fundamental frequency sound pressures before the propagation distance z so as to solve a second harmonic sound field in the space.
Examples of the invention are given below with specific numerical values:
example parameter settings are as follows: density rho of water medium is 1000kg/m3The nonlinear coefficient β was 3.5, the excitation frequency was 100kHz, the sound pressure amplitude was 10kpa, and the relationship between the absorption and frequency of the underwater sound wave was α -2.17 × 10-7f2dB/m (f is unit of kHz), the sound velocities of the fundamental and second harmonic can be considered equal, i.e. c1＝c2＝1500m/s。
Fig. 1 shows the variation of the fundamental frequency solution with propagation distance before and after approximation with weak non-linearity.
As can be seen from fig. 1, under weak non-linear conditions, the approximation of the formula (7) is reasonable.
FIG. 2 shows the modified fundamental frequency solution in comparison to the linear approximation, Bessel-Fubini solution.
Fig. 3 shows a comparison of the second harmonic with the solution of the Burgers equation under weak non-linear conditions.
As can be seen from FIG. 3, under the condition that the nonlinear effect is much larger than the dissipative effect, the numerical solution of the model is well matched with the Bessel-Fubini solution, so that the correctness of the model is verified.
The nonlinear coefficient was changed to β -350, and the remaining parameters were unchanged.
Fig. 4 shows a comparison of the second harmonic with the solution of the Burgers equation under strongly nonlinear conditions.
As can be seen from fig. 4, this model is also applicable under strong non-linear conditions.
Replacing the pure water medium with the bubble-containing water medium, and assuming that the bubbles are singly distributed, the balance radius R of the bubbles010um, the ratio of the volume of the bubbles is 1 × 10-6Giving the excitation frequency at the bubble resonance frequency (bubble resonance frequency f)0＝3.4×105Hz) near the water medium containing bubbles, the excitation frequency f is 3.32 × 105Hz, sound pressure p is 10kPa, and non-linearity coefficient β is 1.52 × 104Fundamental acoustic velocity c11347m/s, attenuation α133.86Np/m, second harmonic speed of sound c21504m/s, attenuation α2＝0.07Np/m。
FIG. 5 shows the second harmonic in bubble-containing aqueous media as a function of distance.
In conclusion, the invention discloses a frequency domain finite difference calculation method under unequal sound velocity and sound attenuation. The method comprises the steps of firstly, taking a space z direction as a main sound propagation direction, and carrying out time dimension and space x and y dimension three-dimensional Fourier transform on a Westervelt equation to obtain a nonlinear wave equation on a frequency domain; calculating a general solution of the frequency domain nonlinear wave equation by using a one-dimensional Green function; weak nonlinear approximation is considered, the fundamental frequency adopts a linear approximation expression, and the corresponding relation between the second harmonic and the fundamental frequency is obtained through solving; replacing the second harmonic of the convolution item in the fundamental frequency expression by using the obtained corresponding relation between the second harmonic and the fundamental frequency, and correcting the fundamental frequency expression; the integral term of the second harmonic general solution is written into a form convenient for numerical solution by Riemann and approximation; determining a space step length based on the wavelength of the sound wave, substituting the fundamental frequency total solution into the Riemann sum, and calculating the Riemann sum; and substituting the Riemann sum into a second harmonic general solution to solve the sound field of the second harmonic in the space. The method considers the inequality of the sound velocity and the sound attenuation of the fundamental frequency and the second harmonic in the medium, accurately describes the nonlinear propagation of each order of harmonic in the dispersion and attenuation medium, and has small calculation amount.
Claims (6)
1. A finite difference calculation method for fundamental frequency and higher harmonic frequency domain of a nonlinear sound field is characterized in that: the method comprises the following steps:
(a) taking the space z direction as a main sound propagation direction, and carrying out time dimension and space x and y dimension three-dimensional Fourier transform on a Westervelt equation to obtain a nonlinear wave equation on a frequency domain;
(b) obtaining a general solution of the frequency domain nonlinear wave equation by using a one-dimensional Green function;
(c) weak nonlinear approximation is considered, the fundamental frequency adopts a linear approximation expression, and the corresponding relation between the second harmonic and the fundamental frequency is obtained through solving;
(d) replacing the second harmonic of the convolution item in the fundamental frequency expression by using the obtained corresponding relation between the second harmonic and the fundamental frequency, and correcting the fundamental frequency expression;
(e) the integral term of the second harmonic general solution is written into a form convenient for numerical solution by Riemann and approximation;
(f) determining a space step length based on the wavelength of the sound wave, substituting the fundamental frequency total solution into the Riemann sum, and calculating the Riemann sum;
(g) and substituting the Riemann sum into a second harmonic general solution to solve the sound field of the second harmonic in the space.
2. The method for calculating the finite difference between the fundamental frequency and the higher harmonic frequency domain of the nonlinear acoustic field according to claim 1, wherein: (a) the frequency domain nonlinear wave equation in (1) is:
in the formula:
w is the angular frequency, ρ is the density of the medium, β is the non-line of the mediumCoefficient of sex, P (r, t) is time-domain sound pressure, P (k)x,kyZ, w) is the frequency domain complex sound pressure after three-dimensional Fourier transform of the time domain sound pressure, the convolution term and kx,kyW correlation, kx,kyWave numbers in x and y directions, w < 0
And w > 0
c (w) is the speed of sound in relation to angular frequency, α (w) is the acoustic attenuation in relation to angular frequency, in Neper/m.
3. The method for calculating the finite difference between the fundamental frequency and the higher harmonic frequency domain of the nonlinear acoustic field according to claim 2, wherein: (b) the general solution in (1) is:
wherein M is β w2/ρc4(w)，
P (0) is the complex sound pressure at z ═ 0.
4. The method for calculating the finite difference between the fundamental frequency and the higher harmonic frequency domain of the nonlinear acoustic field according to claim 3, wherein: (c) the corresponding relation between the second harmonic and the fundamental frequency in (1) is as follows:
in the formula (I), the compound is shown in the specification,
w1、w2negative angular frequencies, c, of fundamental and second harmonics, respectively1、c2The speed of sound of the fundamental and second harmonic, respectively.
5. The method for calculating the finite difference between the fundamental frequency and the higher harmonic frequency domain of the nonlinear acoustic field according to claim 4, wherein: (d) the modified fundamental frequency total solution in (1) is:
in the formula:
6. the method for calculating the finite difference between the fundamental frequency and the higher harmonic frequency domain of the nonlinear acoustic field according to claim 5, wherein: (g) the second harmonic is given by:
and obtaining second harmonic sound pressure at the point by transmitting N fundamental frequency sound pressures before the distance z, and further obtaining a second harmonic sound field in the space.
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