CN102539541B

CN102539541B - Method for non-contact wave velocity extraction of Rayleigh wave of anisotropic blocky material

Info

Publication number: CN102539541B
Application number: CN 201110427881
Authority: CN
Inventors: 何存富; 吕炎; 宋国荣; 柳艳丽; 高忠阳; 吴斌
Original assignee: Beijing University of Technology
Current assignee: Beijing University of Technology
Priority date: 2011-12-19
Filing date: 2011-12-19
Publication date: 2013-11-06
Anticipated expiration: 2031-12-19
Also published as: CN102539541A

Abstract

The invention discloses a method for the non-contact wave velocity extraction of a Rayleigh wave of an anisotropic blocky material, and the method belongs to the technical field of nondestructive examination. In the nondestructive examination for mainly measuring the wave velocity of an acoustic wave, a V(z) curve formed by the interference of a leaky surface wave and a directly reflected wave, namely a longitudinal wave, comprises much information at the microstructure aspect of the material; the method is based on a defocusing measurement system; a wide-frequency pulse is utilized as an excitation source; an ultrasonic wave comprising a plurality of frequency components is received; and the V(z) curve of the material and an oscillating period thereof are obtained through an improved two-dimensional Fourier transform technique, so as to achieve the extraction of the wave velocity of the Lamb wave of the blocky material. By using the method, the wave velocities of the Rayleigh waves of different materials can be extracted; the wave velocity of the Rayleigh wave can be extracted in a wide frequency scope; a single-frequency point-by-point way is replaced; the wave velocities of the Rayleigh waves in different frequency ranges can be extracted; an averaged value is selected as the wave velocity of the Rayleigh wave of the material; and the random error caused by an accidental factor in a single-frequency extraction process is avoided.

Description

A method of Rayleigh wave non-contact wave velocity extraction for isotropic bulk materials

技术领域 technical field

本发明属于无损检测领域，具体涉及一种对各向同性块体材料瑞利波的波速提取方法。 The invention belongs to the field of non-destructive testing, and in particular relates to a wave velocity extraction method for Rayleigh waves of isotropic block materials. the

背景技术 Background technique

随着材料科学的不断向前发展，各种功能型材料不断涌现，但受到制备工艺的影响，很多新型材料的几何尺寸非常有限，例如金属玻璃、块体纳米材料等。因此，采用拉伸等破坏性传统力学性能测试的方法将无法满足新型材料的需求。在以测量声波波速为主的非破坏性检测中，由漏表面波和直接反射波的干涉所形成的V(z)曲线包含材料微结构方面的许多信息，以超声显微镜作为波速测量工具，可以应用于检测晶体结构、弹性模量、残余应力、内部缺陷等材料机械性质，使得超声显微镜在材料力学特性测试和定量无损检测等方面获得了越来越广泛的应用。 With the continuous development of material science, various functional materials are emerging, but affected by the preparation process, the geometric size of many new materials is very limited, such as metallic glass, bulk nanomaterials, etc. Therefore, the method of testing destructive traditional mechanical properties such as tensile will not be able to meet the needs of new materials. In the non-destructive testing mainly measuring the acoustic wave velocity, the V(z) curve formed by the interference of the leakage surface wave and the direct reflection wave contains a lot of information on the microstructure of the material. Using an ultrasonic microscope as a wave velocity measurement tool, it can Applied to the detection of mechanical properties of materials such as crystal structure, elastic modulus, residual stress, and internal defects, ultrasonic microscopy has been more and more widely used in the testing of mechanical properties of materials and quantitative non-destructive testing. the

利用超声波对材料弹性性质进行测量是无损检测领域很有前景的测量方法之一。在各向同性均质材料中，表面波(Surface acoustic wave，SAW)又称为瑞利波(Rayleigh SAW)，其波动行为包含了大量材料特性的信息，因此，通过测量块体材料的表面波波速与纵波波速即可反演出材料的弹性性质。 Using ultrasound to measure the elastic properties of materials is one of the promising measurement methods in the field of non-destructive testing. In isotropic homogeneous materials, surface acoustic wave (SAW) is also called Rayleigh wave (Rayleigh SAW), and its wave behavior contains a large amount of information on material properties. Therefore, by measuring the surface acoustic wave of bulk materials The wave velocity and the longitudinal wave velocity can be used to invert the elastic properties of the material. the

为了达到上述目的，波速的精确提取显得尤为必要。目前对于瑞利波波速提取大多数采用单频逐点提取的方式，通过测量V(z)曲线中的振荡周期Vz来确定表面波的波速，但其缺点是单频波速提取并不适用于宽频脉冲信号的测量。因此，需要开发出一套基于宽频脉冲信号的表面波波速提取方法。 In order to achieve the above purpose, the accurate extraction of wave velocity is particularly necessary. At present, most of the Rayleigh wave velocity extraction methods are single-frequency point-by-point extraction, and the wave velocity of the surface wave is determined by measuring the oscillation period Vz in the V(z) curve, but its disadvantage is that the single-frequency wave velocity extraction is not suitable for broadband Measurement of pulsed signals. Therefore, it is necessary to develop a set of surface wave velocity extraction methods based on broadband pulse signals. the

发明内容 Contents of the invention

本发明的目的是为了解决各向同性块体材料瑞利波宽频连续波速提取的问题，提出一种先进的材料波速提取方法。 The purpose of the present invention is to solve the problem of extracting Rayleigh wave broadband continuous wave velocity of isotropic bulk material, and propose an advanced material wave velocity extraction method. the

步骤1)：确立波速提取的公式。 Step 1): Establish the formula for wave velocity extraction. the

这里需要说明的是，由于水的负载效应，漏表面波与表面波的波速并不完全一致，但由于被测材料的密度远大于水的密度，两者之间的差异是可以忽略的。之后的阐述中将不再区分表面波和漏表面波。在波速提取的过程中，依据V(z)曲线理论，可根据如下公式进行波速的计算： What needs to be explained here is that due to the loading effect of water, the wave speeds of leaky surface waves and surface waves are not exactly the same, but because the density of the measured material is much greater than that of water, the difference between the two is negligible. In the following elaboration, no distinction will be made between surface waves and leaky surface waves. In the process of wave velocity extraction, according to the V(z) curve theory, the wave velocity can be calculated according to the following formula:

${v v}_{SAW SAW} = = {v v}_{w w} \cdot \cdot {[[11 - - {((11 - - \frac{{v v}_{w w}}{22 \cdot \cdot f f \cdot \cdot Vz vz}))}^{22}]]}^{- - 11 / / 22}$

其中：Vz为V(z)曲线振荡周期，v_w为水中的超声波波速，f为换能器的激励频率，v_SAW为材料的表面波波速。测量被测材料的V(z)曲线振荡周期是波速提取的关键。 Among them: Vz is the oscillation period of the V(z) curve, v _w is the ultrasonic wave velocity in water, f is the excitation frequency of the transducer, and v _SAW is the surface wave velocity of the material. Measuring the oscillation period of the V(z) curve of the tested material is the key to extracting the wave velocity.

步骤2)：搭建测试系统。 Step 2): Build a test system. the

为了方便散焦步进测量，搭建了一套进行散焦步进测量的测试系统，如图1所示。该测试系统主要包括：试样1、水槽与水2、换能器3、移动平台4、脉冲激励/接收仪5、示波器6、GPIB总线7、PXI总控制系统8、移动伺服马达9、旋转轴10。其中，在移动平台4下面安装换能器3，换能器3与脉冲激励/接收仪5相连，脉冲激励/接收仪5与示波器6相连，示波器6通过GPIB总线7与PXI总控制系统8相连，PXI总控制系统8与移动伺服马达9相连，同时PXI总控制系统8与旋转轴10相连。 In order to facilitate the step-by-step defocus measurement, a test system for step-by-step defocus measurement is built, as shown in Figure 1. The test system mainly includes: sample 1, water tank and water 2, transducer 3, mobile platform 4, pulse excitation/receiver 5, oscilloscope 6, GPIB bus 7, PXI total control system 8, mobile servo motor 9, rotary axis 10. Among them, the transducer 3 is installed under the mobile platform 4, the transducer 3 is connected to the pulse excitation/reception instrument 5, the pulse excitation/reception instrument 5 is connected to the oscilloscope 6, and the oscilloscope 6 is connected to the PXI general control system 8 through the GPIB bus 7 , The PXI total control system 8 is connected with the moving servo motor 9 , and the PXI total control system 8 is connected with the rotating shaft 10 at the same time. the

步骤3：聚焦面数据采集。 Step 3: Focus plane data collection. the

将块体被测试样置于换能器的聚焦面，脉冲激励/接收仪5在发出一个带宽为10-200MHz的脉冲后转换为接收状态，当接收到反射信号后，将信号传输进示波器6，示波器的采样频率为f_S，f_S为0.5-5GHz，采样点数为N_s，N_s的取值范围为10000-100000点。经过示波器的低通滤波后，通过GPIB总线7存储进PXI总控制系统8。 Place the block to be tested on the focal plane of the transducer, the pulse excitation/receiver 5 switches to the receiving state after sending out a pulse with a bandwidth of 10-200MHz, and transmits the signal into the oscilloscope 6 after receiving the reflected signal , the sampling frequency of the oscilloscope is f _S , f _S is 0.5-5GHz, the number of sampling points is N _s , and the value range of N _s is 10000-100000 points. After being low-pass filtered by the oscilloscope, it is stored into the PXI general control system 8 through the GPIB bus 7 .

步骤4)：散焦测量。 Step 4): Defocus measurement. the

将换能器垂直向下移动一个距离Vz₀，Vz₀的取值范围为1-50μm，待移动完成后进行数据采集，采样频率为f_S，采样点数为N_s。采集结束后再将换能器垂直向下移动Vz₀进行数据采集，如此循环往复，共移动距离z，z的取值范围为2-20mm，因此将得到M组电压数据，M由z与Vz₀共同决定，为40-20000组。 Move the transducer vertically downward for a distance Vz ₀ , the value range of Vz ₀ is 1-50 μm, and collect data after the movement is completed, the sampling frequency is f _S , and the number of sampling points is N _s . After the collection is completed, move the transducer vertically downward to Vz ₀ for data collection. In this way, the total moving distance is z, and the value range of z is 2-20mm. Therefore, M sets of voltage data will be obtained, and M is composed of z and Vz ₀ co-decision, for 40-20000 groups.

步骤5)：时域傅里叶变换。 Step 5): Time domain Fourier transform. the

将所有数据沿散焦距离排列好，对测得的数据进行时域傅里叶变换： Arrange all the data along the defocus distance, and perform time-domain Fourier transform on the measured data:

${A A}_{i i} [[k k]] = = {Σ Σ}_{n no = = 00}^{{N N}_{s the s} - - 11} {x x}_{i i} [[n no]] {e e}^{- - j j 22 πnk πnk / / {N N}_{s the s}}$

其中：A_i为时域傅里叶变换后的频谱值，x_i代表一组电压数据，i＝0，1，2L M-1，k＝0，1，2L N_s-1，j代表虚部。 Among them: A _i is the frequency spectrum value after time-domain Fourier transform, x _i represents a set of voltage data, i=0, 1, 2L M-1, k=0, 1, 2L N _s -1, j represents virtual department.

步骤6)：空间傅里叶变换。 Step 6): Spatial Fourier transform. the

为了得到精确的振荡周期Vz，需要对时域傅里叶变换的结果再进行沿散焦距离方向的空间傅里叶变换，将散焦距离z变换至z^-1域： In order to obtain an accurate oscillation period Vz, it is necessary to perform a spatial Fourier transform on the result of the time-domain Fourier transform along the direction of the defocus distance, and transform the defocus distance z to the z ^-1 domain:

${B B}_{i i} [[k k]] = = {Σ Σ}_{m m = = 00}^{M m - - 11} {A A}_{m m} [[k k]] {e e}^{- - j j 22 πmi πmi / / M m}$

其中：B_i为空间傅里叶变换后的频谱值，A_m代表沿散焦方向的时域傅里叶变换的频谱值，i＝0，1，2L M-1，k＝0，1，2L N_s-1，j代表虚部。沿z^-1域的曲线峰值即为振荡周期Vz的倒数。 Wherein: B _i is the spectral value after the spatial Fourier transform, A _m represents the spectral value of the time-domain Fourier transform along the defocus direction, i=0,1,2L M-1, k=0,1, 2L N _s -1, j represents the imaginary part. The peak value of the curve along the z ^-1 domain is the reciprocal of the oscillation period Vz.

步骤7)：模态追踪。 Step 7): Modality Tracking. the

对1-100MHz范围内的峰值进行追踪，即可找出该频率段连续的振荡周期Vz值。 By tracking the peak value within the range of 1-100MHz, the continuous oscillation period Vz value of this frequency segment can be found. the

步骤8)：波速提取。 Step 8): wave velocity extraction. the

若使用的耦合液为水，则将水中的超声波波速v_W，每一个峰值对应的频率f与振荡周期Vz代入步骤1)中所示公式，即可得到该频率段内连续的表面波波速v_SAW。 If the coupling liquid used is water, then substituting the ultrasonic wave velocity v _W in water, the frequency f corresponding to each peak value, and the oscillation period Vz into the formula shown in step 1), the continuous surface wave velocity v in this frequency range can be obtained _saw .

本发明具有以下优点：1)可对不同材料的瑞利波波速进行提取；2)可在宽频范围内对瑞利波波速进行提取，取代单频逐点的方式；3)可对不同频率段内的瑞利波波速进行提取，选择平均后的值作为材料的瑞利波波速，避免了单频提取时由于偶然因素造成的随机误差。 The invention has the following advantages: 1) the Rayleigh wave velocity of different materials can be extracted; 2) the Rayleigh wave velocity can be extracted in a wide frequency range, replacing the single-frequency point-by-point method; 3) different frequency bands can be extracted The Rayleigh wave velocity in the material is extracted, and the average value is selected as the Rayleigh wave velocity of the material, which avoids random errors caused by accidental factors during single-frequency extraction. the

附图说明 Description of drawings

图1：散焦测量系统示意图； Figure 1: Schematic diagram of the defocus measurement system;

图2：表面波传播示意图； Figure 2: Schematic diagram of surface wave propagation;

图3：聚焦面时域波形图； Figure 3: Time-domain waveform diagram of the focal plane;

图4：不同散焦距离下的时域波形图； Figure 4: Time-domain waveforms at different defocus distances;

图5：时域傅里叶变换图； Figure 5: Time-domain Fourier transform diagram;

图6：7.5MHz频率下V(z)振荡曲线图； Figure 6: V(z) oscillation curve at 7.5MHz frequency;

图7：空间傅里叶变换图； Figure 7: Spatial Fourier transform map;

图8：7.5MHz频率下z^-1域曲线图； Figure 8: z ^-1 domain curve at 7.5MHz frequency;

图9：宽频模态追踪图； Figure 9: Broadband mode tracking diagram;

图10：表面波波速提取图； Figure 10: Surface wave velocity extraction diagram;

具体实施方式 Detailed ways

以下结合具体实例对本发明的内容做进一步的详细说明： Below in conjunction with specific example, content of the present invention is described in further detail:

在单频激励/接收的情况下，图2所示的漏表面波传播示意图中，上表面的直接反射回波I传播的时间与漏表面波L的传播时间分别为： In the case of single-frequency excitation/reception, in the schematic diagram of leaky surface wave propagation shown in Figure 2, the propagation time of the direct reflection echo I on the upper surface and the propagation time of the leaky surface wave L are respectively:

${t t}_{11} = = \frac{22 ((R R - - Vz vz))}{{v v}_{w w}} - - - - - - ((11))$

${t t}_{22} = = \frac{22 ((R R - - \frac{Vz Vz}{cos cos {θ θ}_{SAW SAW}}))}{{v v}_{w w}} + + \frac{22 \cdot &Center Dot; Vz vz \cdot &Center Dot; tan the tan {θ θ}_{SAW SAW}}{{v v}_{SAW SAW}} - - - - - - ((22))$

其中R为聚焦半径，Vz为散焦距离，v_w为水的超声波波速，θ_SAW为产生表面波的瑞利角，v_SAW为材料的表面波波速。因此两者的时间差为： Where R is the focusing radius, Vz is the defocusing distance, v _w is the ultrasonic wave velocity of water, θ _SAW is the Rayleigh angle of the surface wave, and v _SAW is the surface wave velocity of the material. So the time difference between the two is:

$Vt Vt = = {t t}_{22} - - {t t}_{11} = = \frac{22 ((11 - - cos cos {θ θ}_{SAW SAW}))}{{v v}_{w w}} \cdot &Center Dot; Vz vz - - - - - - ((33))$

即： Right now:

$cos cos {θ θ}_{SAW SAW} = = 11 - - \frac{{v v}_{w w} \cdot &Center Dot; Vt Vt}{22 \cdot \cdot Vz vz} - - - - - - ((44))$

将Snell定律： Will Snell's law:

$\sin θ_{SAW} = \frac{v_{w}}{v_{SAW}}$ 或 $θ_{SAW} = \sin^{- 1} (\frac{v_{w}}{v_{SAW}})$ $\sin θ_{SAW} = \frac{v_{w}}{v_{SAW}}$ or $θ_{SAW} = \sin^{- 1} (\frac{v_{w}}{v_{SAW}})$

代入(4)后，可得： After substituting into (4), we can get:

$\frac{{v v}_{w w}}{{v v}_{SAW SAW}} = = \sqrt{11 - - {((11 - - \frac{{v v}_{w w}}{22} \cdot &Center Dot; \frac{Vt Vt}{Vz Vz}))}^{22}} - - - - - - ((55))$

此时如果Vz恰为一个V(z)曲线的振荡周期时，1/Vt则为换能器的激励频率f。如果Vz能够确定，便可使用如下公式进行表面波波速的计算： At this time, if Vz is just an oscillation period of a V(z) curve, 1/Vt is the excitation frequency f of the transducer. If Vz can be determined, the following formula can be used to calculate the surface wave velocity:

${v v}_{SAW SAW} = = {v v}_{w w} \cdot &Center Dot; {[[11 - - {((11 - - \frac{{v v}_{w w}}{22 \cdot &Center Dot; f f \cdot &Center Dot; Vz Vz}))}^{22}]]}^{- - 11 / / 22} - - - - - - ((66))$

因此，测量被测材料的V(z)曲线振荡周期成为波速提取的重点。 Therefore, measuring the oscillation period of the V(z) curve of the tested material becomes the focus of wave velocity extraction. the

步骤2)：搭建测试系统。 Step 2): Build a test system. the

为了方便散焦步进测量，搭建了一套进行散焦步进测量的测试系统，如图1所示。该测试系统主要包括：试样1、水槽与水2、换能器3、移动平台4、脉冲激励/接收仪5、示波器6、GPIB总线7、PXI总控制系统8、移动伺服马达9、旋转轴10。其中，在移动平台4下面安装换能器3，换能器3与脉冲激励/接收仪5相连，脉冲激励/接收仪5与示波器6相连，示波器6通过GPIB总线7与PXI总控制系统8相连，PXI总控制系统8与移动伺服马达9相连，同时PXI总控制系统8与旋转轴10相连。 In order to facilitate the step-by-step defocus measurement, a test system for step-by-step defocus measurement is built, as shown in Figure 1. The test system mainly includes: sample 1, sink and water 2, transducer 3, mobile platform 4, pulse excitation/receiver 5, oscilloscope 6, GPIB bus 7, PXI total control system 8, mobile servo motor 9, rotary axis 10. Among them, the transducer 3 is installed under the mobile platform 4, the transducer 3 is connected to the pulse excitation/receiving instrument 5, the pulse excitation/reception instrument 5 is connected to the oscilloscope 6, and the oscilloscope 6 is connected to the PXI general control system 8 through the GPIB bus 7 , The PXI total control system 8 is connected with the moving servo motor 9 , and the PXI total control system 8 is connected with the rotating shaft 10 at the same time. the

步骤3)：聚焦面数据采集。 Step 3): Acquisition of focal plane data. the

以长方体碳化钨为被测试样，其尺寸为40mm×40mm×10mm，将换能器3聚焦到试样的上表面，通过脉冲激励/接收仪5在发出一个带宽为10-200MHz的脉冲后转换为接收状态，当接收到反射信号后，将信号传输进示波器6，示波器的采样频率f_S＝2.5GHz，采样点数N_s＝10000。经过示波器的低通滤波后，通过GPIB总线7存储进PXI总控制系统，聚焦面的时域波形如图3所示。 Take cuboid tungsten carbide as the sample to be tested, its size is 40mm×40mm×10mm, focus the transducer 3 on the upper surface of the sample, and send out a pulse with a bandwidth of 10-200MHz through the pulse excitation/receiver 5. In the receiving state, when the reflected signal is received, the signal is transmitted to the oscilloscope 6 , the sampling frequency of the oscilloscope is f _S =2.5 GHz, and the number of sampling points N _s =10000. After the low-pass filtering of the oscilloscope, it is stored into the PXI general control system through the GPIB bus 7, and the time domain waveform of the focal plane is shown in Figure 3.

步骤4)：散焦测量。 Step 4): Defocus measurement. the

将换能器朝试样方向移动Vz₀＝10μm，待移动完成后进行电压数据采集，采集结束后再将换能器朝试样方向移动Vz₀＝10μm进行数据采集，采样频率f_S＝2.5GHz，采样点数N_s＝10000，如此循环往复，共移动4mm，因此将得到400组电压数据，将聚焦面的电压数据包含在内共得到M＝401组电压数据。将所有数据沿散焦距离排列好，如表1所示，可得到最终的时域波形图。如图4所示。 Move the transducer toward the sample by Vz ₀ =10 μm, collect voltage data after the movement is completed, and then move the transducer toward the sample by Vz ₀ =10 μm for data acquisition, sampling frequency f _S =2.5 GHz, the number of sampling points N _s =10000, and so on, moving 4mm in total, so 400 sets of voltage data will be obtained, and a total of M=401 sets of voltage data will be obtained including the voltage data of the focal plane. Arrange all the data along the defocus distance, as shown in Table 1, to get the final time domain waveform. As shown in Figure 4.

表1电压数据示意图 Table 1 Schematic diagram of voltage data

步骤5)：时域傅里叶变换。 Step 5): Time domain Fourier transform. the

将测得的数据进行时域傅里叶变换。 The measured data were subjected to time-domain Fourier transform. the

其中：A_i为时域傅里叶变换后的频谱值，x_i代表一组电压数据，i＝0，1，2L M-1， Wherein: A _i is the frequency spectrum value after time-domain Fourier transform, x _i represents a group of voltage data, i=0,1,2L M-1,

k＝0，1，2L N_s-1，j代表虚部，N_s＝10000，即： k=0, 1, 2L N _s -1, j represents the imaginary part, N _s =10000, namely:

x₀[0]＝-0.008985937，x₀[1]＝-0.007846875，x₀[2]＝-0.007509375，L，x₀[9999]＝-0.011221875 _x0 [0]=-0.008985937, _x0 [1]=-0.007846875, _x0 [2]=-0.007509375, L, _x0 [9999]=-0.011221875

x₁[0]＝-0.006519375，x₁[1]＝-0.007625000，x₁[2]＝-0.007091250，L，x₁[9999]＝-0.011399375 _x1 [0]=-0.006519375, _x1 [1]=-0.007625000, _x1 [2]=-0.007091250, L, _x1 [9999]=-0.011399375

x₂[0]＝-0.007612500，x₂[1]＝-0.009487500，x₂[2]＝-0.009637500，L，x₂[9999]＝-0.011362500 _x2 [0]=-0.007612500, _x2 [1]=-0.009487500, _x2 [2]=-0.009637500, L, _x2 [9999]=-0.011362500

L L

x₄₀₀[0]＝-0.018224968，x₄₀₀[1]＝-0.018341468，x₄₀₀[2]＝-0.018210406，L，x₄₀₀[9999]＝-0.008985062 _x400 [0]=-0.018224968, _x400 [1]=-0.018341468, _x400 [2]=-0.018210406, L, _x400 [9999]=-0.008985062

${A A}_{00} [[00]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{00} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 00 / / 1000010000} = = {x x}_{00} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 00 / / 1000010000} + + {x x}_{00} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 00 / / 1000010000}$

$+ + {x x}_{00} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 00 / / 1000010000} + + L L + + {x x}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 99999999 \cdot &Center Dot; 00 / / 1000010000}$

${A A}_{00} [[11]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{00} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 11 / / 1000010000} = = {x x}_{00} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot \cdot 11 / / 1000010000} + + {x x}_{00} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 11 / / 1000010000}$

$+ + {x x}_{00} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot &Center Dot; 11 / / 1000010000} + + L L + + {x x}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot &Center Dot; 11 / / 1000010000}$

${A A}_{00} [[22]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{00} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 22 / / 1000010000} = = {x x}_{00} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 22 / / 1000010000} + + {x x}_{00} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 22 / / 1000010000}$

$+ + {x x}_{00} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 22 / / 1000010000} + + L L + + {x x}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot &Center Dot; 22 / / 1000010000}$

M M

${A A}_{00} [[99999999]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{00} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 99999999 / / 1000010000} = = {x x}_{00} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot &Center Dot; 99999999 / / 1000010000} + + {x x}_{00} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 99999999 / / 1000010000}$

$+ + {x x}_{00} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot &Center Dot; 99999999 / / 1000010000} + + L L + + {x x}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot &Center Dot; 99999999 / / 1000010000}$

${A A}_{11} [[00]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{11} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 00 / / 1000010000} = = {x x}_{11} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 00 / / 1000010000} + + {x x}_{11} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot &Center Dot; 00 / / 1000010000}$

$+ + {x x}_{11} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot &Center Dot; 00 / / 1000010000} + + L L + + {x x}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 99999999 \cdot &Center Dot; 00 / / 1000010000}$

${A A}_{11} [[11]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{11} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 11 / / 1000010000} = = {x x}_{11} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot \cdot 11 / / 1000010000} + + {x x}_{11} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 11 / / 1000010000}$

$+ + {x x}_{11} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot \cdot 11 / / 1000010000} + + L L + + {x x}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot &Center Dot; 11 / / 1000010000}$

${A A}_{11} [[22]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{11} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 22 / / 1000010000} = = {x x}_{11} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 22 / / 1000010000} + + {x x}_{11} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 22 / / 1000010000}$

$+ + {x x}_{11} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 22 / / 1000010000} + + L L + + {x x}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot &Center Dot; 22 / / 1000010000}$

M M

${A A}_{11} [[99999999]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{11} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 99999999 / / 1000010000} = = {x x}_{11} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 99999999 / / 1000010000} + + {x x}_{11} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 99999999 / / 1000010000}$

$+ + {x x}_{11} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 99999999 / / 1000010000} + + L L + + {x x}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 99999999 \cdot \cdot 99999999 / / 1000010000}$

${A A}_{22} [[00]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{22} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 00 / / 1000010000} = = {x x}_{22} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 00 / / 1000010000} + + {x x}_{22} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 00 / / 1000010000}$

$+ + {x x}_{22} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 00 / / 1000010000} + + L L + + {x x}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot &Center Dot; 00 / / 1000010000}$

${A A}_{22} [[11]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{22} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 11 / / 1000010000} = = {x x}_{22} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot &Center Dot; 11 / / 1000010000} + + {x x}_{22} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot &Center Dot; 11 / / 1000010000}$

$+ + {x x}_{22} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 11 / / 1000010000} + + L L + + {x x}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot \cdot 11 / / 1000010000}$

${A A}_{22} [[22]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{22} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 22 / / 1000010000} = = {x x}_{22} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 22 / / 1000010000} + + {x x}_{22} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 22 / / 1000010000}$

$+ + {x x}_{22} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 22 / / 1000010000} + + L L + + {x x}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot &Center Dot; 22 / / 1000010000}$

M M

${A A}_{22} [[99999999]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{22} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 99999999 / / 1000010000} = = {x x}_{22} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot \cdot 99999999 / / 1000010000} + + {x x}_{22} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 99999999 / / 1000010000}$

$+ + {x x}_{22} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 99999999 / / 1000010000} + + L L + + {x x}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 99999999 \cdot \cdot 99999999 / / 1000010000}$

M M

${A A}_{400400} [[00]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{400400} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 00 / / 1000010000} = = {x x}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 00 / / 1000010000} + + {x x}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 00 / / 1000010000}$

$+ + {x x}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 00 / / 1000010000} + + L L + + {x x}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 99999999 \cdot \cdot 00 / / 1000010000}$

${A A}_{400400} [[11]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{400400} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 11 / / 1000010000} = = {x x}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 11 / / 1000010000} + + {x x}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 11 / / 1000010000}$

$+ + {x x}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 11 / / 1000010000} + + L L + + {x x}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 99999999 \cdot &Center Dot; 11 / / 1000010000}$

${A A}_{400400} [[22]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{400400} [[n no]] {e e}^{- - j j 22 πn πn \cdot &Center Dot; 22 / / 1000010000} = = {x x}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 22 / / 1000010000} + + {x x}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 22 / / 1000010000}$

$+ + {x x}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 22 / / 1000010000} + + L L + + {x x}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 99999999 \cdot \cdot 22 / / 1000010000}$

M M

${A A}_{400400} [[99999999]] = = {Σ Σ}_{n no = = 00}^{99999999} {x x}_{400400} [[n no]] {e e}^{- - j j 22 πn πn \cdot \cdot 99999999 / / 1000010000} = = {x x}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot &Center Dot; 99999999 / / 1000010000} + + {x x}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 99999999 / / 1000010000}$

$+ + {x x}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 99999999 / / 1000010000} + + L L + + {x x}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 99999999 \cdot \cdot 99999999 / / 1000010000}$

所得A_i[k]，i＝0，1，2L M-1，k＝0，1，2L N_s-1，如表2、图5所示。 The obtained A _i [k], i=0, 1, 2L M-1, k=0, 1, 2L N _s -1, are shown in Table 2 and Fig. 5 .

表2 A_i[k]数据示意图 Table 2 Schematic diagram of A _i [k] data

特定频率下沿散焦距离的振荡曲线即为V(z)曲线，其振荡周期即为Vz。例如，7.5MHz频率下的振荡曲线如图6所示。 The oscillation curve along the defocus distance at a specific frequency is the V(z) curve, and its oscillation period is Vz. For example, the oscillation curve at a frequency of 7.5MHz is shown in Figure 6. the

步骤6)：空间傅里叶变换。 Step 6): Spatial Fourier transform. the

其中：B_i为空间傅里叶变换后的频谱值，A_m代表沿散焦方向的时域傅里叶变换的频谱值，i＝0，1，2L M-1，k＝0，1，2L N_s-1，M＝401，j代表虚部，即： Wherein: B _i is the spectral value after the spatial Fourier transform, A _m represents the spectral value of the time-domain Fourier transform along the defocus direction, i=0,1,2L M-1, k=0,1, 2L N _s -1, M=401, j represents the imaginary part, namely:

${B B}_{00} [[00]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[00]] {e e}^{- - j j 22 π π \cdot \cdot m m \cdot \cdot 00 / / 401401} = = {A A}_{00} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 00 / / 401401} + + {A A}_{11} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 00 / / 401401}$

$+ + {A A}_{22} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 00 / / 401401} + + L L + + + + {A A}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 00 / / 401401}$

${B B}_{11} [[00]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot \cdot 11 / / 401401} = = {A A}_{00} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 11 / / 401401} + + {A A}_{11} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 11 / / 401401}$

$+ + {A A}_{22} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot \cdot 11 / / 401401} + + L L + + + + {A A}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 400400 \cdot \cdot 11 / / 401401}$

${B B}_{22} [[00]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 22 / / 401401} = = {A A}_{00} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 22 / / 401401} + + {A A}_{11} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 22 / / 401401}$

$+ + {A A}_{22} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 22 / / 401401} + + L L + + + + {A A}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 22 / / 401401}$

M M

${B B}_{400400} [[00]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 400400 / / 401401} = = {A A}_{00} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot &Center Dot; 400400 / / 401401} + + {A A}_{11} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 400400 / / 401401}$

$+ + {A A}_{22} [[00]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot &Center Dot; 400400 / / 401401} + + L L + + + + {A A}_{400400} [[00]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 400400 / / 401401}$

${B B}_{00} [[11]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 00 / / 401401} = = {A A}_{00} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 00 / / 401401} + + {A A}_{11} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 00 / / 401401}$

$+ + {A A}_{22} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 00 / / 401401} + + L L + + + + {A A}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 400400 \cdot \cdot 00 / / 401401}$

${B B}_{11} [[11]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[11]] {e e}^{- - j j 22 π π \cdot \cdot m m \cdot &Center Dot; 11 / / 401401} = = {A A}_{00} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 11 / / 401401} + + {A A}_{11} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 11 / / 401401}$

$+ + {A A}_{22} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot \cdot 11 / / 401401} + + L L + + + + {A A}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 11 / / 401401}$

${B B}_{22} [[11]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 22 / / 401401} = = {A A}_{00} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 22 / / 401401} + + {A A}_{11} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 22 / / 401401}$

$+ + {A A}_{22} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 22 / / 401401} + + L L + + + + {A A}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 22 / / 401401}$

M M

${B B}_{400400} [[11]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[11]] {e e}^{- - j j 22 π π \cdot \cdot m m \cdot &Center Dot; 400400 / / 401401} = = {A A}_{00} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 400400 / / 401401} + + {A A}_{11} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 400400 / / 401401}$

$+ + {A A}_{22} [[11]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 400400 / / 401401} + + L L + + + + {A A}_{400400} [[11]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot \cdot 400400 / / 401401}$

${B B}_{00} [[22]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 00 / / 401401} = = {A A}_{00} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 00 / / 401401} + + {A A}_{11} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 00 / / 401401}$

$+ + {A A}_{22} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 00 / / 401401} + + L L + + + + {A A}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 00 / / 401401}$

${B B}_{11} [[22]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 11 / / 401401} = = {A A}_{00} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot &Center Dot; 11 / / 401401} + + {A A}_{11} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 11 / / 401401}$

$+ + {A A}_{22} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 11 / / 401401} + + L L + + + + {A A}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 11 / / 401401}$

${B B}_{22} [[22]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[22]] {e e}^{- - j j 22 π π \cdot \cdot m m \cdot \cdot 22 / / 401401} = = {A A}_{00} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot &Center Dot; 22 / / 401401} + + {A A}_{11} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 22 / / 401401}$

$+ + {A A}_{22} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot &Center Dot; 22 / / 401401} + + L L + + + + {A A}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 400400 \cdot \cdot 22 / / 401401}$

M M

${B B}_{400400} [[22]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[22]] {e e}^{- - j j 22 π π \cdot \cdot m m \cdot &Center Dot; 400400 / / 401401} = = {A A}_{00} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 400400 / / 401401} + + {A A}_{11} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 400400 / / 401401}$

$+ + {A A}_{22} [[22]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 400400 / / 401401} + + L L + + + + {A A}_{400400} [[22]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot \cdot 400400 / / 401401}$

M M

${B B}_{00} [[99999999]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 00 / / 401401} = = {A A}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 00 / / 401401} + + {A A}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot &Center Dot; 00 / / 401401}$

$+ + {A A}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot &Center Dot; 00 / / 401401} + + L L + + + + {A A}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 00 / / 401401}$

${B B}_{11} [[99999999]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot \cdot 11 / / 401401} = = {A A}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 11 / / 401401} + + {A A}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 11 \cdot \cdot 11 / / 401401}$

$+ + {A A}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 11 / / 401401} + + L L + + + + {A A}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 400400 \cdot &Center Dot; 11 / / 401401}$

${B B}_{22} [[99999999]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; m m \cdot &Center Dot; 22 / / 401401} = = {A A}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 00 \cdot &Center Dot; 22 / / 401401} + + {A A}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot &Center Dot; 22 / / 401401}$

$+ + {A A}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 22 \cdot &Center Dot; 22 / / 401401} + + L L + + + + {A A}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 400400 \cdot \cdot 22 / / 401401}$

M M

${B B}_{400400} [[99999999]] = = {Σ Σ}_{m m = = 00}^{400400} {A A}_{m m} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot m m \cdot \cdot 400400 / / 401401} = = {A A}_{00} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 00 \cdot \cdot 400400 / / 401401} + + {A A}_{11} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 11 \cdot \cdot 400400 / / 401401}$

$+ + {A A}_{22} [[99999999]] {e e}^{- - j j 22 π π \cdot \cdot 22 \cdot \cdot 400400 / / 401401} + + L L + + + + {A A}_{400400} [[99999999]] {e e}^{- - j j 22 π π \cdot &Center Dot; 400400 \cdot &Center Dot; 400400 / / 401401}$

所得B_i[k]，i＝0，1，2L M-1，k＝0，1，2L N_s-1，如表3、图7所示。 The obtained B _i [k], i=0, 1, 2L M-1, k=0, 1, 2L N _s -1, are shown in Table 3 and Figure 7 .

表3 B_i[k]数据示意图 Table 3 Schematic diagram of B _i [k] data

特定频率下沿z^-1域的曲线峰值即为振荡周期Vz的倒数。例如，7.5MHz频率下z^-1域的曲线如图8所示。 The peak value of the curve along the z ^-1 domain at a specific frequency is the reciprocal of the oscillation period Vz. For example, the z ^-1 domain plot at 7.5MHz is shown in Figure 8.

步骤7)：模态追踪。 Step 7): Modality Tracking. the

对2.5-22.5MHz范围内的峰值进行追踪，即可找出该频率段连续的Vz值，如图9所示。 By tracking the peak value in the range of 2.5-22.5MHz, the continuous Vz value in this frequency range can be found, as shown in Figure 9. the

步骤8)：波速提取。 Step 8): wave velocity extraction. the

将水中的超声波波速v_W＝1500m/s，每一个峰值对应的频率与Vz带入公式(6)，即可得到该频率段内连续的表面波波速。碳化钨的理论表面波波速为2680m/s，测得的表面波平均波速为2668m/s，两者误差仅为12m/s，提取精度很高。如图10所示。 Substituting the ultrasonic wave velocity v _W in water = 1500m/s, the frequency corresponding to each peak and Vz into the formula (6), the continuous surface wave velocity in this frequency range can be obtained. The theoretical surface wave velocity of tungsten carbide is 2680m/s, and the measured average surface wave velocity is 2668m/s. The error between the two is only 12m/s, and the extraction accuracy is very high. As shown in Figure 10.

Claims

1. A method for isotropic block material Rayleigh wave non-contact wave velocity extraction is characterized in that the method is carried out according to the following steps:

Step 1): Establish the formula for wave velocity extraction;

In the process of wave velocity extraction, according to the V(z) curve theory, the wave velocity is calculated according to the following formula:

{v v}_{SAW SAW} = = {v v}_{w w} \cdot &Center Dot; {[[11 - - {((11 - - \frac{{v v}_{w w}}{22 \cdot &Center Dot; f f \cdot \cdot Δz Δz}))}^{22}]]}^{- - 11 / / 22}

Where: Δz is the oscillation period of the V(z) curve, v _w is the ultrasonic wave velocity of water, f is the excitation frequency of the transducer, and v _SAW is the surface wave velocity of the material;

Step 2): Build a test system;

The test system includes: sample (1), sink and water (2), transducer (3), mobile platform (4), pulse excitation/receiver (5), oscilloscope (6), GPIB bus (7) , PXI general control system (8), moving servo motor (9), and rotating shaft (10); among them, the transducer (3) is installed under the mobile platform (4), and the transducer (3) is connected with the pulse excitation/reception The pulse excitation/receiving instrument (5) is connected to the oscilloscope (6), the oscilloscope (6) is connected to the PXI total control system (8) through the GPIB bus (7), and the PXI total control system (8) is connected to the mobile The servo motor (9) is connected, and the PXI general control system (8) is connected with the rotating shaft (10);

Step 3): data collection of focal plane;

Place the sample on the focal plane of the transducer, the pulse excitation/reception instrument (5) switches to the receiving state after sending out a 10-200MHz pulse, and transmits the signal into the oscilloscope (6) after receiving the reflected signal, The sampling frequency of the oscilloscope is f _S , f _S is 0.5-5GHz, and the number of sampling points is N _s ; after low-pass filtering by the oscilloscope, it is stored into the PXI total control system (8) through the GPIB bus (7);

Step 4): defocus measurement;

Move the transducer vertically downward by a distance Δz ₀ , the value range of Δz ₀ is 1-50 μm, and collect data after the movement is completed, the sampling frequency is f _S , the number of sampling points is N _s , and the value range of N _s It is 10000-100000 points; after the collection is completed, move the transducer vertically downward by Δz ₀ for data collection, and so on and on, the total moving distance z, the value range of z is 2-20mm, so M groups of voltage data will be obtained , M is determined jointly by z and Δz ₀ , which is 40-20000 groups;

Step 5): time-domain Fourier transform;

Arrange all the data along the defocus distance, and perform time-domain Fourier transform on the measured data:

{A A}_{i i} [[k k]] = = {Σ Σ}_{n no = = 00}^{{N N}_{s the s} - - 11} {x x}_{i i} [[n no]] {e e}^{- - j j 22 πnk πnk / / {N N}_{s the s}}

Among them: A _i is the spectrum value after time-domain Fourier transform, x _i represents a set of voltage data, i=0,1,2...M-, k=0,1,2...N _s -1, j represents imaginary part;

Step 6): Spatial Fourier Transform

In order to obtain an accurate oscillation period Δz, it is necessary to perform a spatial Fourier transform on the result of the time-domain Fourier transform along the direction of the defocus distance, and transform the defocus distance z to the z ^-1 domain:

{B B}_{i i} [[k k]] = = {Σ Σ}_{m m = = 00}^{M m - - 11} {A A}_{m m} [[k k]] {e e}^{- - j j 22 πmi πmi / / M m}

Among them: B _i is the spectral value after spatial Fourier transform, A _m represents the spectral value of time-domain Fourier transform along the defocus direction, i=0, 1, 2...M-1, k=0, 1 , 2...N _s -1, j represents the imaginary part; the peak value of the curve along the z ^-1 domain is the reciprocal of the oscillation period Δz;

Step 7): Modal Tracking

By tracking the peak value in the range of 1-100MHz, you can find out the Δz value of the continuous oscillation period in this frequency segment;

Step 8): Wave Velocity Extraction

Substitute the wave velocity of water, the frequency f and Δz corresponding to each peak value into the formula shown in step 1) to obtain the continuous surface wave velocity v _SAW .