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

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

The method that the contactless velocity of wave of a kind of isotropy block materials R wave extracts

Technical field

The invention belongs to field of non destructive testing, be specifically related to a kind of velocity of wave extracting method to isotropy block materials R wave.

Background technology

Along with constantly advancing of material science, various functional form materials continue to bring out, but are subject to preparation technology's impact, and the physical dimension of a lot of new materials is very limited, such as metallic glass, block nanometer material etc.Therefore, adopt the method for the destructive traditional mechanics performance tests such as stretching can't satisfy the demand of new material.In detecting take the measurement acoustic velocity as main non-destructive, the many information that comprised the material microstructure aspect by the formed V of interference (z) curve of leaky surface wave and direct reflection wave, with ultrasonic microscope as the velocity of wave survey instrument, can be applied to detect the material mechanical character such as crystal structure, elastic modulus, unrelieved stress, inherent vice, make ultrasonic microscope obtain to use more and more widely at aspects such as characteristic of material mechanics test and quantitative Non-Destructive Testings.

Measurement is one of very promising measuring method of field of non destructive testing to elastic properties of materials character to utilize ultrasound wave.In the isotropy homogeneous material, surface wave (Surface acoustic wave, SAW) be called again R wave (Rayleigh SAW), its fluctuation behavior has comprised the information of lot of materials characteristic, therefore, by the surface wave velocity of wave of measuring block materials and the elastic property that longitudinal wave velocity can be finally inversed by material.

In order to achieve the above object, the accurate extraction of velocity of wave seems particularly necessary.Present for the most of modes that adopt the single-frequency pointwise to extract of R wave velocity of wave extraction, determine the velocity of wave of surface wave by Vz oscillation period in measurement V (z) curve, but its shortcoming is the measurement that the single-frequency velocity of wave extracted and be not suitable for the wideband pulse signal.Therefore, need to develop a cover based on the surface wave velocity of wave extracting method of wideband pulse signal.

Summary of the invention

The objective of the invention is to propose a kind of advanced person's material velocity of wave extracting method in order to solve the problem of the continuous velocity of wave extraction of isotropy block materials R wave wideband.

Step 1): establish the formula that velocity of wave extracts.

Here need to prove, due to the load effect of water, the velocity of wave of leaky surface wave and surface wave and not quite identical, but due to the density of the measured material density much larger than water, difference between the two is negligible.To no longer distinguish surface wave and leaky surface wave in elaboration afterwards.In the process that velocity of wave extracts, according to V (z) curve theory, can carry out according to following formula the calculating of velocity of wave:

$v_{\mathrm{SAW}}=v_w\·{[1-{(1-\frac{v_w}{2\·f\·\mathrm{Vz}})}^2]}^{-1/2}$

Wherein: Vz is V (z) curve oscillation period, v _wBe the ultrasonic velocity in water, f is the excitation frequency of transducer, v _SAWSurface wave velocity of wave for material.Be the key that velocity of wave extracts V (z) curve oscillation period of measuring measured material.

Step 2): test system building.

In order conveniently to defocus stepping measurement, built the test macro that a cover defocuses stepping measurement, as shown in Figure 1.This test macro mainly comprises: sample 1, tank and water 2, transducer 3, mobile platform 4, pulse excitation/receiving instrument 5, oscillograph 6, gpib bus 7, PXI general control system 8, shift servo motor 9, turning axle 10.Wherein, transducer 3 is installed below mobile platform 4, transducer 3 is connected with pulse excitation/receiving instrument 5, pulse excitation/receiving instrument 5 is connected with oscillograph 6, oscillograph 6 is connected with PXI general control system 8 by gpib bus 7, PXI general control system 8 is connected with shift servo motor 9, and PXI general control system 8 is connected with turning axle 10 simultaneously.

Step 3: focusing surface data acquisition.

The block tested sample is placed in the focusing surface of transducer, pulse excitation/receiving instrument 5 is converted to accepting state after the pulse that to send a bandwidth be 10-200MHz, after receiving reflected signal, signal is transmitted into oscillograph 6, and oscillographic sample frequency is f _S, f _SBe 0.5-5GHz, sampling number is N _s, N _sSpan be the 10000-100000 point.Through after oscillographic low-pass filtering, be stored into PXI general control system 8 by gpib bus 7.

Step 4): defocus measurement.

Transducer is moved one vertically downward apart from Vz ₀, Vz ₀Span be 1-50 μ m, carry out the data collection after mobile completing, sample frequency is f _S, sampling number is N _sAfter gather finishing again with transducer mobile Vz vertically downward ₀Carry out data acquisition, so move in circles, be total to displacement z, the span of z is 2-20mm, therefore will obtain M group voltage data, and M is by z and Vz ₀The common decision is the 40-20000 group.

Step 5): the time domain Fourier transform.

All data are arranged along defocus distance, the data that record are carried out the time domain Fourier transform:

$A_i\left[k\right]=\underset{n=0}{\overset{N_s-1}{\mathrm{\&Sigma;}}}{x}_i\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;nk}/N_s}$

Wherein: A _iBe the spectrum value after the time domain Fourier transform, x _iRepresent one group of voltage data, i=0,1,2L M-1, k=0,1,2L N _s-1, j represents imaginary part.

Step 6): spatial fourier transform.

In order to obtain accurate oscillation period of Vz, need to carry out again along the spatial fourier transform of defocus distance direction the result of time domain Fourier transform, defocus distance z is converted into z ^-1The territory:

$B_i\left[k\right]=\underset{m=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{A}_m\left[k\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mi}/M}$

Wherein: B _iBe the spectrum value after spatial fourier transform, A _mRepresent along the spectrum value that defocuses the time domain Fourier transform of direction, i=0,1,2L M-1, k=0,1,2L N _s-1, j represents imaginary part.Along z ^-1The peak of curve in territory is the inverse of Vz oscillation period.

Step 7): mode is followed the trail of.

Peak value in the 1-100MHz scope is followed the trail of, can be found out continuous Vz value oscillation period of this frequency band.

Step 8): velocity of wave extracts.

If the coupling liquid of using is water, with the ultrasonic velocity v in water _W, the frequency f that each peak value is corresponding and Vz substitution oscillation period step 1) shown in formula, can obtain continuous surface wave velocity of wave v in this frequency band _SAW

The present invention has the following advantages: 1) can the R wave velocity of wave of different materials be extracted; 2) can extract the R wave velocity of wave in wide frequency range, replace the mode of single-frequency pointwise; 3) can the R wave velocity of wave in the different frequency section be extracted, select value after average as the R wave velocity of wave of material, the stochastic error that has caused due to accidentalia when having avoided single-frequency to extract.

Description of drawings

Fig. 1: defocus the measuring system schematic diagram;

Fig. 2: surface wave propagation schematic diagram;

Fig. 3: focusing surface time domain waveform figure;

Fig. 4: the time domain waveform figure under different defocus distance;

Fig. 5: time domain Fourier transform figure;

Fig. 6: V under the 7.5MHz frequency (z) oscillating curve figure;

Fig. 7: spatial fourier transform figure;

Fig. 8: z under the 7.5MHz frequency ^-1The territory curve map;

Fig. 9: wideband mode tracking map;

Figure 10: the surface wave velocity of wave extracts figure;

Embodiment

Below in conjunction with instantiation, content of the present invention is described in further detail:

Step 1): establish the formula that velocity of wave extracts.

In the situation that the single-frequency excitation/receiving, leaky surface wave shown in Figure 2 is propagated in schematic diagram, and the time that the direct reflection echo I of upper surface propagates and the travel-time of leaky surface wave L are respectively:

${\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=\frac{2(R-\mathrm{Vz})}{v_w}---\left(1\right)$

${\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=\frac{2(R-\frac{\mathrm{Vz}}{\cos {\mathrm{\&theta;}}_{\mathrm{SAW}}})}{v_w}+\frac{2\&CenterDot;\mathrm{Vz}\&CenterDot;\tan {\mathrm{\&theta;}}_{\mathrm{SAW}}}{v_{\mathrm{SAW}}}---\left(2\right)$

Wherein R is focused radius, and Vz is defocus distance, v _wBe the ultrasonic velocity of water, θ _SAWFor producing the Rayleigh angle of surface wave, v _SAWSurface wave velocity of wave for material.Therefore both mistimings are:

$\mathrm{Vt}=t_2-{\mathrm{<figure-callout\; id="1"\; label="t"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=\frac{2(1-\cos {\mathrm{\&theta;}}_{\mathrm{SAW}})}{v_w}\&CenterDot;\mathrm{Vz}---\left(3\right)$

That is:

$\cos {\mathrm{\&theta;}}_{\mathrm{SAW}}=1-\frac{v_w\&CenterDot;\mathrm{<figure-callout\; id="2"\; label="Vt"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">Vt</figure-callout>}}{2\&CenterDot;\mathrm{Vz}}---\left(4\right)$

With the Snell law:

$\sin {\mathrm{\&theta;}}_{\mathrm{SAW}}=\frac{v_w}{v_{\mathrm{SAW}}}$ Or ${\mathrm{\&theta;}}_{\mathrm{SAW}}={\sin }^{-1}\left(\frac{v_w}{v_{\mathrm{SAW}}}\right)$

After substitution (4), can get:

$\frac{v_w}{v_{\mathrm{SAW}}}=\sqrt{1-{(1-\frac{v_w}{2}\&CenterDot;\frac{\mathrm{Vt}}{\mathrm{Vz}})}^2}---\left(5\right)$

If when this moment, Vz was just the oscillation period of a V (z) curve, 1/Vt was the excitation frequency f of transducer.If Vz can determine, just can use following formula to carry out the calculating of surface wave velocity of wave:

$v_{\mathrm{SAW}}=v_w\&CenterDot;{[1-{(1-\frac{v_w}{2\&CenterDot;f\&CenterDot;\mathrm{Vz}})}^2]}^{-1/2}---\left(6\right)$

Therefore, V (z) curve of measurement measured material becomes the emphasis of velocity of wave extraction oscillation period.

Step 2): test system building.

In order conveniently to defocus stepping measurement, built the test macro that a cover defocuses stepping measurement, as shown in Figure 1.This test macro mainly comprises: sample 1, tank and water 2, transducer 3, mobile platform 4, pulse excitation/receiving instrument 5, oscillograph 6, gpib bus 7, PXI general control system 8, shift servo motor 9, turning axle 10.Wherein, transducer 3 is installed below mobile platform 4, transducer 3 is connected with pulse excitation/receiving instrument 5, pulse excitation/receiving instrument 5 is connected with oscillograph 6, oscillograph 6 is connected with PXI general control system 8 by gpib bus 7, PXI general control system 8 is connected with shift servo motor 9, and PXI general control system 8 is connected with turning axle 10 simultaneously.

Step 3): the focusing surface data acquisition.

Take the rectangular parallelepiped tungsten carbide as tested sample, it is of a size of 40mm * 40mm * 10mm, transducer 3 is focused on the upper surface of sample, be converted to accepting state by pulse excitation/receiving instrument 5 after the pulse that to send a bandwidth be 10-200MHz, after receiving reflected signal, signal is transmitted into oscillograph 6 oscillographic sample frequency f _S=2.5GHz, sampling number N _s=10000.Through after oscillographic low-pass filtering, be stored into the PXI general control system by gpib bus 7, the time domain waveform of focusing surface is as shown in Figure 3.

Step 4): defocus measurement.

Transducer is moved Vz towards the sample direction ₀=10 μ m carry out the voltage data collection after mobile completing, collection is moved Vz with transducer towards the sample direction after finishing again ₀=10 μ m carry out data acquisition, sample frequency f _S=2.5GHz, sampling number N _s=10000, so move in circles, altogether mobile 4mm, therefore will obtain 400 groups of voltage datas, the voltage data of focusing surface is included obtain altogether M=401 group voltage data.All data are arranged along defocus distance, as shown in table 1, can obtain final time domain waveform figure.As shown in Figure 4.

Table 1 voltage data schematic diagram

Step 5): the time domain Fourier transform.

The data that record are carried out the time domain Fourier transform.

$A_i\left[k\right]=\underset{n=0}{\overset{N_s-1}{\mathrm{\&Sigma;}}}{x}_i\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;nk}/N_s}$

Wherein: A _iBe the spectrum value after the time domain Fourier transform, x _iRepresent one group of voltage data, i=0,1,2L M-1,

K=0,1,2L N _s-1, j represents imaginary part, N _s=10000, that is:

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

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

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

L

x ₄₀₀[0]＝-0.018224968，x ₄₀₀[1]＝-0.018341468，x ₄₀₀[2]＝-0.018210406，L，x ₄₀₀[9999]＝-0.008985062

$A_0\left[0\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_0\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;0/10000}=x_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/10000}+x_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/10000}$

$+x_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/10000}+L+x_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;0/10000}$

$A_0\left[1\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_0\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;1/10000}=x_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/10000}+x_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/10000}$

$+x_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/10000}+L+x_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;1/10000}$

$A_0\left[2\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_0\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;2/10000}=x_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/10000}+x_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/10000}$

$+x_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/10000}+L+x_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;2/10000}$

M

$A_0\left[9999\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_0\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;9999/10000}=x_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;9999/10000}+x_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;9999/10000}$

$+x_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;9999/10000}+L+x_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;9999/10000}$

$A_1\left[0\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_1\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;0/10000}=x_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/10000}+x_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/10000}$

$+x_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/10000}+L+x_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;0/10000}$

$A_1\left[1\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_1\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;1/10000}=x_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/10000}+x_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/10000}$

$+x_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/10000}+L+x_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;1/10000}$

$A_1\left[2\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_1\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;2/10000}=x_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/10000}+x_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/10000}$

$+x_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/10000}+L+x_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;2/10000}$

M

$A_1\left[9999\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_1\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;9999/10000}=x_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;9999/10000}+x_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;9999/10000}$

$+x_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;9999/10000}+L+x_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;9999/10000}$

$A_2\left[0\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_2\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;0/10000}=x_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/10000}+x_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/10000}$

$+x_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/10000}+L+x_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;0/10000}$

$A_2\left[1\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_2\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;1/10000}=x_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/10000}+x_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/10000}$

$+x_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/10000}+L+x_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;1/10000}$

$A_2\left[2\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_2\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;2/10000}=x_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/10000}+x_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/10000}$

$+x_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/10000}+L+x_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;2/10000}$

M

$A_2\left[9999\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_2\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;9999/10000}=x_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;9999/10000}+x_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;9999/10000}$

$+x_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;9999/10000}+L+x_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;9999/10000}$

M

$A_{400}\left[0\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_{400}\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;0/10000}=x_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/10000}+x_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/10000}$

$+x_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/10000}+L+x_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;0/10000}$

$A_{400}\left[1\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_{400}\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;1/10000}=x_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/10000}+x_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/10000}$

$+x_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/10000}+L+x_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;1/10000}$

$A_{400}\left[2\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_{400}\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;2/10000}=x_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/10000}+x_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/10000}$

$+x_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/10000}+L+x_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;2/10000}$

M

$A_{400}\left[9999\right]=\underset{n=0}{\overset{9999}{\mathrm{\&Sigma;}}}{x}_{400}\left[n\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;n}\&CenterDot;9999/10000}=x_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;9999/10000}+x_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;9999/10000}$

$+x_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;9999/10000}+L+x_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;9999\&CenterDot;9999/10000}$

Gained A _i[k], i=0,1,2L M-1, k=0,1,2L N _s-1, as table 2, shown in Figure 5.

Table 2 A _i[k] schematic diagram data

Oscillating curve along defocus distance under characteristic frequency is V (z) curve, is Vz its oscillation period.For example, the oscillating curve under the 7.5MHz frequency as shown in Figure 6.

Step 6): spatial fourier transform.

In order to obtain accurate oscillation period of Vz, need to carry out again along the spatial fourier transform of defocus distance direction the result of time domain Fourier transform, defocus distance z is converted into z ^-1The territory:

$B_i\left[k\right]=\underset{m=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{A}_m\left[k\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;mi}/M}$

Wherein: B _iBe the spectrum value after spatial fourier transform, A _mRepresent along the spectrum value that defocuses the time domain Fourier transform of direction, i=0,1,2L M-1, k=0,1,2L N _s-1, M=401, j represents imaginary part, that is:

$B_0\left[0\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;0/401}=A_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/401}+A_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/401}$

$+A_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/401}+L++A_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;0/401}$

$B_1\left[0\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;1/401}=A_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/401}+A_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/401}$

$+A_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/401}+L++A_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;1/401}$

$B_2\left[0\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;2/401}=A_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/401}+A_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/401}$

$+A_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/401}+L++A_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;2/401}$

M

$B_{400}\left[0\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;400/401}=A_0\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;400/401}+A_1\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;400/401}$

$+A_2\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;400/401}+L++A_{400}\left[0\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;400/401}$

$B_0\left[1\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;0/401}=A_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/401}+A_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/401}$

$+A_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/401}+L++A_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;0/401}$

$B_1\left[1\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;1/401}=A_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/401}+A_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/401}$

$+A_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/401}+L++A_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;1/401}$

$B_2\left[1\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;2/401}=A_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/401}+A_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/401}$

$+A_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/401}+L++A_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;2/401}$

M

$B_{400}\left[1\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;400/401}=A_0\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;400/401}+A_1\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;400/401}$

$+A_2\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;400/401}+L++A_{400}\left[1\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;400/401}$

$B_0\left[2\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;0/401}=A_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/401}+A_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/401}$

$+A_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/401}+L++A_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;0/401}$

$B_1\left[2\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;1/401}=A_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/401}+A_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/401}$

$+A_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/401}+L++A_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;1/401}$

$B_2\left[2\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;2/401}=A_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/401}+A_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/401}$

$+A_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/401}+L++A_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;2/401}$

M

$B_{400}\left[2\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;400/401}=A_0\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;400/401}+A_1\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;400/401}$

$+A_2\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;400/401}+L++A_{400}\left[2\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;400/401}$

M

$B_0\left[9999\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;0/401}=A_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;0/401}+A_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;0/401}$

$+A_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;0/401}+L++A_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;0/401}$

$B_1\left[9999\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;1/401}=A_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;1/401}+A_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;1/401}$

$+A_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;1/401}+L++A_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;1/401}$

$B_2\left[9999\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;2/401}=A_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;2/401}+A_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;2/401}$

$+A_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;2/401}+L++A_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;2/401}$

M

$B_{400}\left[9999\right]=\underset{m=0}{\overset{400}{\mathrm{\&Sigma;}}}{A}_m\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;m\&CenterDot;400/401}=A_0\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;0\&CenterDot;400/401}+A_1\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;1\&CenterDot;400/401}$

$+A_2\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;2\&CenterDot;400/401}+L++A_{400}\left[9999\right]e^{-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HDA0000122101780000011.png,HDA0000122101780000012.png"\; state="\{\{state\}\}">j</figure-callout>}2\mathrm{\&pi;}\&CenterDot;400\&CenterDot;400/401}$

Gained B _i[k], i=0,1,2L M-1, k=0,1,2L N _s-1, as table 3, shown in Figure 7.

Table 3 B _i[k] schematic diagram data

Under characteristic frequency along z ^-1The peak of curve in territory is the inverse of Vz oscillation period.For example, z under the 7.5MHz frequency ^-1The curve in territory as shown in Figure 8.

Step 7): mode is followed the trail of.

Peak value in the 2.5-22.5MHz scope is followed the trail of, can be found out the continuous Vz value of this frequency band, as shown in Figure 9.

Step 8): velocity of wave extracts.

With the ultrasonic velocity v in water _W=1500m/s, the frequency that each peak value is corresponding and Vz bring formula (6) into, can obtain continuous surface wave velocity of wave in this frequency band.The theoretical surface wave-wave speed of tungsten carbide is 2680m/s, and the average velocity of wave of the surface wave that records is 2668m/s, and both errors are only 12m/s, and extraction accuracy is very high.As shown in figure 10.

The present invention has the following advantages: 1) can the R wave velocity of wave of different materials be extracted; 2) can extract the R wave velocity of wave in wide frequency range, replace the mode of single-frequency pointwise; 3) can the R wave velocity of wave in the different frequency section be extracted, select value after average as the R wave velocity of wave of material, the stochastic error that has caused due to accidentalia when having avoided single-frequency to extract.

Claims (1)

1. the method extracted of the contactless velocity of wave of an isotropy block materials R wave is characterized in that the method carries out in accordance with the following steps:

Step 1): establish the formula that velocity of wave extracts;

In the process that velocity of wave extracts, according to V (z) curve theory, carry out the calculating of velocity of wave according to following formula:

$v_{\mathrm{SAW}}=v_w\&CenterDot;{[1-{(1-\frac{v_w}{2\&CenterDot;f\&CenterDot;\mathrm{\&Delta;z}})}^2]}^{-1/2}$

Wherein: Δ z is V (z) curve oscillation period, v _wBe the ultrasonic velocity of water, f is the excitation frequency of transducer, v _SAWSurface wave velocity of wave for material;

Step 2): test system building;

This test macro comprises: sample (1), tank and water (2), transducer (3), mobile platform (4), pulse excitation/receiving instrument (5), oscillograph (6), gpib bus (7), PXI general control system (8), shift servo motor (9), turning axle (10); Wherein, transducer (3) is installed below mobile platform (4), transducer (3) is connected with pulse excitation/receiving instrument (5), pulse excitation/receiving instrument (5) is connected with oscillograph (6), oscillograph (6) is connected with PXI general control system (8) by gpib bus (7), PXI general control system (8) is connected with shift servo motor (9), and PXI general control system (8) is connected with turning axle (10) simultaneously;

Step 3): focusing surface data acquisition;

Sample is placed in the focusing surface of transducer, pulse excitation/receiving instrument (5) is converted to accepting state after the pulse of sending a 10-200MHz, after receiving reflected signal, signal is transmitted into oscillograph (6), and oscillographic sample frequency is f _S, f _SBe 0.5-5GHz, sampling number is N _sThrough after oscillographic low-pass filtering, be stored into PXI general control system (8) by gpib bus (7);

Step 4): defocus measurement;

Transducer is moved a distance, delta z vertically downward ₀, Δ z ₀Span be 1-50 μ m, carry out the data collection after mobile completing, sample frequency is f _S, sampling number is N _s, N _sSpan be the 10000-100000 point; After gather finishing again with transducer mobile Δ z vertically downward ₀Carry out data acquisition, so move in circles, be total to displacement z, the span of z is 2-20mm, therefore will obtain M group voltage data, and M is by z and Δ z ₀The common decision is the 40-20000 group;

Step 5): time domain Fourier transform;

All data are arranged along defocus distance, the data that record are carried out the time domain Fourier transform:

$A_i\left[k\right]=\underset{n=0}{\overset{N_s-1}{\mathrm{\&Sigma;}}}{x}_i\left[n\right]e^{-j2\mathrm{\&pi;nk}/N_s}$

Wherein: A _iBe the spectrum value after the time domain Fourier transform, x _iRepresent one group 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 accurate oscillation period of Δ z, need to carry out again along the spatial fourier transform of defocus distance direction the result of time domain Fourier transform, defocus distance z is converted into z ^-1The territory:

$B_i\left[k\right]=\underset{m=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{A}_m\left[k\right]e^{-j2\mathrm{\&pi;mi}/M}$

Wherein: B _iBe the spectrum value after spatial fourier transform, A _mRepresent along the spectrum value that defocuses the time domain Fourier transform of direction, i=0,1,2 ... M-1, k=0,1,2 ... N _s-1, j represents imaginary part; Along z ^-1The peak of curve in territory is the inverse of Δ z oscillation period;

Step 7): mode is followed the trail of

Peak value in the 1-100MHz scope is followed the trail of, can be found out continuous Δ z value oscillation period of this frequency band;

Step 8): velocity of wave extracts

Formula shown in the frequency f that velocity of wave, each peak value of water is corresponding and Δ z substitution step 1) namely obtains continuous surface wave velocity of wave v _SAW

Images