Measurement of the full wave field on a shell using a single-point laser Doppler vibrometer

Figure 1 shows the schematic of scanning an area on a curved shell with a single-point LDV translated on the $x^{\prime} y^{\prime}$ plane at a constant angle. We set equal spatial intervals for scanning along the $x^{\prime}$ and $y^{\prime}$ axis, denoted as $\Delta x^{\prime}$ and $\Delta y^{\prime}$ respectively. The measured velocity ${v_i}$ of each point obtained from LDV is a projection of the actual velocity onto the laser direction unit vector ${\boldsymbol{e}_i}$ ($i$ denotes the ordinal number of the direction of laser). After scanning the entire region in one direction, the intervals $\Delta x$ and $\Delta y$ between two adjacent measured points on the surfaces exhibit irregular variations as the direction of the laser changes. Additional processing is required for the calculation of 3D wave fields based solely on data of the measured points.

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In fact, the velocity field is a continuous function of coordinates, allowing us to perform interpolation on discrete points and obtain a continuous velocity function ${v_i}\left( {x,y,t} \right)$ for the entire area. For the interpolation, the demanded coordinates of the measured points can be solved with the known geometry of the shells and the incident angles of laser beams. Considering the relatively simple geometry of the shell, the surface equation can be represented as $z = f\left( {x,y} \right)$. For one of the laser directions $i$, the line equations of the measured lasers are represented as

$$\begin{equation}\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {x = {x_0} + \left( {m - 1} \right)\Delta x^{\prime} + {e_{xi}}l} \\ {y = {y_0} + \left( {n - 1} \right)\Delta y^{\prime} + {e_{yi}}l} \\ {z = {z_0} + {e_{zi}}l} \end{array}} \right.} \end{array}\end{equation} \tag{ 1 }$$

where $\left( {{x_0},{y_0},{z_0}} \right)$ denotes the first point's coordinate, $m$ and $n$ are the ordinal of measured points along the $x$ axis and $y$ axis respectively, $l$ is the parameter in the parametric equation of a line. The mapped coordinates $\left( {x,y} \right)$ of the measured points can be solved by the surface equation and equation (1), and the continuous measured velocity field ${v_i}\left( {x,y,t} \right)$ of the target area can be obtained by interpolation.

The LDV measures the velocity of a single point in the direction aligned with the laser beam [15, 23]. Therefore, the velocity components for an individual point can be written as:

$$\begin{equation}{v_i} = v \cdot {{\boldsymbol{e}}_i} = {v_x}{\cos}{\alpha _i} + {v_y}{\cos}{\beta _i} + {v_z}{\cos}{\gamma _i}\end{equation} \tag{ 2 }$$

where $i = 1,2,3$ denotes three incident lasers in different directions. ${v_x},{\text{ }}{v_y},{\text{ }}{v_z}$ represent the velocity components along the orthogonal axes in a Cartesian coordinate system and ${\alpha _i},{\text{ }}{\beta _i},{\text{ }}{\gamma _i}$ represent the angles between the laser's unit vector ${\boldsymbol{e}_i}$ and $x,\,y,\,z$ axis respectively. Considering that there are three unknown components (${v_x},{v_y},{v_z}$) to be solved, measurements with lasers in three directions are required. The vector relations between the components of velocity and the measured velocities in three directions can be written in matrix form:

$$\begin{equation}\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {{\text{cos}}{\alpha _1}}&{{\text{cos}}{\beta _1}}&{{\text{cos}}{\gamma _1}} \\ {{\text{cos}}{\alpha _2}}&{{\text{cos}}{\beta _2}}&{{\text{cos}}{\gamma _2}} \\ {{\text{cos}}{\alpha _3}}&{{\text{cos}}{\beta _3}}&{{\text{cos}}{\gamma _3}} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{v_x}} \\ {{v_y}} \\ {{v_z}} \end{array}} \right\}} \end{array}\end{equation} \tag{ 3 }$$

where ${\boldsymbol{e}_1},{\boldsymbol{e}_2},{\boldsymbol{e}_3}$ are non-coplanar to ensure that the direction matrix can be inversed. The orthogonal components of 3D velocity at this point can be figured:

$$\begin{equation}\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {{v_x}} \\ {{v_y}} \\ {{v_z}} \end{array}} \right\} = {{\left[ {\begin{array}{*{20}{c}} {{\text{cos}}{\alpha _1}}&{{\text{cos}}{\beta _1}}&{{\text{cos}}{\gamma _1}} \\ {{\text{cos}}{\alpha _2}}&{{\text{cos}}{\beta _2}}&{{\text{cos}}{\gamma _2}} \\ {{\text{cos}}{\alpha _3}}&{{\text{cos}}{\beta _3}}&{{\text{cos}}{\gamma _3}} \end{array}} \right]}^{ - 1}}\left\{ {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right\}} \end{array}\end{equation} \tag{ 4 }$$

and the velocity fields can be solved by

$$\begin{align} \left\{\begin{array}{*{20}{c}} {{v_x}\left( {x,y,t} \right)} \\ {{v_y}\left( {x,y,t} \right)} \\ {{v_z}\left( {x,y,t} \right)} \end{array}\right\} & = {{\left[\begin{array}{*{20}{c}} {{\text{cos}}{\alpha _1}}& {{\text{cos}}{\beta _1}}& {{\text{cos}}{\gamma _1}} \\ {{\text{cos}}{\alpha _2}}& {{\text{cos}}{\beta _2}}& {{\text{cos}}{\gamma _2}} \\ {{\text{cos}}{\alpha _3}}& {{\text{cos}}{\beta _3}}& {{\text{cos}}{\gamma _3}} \end{array}\right]}^{ - 1}} \left\{\begin{array}{*{20}{c}} {{v_1}\left( {x,y,t} \right)} \\ {{v_2}\left( {x,y,t} \right)} \\ {{v_3}\left( {x,y,t} \right)} \end{array}\right\} \end{align} \tag{ 5 }$$

where ${v_x}\left( {x,y,t} \right)$, ${v_y}\left( {x,y,t} \right)$, ${v_z}\left( {x,y,t} \right)$ denote the velocity field along the axes in the Cartesian coordinate system. Using these components, a time-domain wave field which is vibrating along an arbitrary direction can be obtained by projection, and the corresponding frequency-domain response can be acquired by the Fourier transform. In addition, the accuracy of the measurement can be affected by the surface roughness, since the intensity of the laser light reflected to the LDV in the original track is reduced due to the diffuse reflection, leading to an increase of noise. However, the measured velocity ${v_i}$ is dependent on both the velocity at the measurement point and the incident angle of the laser, indicating that there will be no distortion in the measured signal. To enhance the signal-to-noise ratio of the measured signal, a thin reflective tape is applied onto the rough surface or transparent objects, thereby amplifying the intensity of the reflected laser.

Figure 2(a) shows the setting up schematic of the scanning system. A RIGOL DG4062 signal generator is used to exert a stimulating signal, which is magnified by a power amplifier to increase the signal-to-noise ratio. A piezoelectric patch driven by the power amplifier is bonded on a fabricated cylindrical tube as stimulus. A Polytec NLV-2500 single-point LDV is mounted on a $xy$ -plane motorized system via a replaceable clamp. The three scanning directions should be non-coplanar to ensure the sollution of three independent orthotropic signal components. Additionally, in order to enhance the signal-to-noise ratio, the angle between the laser and the normal directions of surfaces of the sample should be less than $45^\circ$ [21]. In this work, the laser is oriented at an angle of $60^\circ$, $90^\circ$, or $120^\circ$ to the $x$ axis in the $xz$ plane by rotating the clamp or at an angle of $60^\circ$ to the $y$ axis in the $yz$ plane by another clamp. The motorized system has a bundle software to control the motors to move the LDV. The voltage signals collected by the LDV are directly proportional to the measured velocity, and the ratio can be set as $5,{\text{ }}10{\text{ or }}20{\text{ mm}}\,{{\text{s}}^{ - {\text{1}}}}\,{{\text{V}}^{ - {\text{1}}}}$ considering the vibration range. The time history of the velocity is stored in a computer via a PC oscilloscope of Pico Scope 4000 series.

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As shown in figure 2(b), in this paper we measured the wave fields on the surface of a cylindrical shell as an example to illustrate the measurement method. The thin-walled cylinder has an external diameter of 100 mm, a wall thickness of 2 mm, and a length of 500 mm and is made of Acrylic. In order to enhance the reflected signal when the laser is oblique incidence to the surface, a reflective tape is affixed to the targeted area, which is 150 mm in length and 60 mm in width. Blue tack gel is clung to both cylinder edges to absorb the waves to prevent reflection. A 5-cycle tone burst signal with a central frequency of $f = 20{\text{ kHz}}$ is triggered 15 times per second to prevent interference between two response signals. When the trigger signal reaches the set voltage, the USB oscilloscope begins to collect time domain voltage signals that are proportional to ${v_i}$. The LDV captures the time-domain signals at each measuring point, then moves to the next point. During the movement of the laser head, the excited wave has been attenuated away. This ensures the initial time of the time domain response of each measuring point is the same with respect to the excitation signal. The motorized system is computed to move the LDV with intervals $\Delta x^{\prime}$ and $\Delta y^{\prime}$ of 3 mm by a cursor control script, and 1071 measuring points covering the target area are measured in each scanning. After scanning the area in one direction, the original clamp is adjusted to change the directions of the incident laser, and repeat the measuring steps until collecting signals from the four directions to generate more combinations to verify the results. The unit direction vectors of the four directions ${\boldsymbol{e}_i},{\text{ }}i = 1,{\text{ }}2,{\text{ }}3,{\text{ }}4$ are presented as follows

$$\begin{align} {\boldsymbol{e}_1} & = \left( {\cos{\alpha _1},\cos{\beta _1},\cos{\gamma _1}} \right) = \left( {\frac{1}{2},0,\frac{{\sqrt 3 }}{2}} \right) \nonumber\\[12pt]{\boldsymbol{e}_2} & = \left( {\cos{\alpha _2},\cos{\beta _2},\cos{\gamma _2}} \right) = \left( {0,0,1} \right)\nonumber\\[12pt] {\boldsymbol{e}_3} & = \left( {\cos{\alpha _3},\cos{\beta _3},\cos{\gamma _3}} \right) = \left( { - \frac{1}{2},0,\frac{{\sqrt 3 }}{2}} \right)\nonumber\\[4pt] {\boldsymbol{e}_4} & = \left( {\cos{\alpha _4},\cos{\beta _4},\cos{\gamma _4}} \right) = \left( {0,\frac{1}{2},\frac{{\sqrt 3 }}{2}} \right) .\end{align} \tag{ 6 }$$

According to the 3D wave measurement theory presented in section 2, the 3D velocity field can be solved by selecting measuring signals of three non-coplanar laser beams from the four measurement directions. Only three cases of all the combinations can be used to calculate the 3D velocity fields of the target area. The case $A{\text{ }}\left( {{\boldsymbol{e}_1},{\text{ }}{\boldsymbol{e}_2},{\text{ }}{\boldsymbol{e}_4}} \right)$ is

$$\begin{align}\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {{v_x}\left( {x,y,t} \right)} \\ {{v_y}\left( {x,y,t} \right)} \\ {{v_z}\left( {x,y,t} \right)} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} 2&{ - \sqrt 3 }&0 \\ 0&{\sqrt 3 }&{ - 2} \\ 0&1&0 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{v_1}\left( {x,y,t} \right)} \\ {{v_2}\left( {x,y,t} \right)} \\ {{v_4}\left( {x,y,t} \right)} \end{array}} \right\}} \end{array}.\end{align} \tag{ 7 }$$

The case $B{\text{ }}\left( {{e_1},{\text{ }}{e_3},{\text{ }}{e_4}} \right){\text{ }}$is

$$\begin{align}\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {{v_x}\left( {x,y,t} \right)} \\ {{v_y}\left( {x,y,t} \right)} \\ {{v_z}\left( {x,y,t} \right)} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0 \\ 1&1&{ - 2} \\ {\sqrt 3 /3}&{\sqrt 3 /3}&0 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{v_1}\left( {x,y,t} \right)} \\ {{v_3}\left( {x,y,t} \right)} \\ {{v_4}\left( {x,y,t} \right)} \end{array}} \right\}} \end{array}.\end{align} \tag{ 8 }$$

The case $C{\text{ }}\left( {{e_2},{\text{ }}{e_3},{\text{ }}{e_4}} \right)$ is

$$\begin{align}\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {{v_x}\left( {x,y,t} \right)} \\ {{v_y}\left( {x,y,t} \right)} \\ {{v_z}\left( {x,y,t} \right)} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} {\sqrt 3 }&{ - 2}&0 \\ {\sqrt 3 }&0&{ - 2} \\ 1&0&0 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{v_2}\left( {x,y,t} \right)} \\ {{v_3}\left( {x,y,t} \right)} \\ {{v_4}\left( {x,y,t} \right)} \end{array}} \right\}} \end{array}.\end{align} \tag{ 9 }$$

The three cases work independently and their results can be used to compare with each other to verify accuracy. The impact of laser measurement error on the orthogonal velocities ${v_x}$, ${v_y}$ and ${v_z}$ can be understood based on above formulas. Taking Case A as an example, the error estimates of ${v_x}$, ${v_y}$ and ${v_z}$ can be expressed as

$$\begin{equation}\begin{array}{*{20}{c}} {\left\{ {\begin{array}{*{20}{c}} {{{{\Delta }}_{x}}} \\ {{{{\Delta }}_{y}}} \\ {{{{\Delta }}_{z}}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} 2&{\sqrt 3 }&0 \\ 0&{\sqrt 3 }&2 \\ 0&1&0 \end{array}} \right]{\text{ }}\left\{ {\begin{array}{*{20}{c}} {{{{\Delta }}_1}} \\ {{{{\Delta }}_2}} \\ {{{{\Delta }}_4}} \end{array}} \right\}} \end{array}\end{equation} \tag{ 10 }$$

where ${{{\Delta }}_{x}}$, ${{{\Delta }}_{y}}$ and ${{{\Delta }}_{z}}$ denote the estimated errors of ${v_x},{\text{ }}{v_y},{\text{ }}{v_z}$, while ${{{\Delta }}_1}$, ${{{\Delta }}_2}$ and ${{{\Delta }}_4}$ denote the measurement error of the laser ${{\boldsymbol{{e}}}_1},{\text{ }}{{\boldsymbol{{e}}}_2},{\text{ }}{{\boldsymbol{{e}}}_4}$. The formula indicates that ${{{\Delta }}_{x}}$ and ${{{\Delta }}_{y}}$ are more sensitive to the measured error of ${{{\Delta }}_2}$, and disturbed by the measurement error of ${{{\Delta }}_1}$ and ${{{\Delta }}_4}$. In addition, the velocity component with a smaller amplitude is more susceptible to the measurement errors.

The data processing procedure is shown in figure 3. The original data collected from the experiment is the time-domain velocity of the spatially scattered points ${v_i}\left( {X,Y,t} \right)$ in four directions described in section 3.1, in which $X\text{, }Y$ represent the discontinuous coordinates. Then, the original signal ${v_i}\left( {X,Y,t} \right)$ is transferred to the continuous field of an arbitrary point ${v_i}\left( {x,y,t} \right)$ in four directions respectively. By equation (4), three sets of measuring cases are chosen to solve the velocity components' field ${v_x}\left( {x,y,t} \right)$, ${v_y}\left( {x,y,t} \right)$, ${v_z}\left( {x,y,t} \right)$, which can be used to solve the wave fields of velocity along any directions in the time domain.

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Figure 4 shows the normalized velocity fields of ${v_z}$ in the time domain solved by case A. Then the components of the aim direction can be obtained by vector composition. For the case of flexural wave, the field of velocity perpendicular to the mid-surface of the shell, $\boldsymbol{v_n}\left( {x,y,t} \right)$ is calculated by:

$$\begin{align} \left\{\begin{array}{*{20}{c}} {{v_{nx}}\left( {x,y,t} \right)} \\ {{v_{ny}}\left( {x,y,t} \right)} \\ {{v_{nz}}\left( {x,y,t} \right)} \end{array}\right\} & = \left[ {{n_x}\left( {x,y} \right){\text{ }}{n_y}\left( {x,y} \right){\text{ }}{n_z}\left( {x,y} \right)} \right]\left\{\begin{array}{*{20}{c}} {{v_x}\left( {x,y,t} \right)} \\ {{v_y}\left( {x,y,t} \right)} \\ {{v_z}\left( {x,y,t} \right)} \end{array}\right\} \left\{\begin{array}{*{20}{c}} {{n_x}\left( {x,y} \right)} \\ {{n_y}\left( {x,y} \right)} \\ {{n_z}\left( {x,y} \right)} \end{array}\right\} \end{align} \tag{ 11 }$$

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where ${n_x}\left( {x,y} \right)$, ${n_y}\left( {x,y} \right)$, ${n_z}\left( {x,y} \right)$ represent the unit normal vectors on the surface.

For the in-plane component, the wave field $\boldsymbol{v_\tau }\left( {x,y,t} \right)$ is calculated by:

$$\begin{equation}\begin{array}{*{20}{c}} {\boldsymbol{v_\tau }\left( {x,y,t} \right) = \boldsymbol{v}\left( {x,y,t} \right) - \boldsymbol{v_n}\left( {x,y,t} \right)} \end{array}.\end{equation} \tag{ 12 }$$

The out-of-plane and in-plane components of wave propagation on the shell in the time domain are presented in figures 5 and 6. The amplitude is normalized with the same scale, and the arrows indicate the amplitude and direction of the vibration. The in-plane components ${v_{\tau x}}$, ${v_{\tau y}}$, ${v_{\tau z}}$ in the time domain are provided in the supplementary material as well. In figure 6(c), the asymmetry of the in-plane velocity ${v_\tau }$ is mainly caused by the velocity components of ${v_x}$, which is totally in plane and also shows asymmetrical in supplementary. The uneven adhesion of the tape amplified ${{{\Delta }}_x}$, thereby enlarging the difference in the wave field on ${v_\tau }$. In contrast, the out-of-plane velocity ${v_n}$ is not sensitive to the error induced by the tape, showing a good symmetry. The drawbacks of reflective tapes can be eliminated in the future by painting reflective particles on sample surfaces instead of bonding reflective tapes.

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To verify the accuracy of the measuring method, we analyzed the velocity result at an arbitrary point on the surface in the time and frequency domain. Figures 7(a)–(c) shows the time-domain velocity components of ${v_x}$, ${v_y}$, ${v_z}$ of a point mapped at $\left( {77, - 30} \right)$ on the $xy$ plane. The results for the three cases are represented by three different line types. After the Fourier transformation, the same amplitude-frequency curves shown in figures 7(d)–(f) of each case are obtained and are basically consistent with each other. We calculate the correlation coefficient among the cases for time and frequency domain, and observe that the smallest values appear in the cases between A and C for ${v_x}$ are 0.9527 in time domain, and 0.9732 in frequency domain. This reveals that the time-domain and frequency-domain results are consistent among the cases. In addition, one can see that the deviations on components ${v_x}$ are greater than those on components ${v_y}$ and ${v_z}$. Compared to ${v_y}$ and ${v_z}$, the measurement error of ${v_x}$ is more sensitive to the measurement error as indicated in section 3.1.

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The frequency domain velocity field is crucial in structural strength analysis and wave propagation research, analyzing wave intensity and other characteristics. In this section, we show the experimental (EXP) velocity field in the frequency domain and compare it with the finite element method (FEM) results to verify the validity of the measurement method. Figure 8(a) shows the EXP normalized velocity field ${v_z}$ at $20{\text{ kHz}}$ for case A. For data processing, we performed fast Fourier transform to the velocity signals and then obtained the signals in the frequency domain. The real part of the frequency-domain signal is the velocity component at a certain frequency. The FEM is conducted with frequency-domain analysis in the solid mechanics module of COMSOL Multiphysics. The model has the same size as the EXP one. The material is set as acrylic with Young's modulus $E = 4{\text{ GPa}}$ with a loss modulus of $0.24{\text{ GPa}}$, Poisson's ratio $\nu = 0.37$ and mass density $\rho = 1200{\text{ kg}}\,{{\text{m}}^{ - 3}}$. The normalized FEM velocity field ${v_z}$ at $20{\text{ kHz}}$ is shown in figure 8(b). The normalized amplitude profiles on the midline $y = 0$ are shown in figure 6(c), and the correlation coefficient between the EXP measurement (yellow dots) and the finite element (blue solid line) is 0.9717, which proves that the measurement is in agreement with the finite element results. The experimentally measured wavelength is $\lambda = 20.2 \text{ mm}$, while the wavelength calculated by the finite element is $\lambda = 20.0 \text{ mm}$, with a difference of only 1%. Although the excitation frequency of the point source is at $20{\text{ kHz}}$, the response can still be obtained for frequencies near the excitation frequency. The $18{\text{ kHz}}$ and $22{\text{ kHz}}$ results can be seen in Supplementary, and their correlation coefficients are 0.9853 and 0.9513, respectively, which illustrated the feasibility of the method.

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Figure 9 shows the velocity field in the normal direction of the curved surface at $20{\text{ kHz}}$ obtained in EXP and FEM. The arrows indicate the direction of vibration, and the length of the arrows indicates the vibration magnitude. We can visually see that the EXP velocity field matches well with that of FEM. In the same way, frequency-domain fields in any other direction are able to be calculated according to the specific situation. The in-plane components ${v_{\tau x}}$, ${v_{\tau y}}$, ${v_{\tau z}}$ at a frequency of $20{\text{ kHz}}$ are provided in the supplementary material as well.

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Scatters placed in the path of wave propagation can block and reflect waves, which are applied to design structures to guide and block waves [24, 25]. As shown in figure 10(a), two 304 stainless steel blocks, 10 mm away from the piezo-electric patch, are bonded on the inner surface of the cylindrical shell. Shown in figure 10(b), there is an angle of $\pm 10^\circ$ between the z axis and each of these two steel blocks with size $2.7 \times 15 \times 10{\text{ mm}}$. The target area covers a range of $60 \times 60{\text{ }}\,{\text{mm}}$ mapped on the $xy$ plane, and we use lasers in direction ${L_1},{\text{ }}{L_3},{\text{ }}{L_4}$ to scanned the area respectively, which correspond to Case B. The measured spatial resolution is set as 3 mm. The excitation of the signal and the reflection-absorption boundary remain in their original settings.

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The measured 3D velocities in the frequency domain are obtained after interpolation and matrix operation in equation (8). The EXP and simulated velocity fields in the normal direction are shown in figure 10(c), which agree with each other. Further, in figure 10(d) we obtain the normalized normal velocities on the profiles $y = 0$ with blocks (red) and without blocks (black). The correlation coefficient for the structure with scatters between the EXP measurement (red dots) and the finite element analysis (red line) is 0.8003. Great agreements can be seen between the simulated and EXP results, showing that the blocks magnify the amplitude of the wave between the blocks. In figure 10(e), we also provide a three-dimensional image by EXP based on the direction and magnitude of the velocity on the cylinder. The wave fields of the three orthogonal components in the experiment and the whole wave field scattered by the two blocks through simulation are offered in the supplementary materials.