## 3 Leaky Lamb Wave Radiation from a Waveguide Plate with Finite Width  
### Rayleigh-Sommerfeld Integral Model  
Figure 4 presents a radiation aperture area with length \(L\) and width \(W\). A leaky Lamb wave is propagated along the \(+x\) axis and is sequentially radiated into the surrounding liquid (\(z>0\)) from the radiation surface \(S\); the leaky Lamb wave radiation from the radiation surface is limited by \(x_{0}\in[-L/2,L/2]\) in an infinite plate in the \(\pm x\) direction. In fact, leaky Lamb wave radiation from the radiation surface at the plate end is used in the practical waveguide sensor. However, the radiation of the backward leaky Lamb wave reflected from the aperture end is not considered in this study; its effect on the main beam generated by the forward wave is assumed to be negligible based on its different radiation angle and diminished energy. An acoustic pressure at a certain point \(p\) can be defined by the Rayleigh-Sommerfeld integral (RSI) [41]:  
\[P(r,\theta,\rho,t)=-\frac{i\rho\omega}{2\pi}\int_{S}v(x_{0},y_{0},t).\frac{e^{ ik_{l}r^{\prime}}}{r^{\prime}}dS(x_{0},y_{0}), \tag{3}\]  
where \(\rho\) is the liquid density, \(k_{l}\) is the wavenumber in the liquid, and \(v(x_{0},y_{0},t)\) is the velocity distribution on the aperture, given by:  
\[v(x_{0},y_{0},t)=V(y_{0})e^{-ax_{0}}.e^{i(k_{p}x_{0}-\omega t)},\ x_{0}\in[-L/2,L/2] \tag{4}\]  
where \(\alpha\) and \(k_{P}\) are the attenuation coefficient and the wavenumber of the leaky Lamb wave, respectively, and \(V(y_{0})\) is the velocity profile of a certain width mode in the width direction. Based on this integral equation, the RSI model was constructed as a computational approach; this model is also called the Rayleigh-Sommerfeld numerical integration (RSNI) [42]. The RSI model computes the integral equation (Equation (3)) without any assumptions and approximations such as the far-field approximation; therefore, this model is known to have computation results of high accuracy [43]. In addition, RSI can provide the better computational speed compared with SAFE or FEM approaches, especially for the high-frequency model that requires both small wavelengths and an integration time step [44]. The underwater beam profile of the leaky wave radiated from the unbaffled aperture shown in Figure 4 was calculated using the constructed RSI model. The water domain for calculating the beam profile was predetermined as \(90\) mm (\(x\)) \(\times\)\(50\) mm (\(y\)) \(\times\)\(90\) mm (\(z\)) and the total number of grid points was \(300\times 167\times 300\), with \(0.3\) mm grid spacing (about 1/5 times smaller than the wavelength of the leaky wave), and \(0.5\) mm source spacing (about 1/5 times smaller than the wavelength of the leaky Lamb wave). The out-of-plane velocity profiles obtained by the SAFE method were applied to the point sources with the curve-fitting technique. The 3D acoustic field by a certain point source on the aperture  
Figure 4: Coordinate system of radiation aperture area for leaky Lamb wave radiation. was calculated; then, the calculation process was repeated for all point sources. The acoustic field of the leaky wave radiated from the aperture was obtained by integrating the individual calculation results. From the calculated acoustic field, the beam profile was reconstructed by making an envelope for the maximum pressure peaks on the grid points; the beam profile was then normalized by the maximum value in the entire calculation domain. All parameters to calculate the radiation beam profiles of the leaky Lamb wave, such as the phase velocity, were determined on the basis of the dispersion curve results shown in Figure 2a. Other model information is presented in Table 1; the attenuation coefficient of the leaky Lamb wave was adopted from the attenuation dispersion curve in the infinite plate; the attenuation coefficient is strictly different depending on the width mode, but its value at 1.0 MHz is assumed to be equal to that of the fundamental flexural mode in the infinite plate.