# A Through-Transmission Ultrasonic Method for the Detection of Ferrite Tile Defects
## 2 Principle of Defect Detection by Liquid-Immersed Ultrasonic Transmission Method
The wave equation of sound wave propagation is \(\
abla^{2}p+(\frac{\omega}{c})^{2}p=0\), then the far-field scattered sound pressure at a distance of r from the defect can be calculated by Equation (3) where j is the imaginary number unit, and \(R(\theta)\) is the directionality function of the scattered sound field [23; 24]:  
\[p_{i}(r,\theta,t)=-p_{0}a\frac{e^{j(\omega t-kr)}}{r}R(\theta) \tag{3}\]  
The size of the \(R(\theta)\) function is determined by the defect size parameter ka, which is a dimensionless parameter characterizing the defect size relative to the wavelength, expressed as shown in Equation (4) where \(P_{l}(x)\) is the Legendre polynomial:  
\[R(\theta)=\frac{1}{ka}\sum_{l=0}^{\infty}b_{l}e^{j\frac{l+1}{2}\pi}P_{l}( \cos\theta) \tag{4}\]  
Figure 1: Focusing liquid immersion ultrasonic transmission detection method; (**a**) system diagram (**b**) the basis for judging defects. \[b_{l}=-\frac{B_{l}}{p_{0}}=(-j)^{l}(2l+1)\frac{\frac{d[j_{l}(k_{l})]}{d(k_{l})}}{ \frac{d[h_{l}^{(2)}(k_{l})]}{d(k_{l})}} \tag{5}\]  
Therefore, in an ideal isotropic medium, the sound pressure \(p_{s}\) received by the probe is approximately:  
\[p_{s}(r,\theta,t)\approx\pi\Big{(}R_{r}^{2}-aR(\theta)\Big{)}\frac{e^{j(\omega t -kr)}}{r} \tag{6}\]  
From the formula: with the increase of the defect size, the scattered acoustic field acoustic wave sound pressure increases. Therefore, defects can be identified from the signal energy when the transmitted signal energy appears to be significantly weakened. The signal energy is defined as:  
\[\mathrm{E}=\int P_{s}^{2}dt=\sum A_{i}^{2} \tag{7}\]  
The signal energy is the integral of the square of the sound pressure received by the probe, and the sampling in practice is the amplitude \(A_{i}\) of the sampling point of the transmitted signal. When an ultrasonic wave encounters the interface between the medium transition in defects and samples, interactions such as attenuation and diffraction occur, which will further reduce the signal energy of the transmitted acoustic wave propagation. Therefore, it can be concluded that when the cross-sectional area of the defect is larger, the sound pressure of the ultrasonic transmitted signal is smaller. By comparing the energy of the acoustic signal, the defects and their sizes in the sample can be detected. Based on the above conclusions, a device for detecting internal and external defects of magnetic tile samples by immersion focused transmission method is developed in this paper. Its characteristics include: the ultrasonic transmission method using one end to emit a specific frequency of ultrasonic waves, the other end to receive the corresponding sound waves, according to the waveform diagram to judge the presence and size of defects; the transmitting probe, the sample under test, and the receiving probe are immersed in the coupler to reduce the loss of ultrasonic waves between the air and the interface of the workpiece under test. This will be beneficial to the acquisition signal processing of transmitted sound waves, including filtering of transmitted wave signals in the time-frequency domain, peak extraction and so on.