# A Through-Transmission Ultrasonic Method for the Detection of Ferrite Tile Defects
## 2 Principle of Defect Detection by Liquid-Immersed Ultrasonic Transmission Method  
The ultrasonic transmission method is a detection method developed based on the loss of ultrasonic waves in the medium, including the phenomena of absorption of sound waves, dissipation, scattering at interfaces, and attenuation of the diffusion of the sound beam. During the measurement, a pair of transmitting probes and receiving transducers are placed on both sides of the sample to be measured. When a piezoelectric crystal inside the transmitting probe is excited by a pulsed voltage, pulsed ultrasonic waves propagate through the sample. A focusing transducer was employed in the transmitting probe for the focus of the ultrasonic beam. This is achieved by applying an acoustic lens, which increases the sound intensity inside the sample and therefore the signal-to-noise ratio. When ultrasound is applied to the non-destructive testing of internal defects, the main consideration is the scattering and attenuation of the defects encountered by the ultrasound waves during propagation. As shown in Figure 1, ultrasonic waves are emitted from the transmitting probe at an angle perpendicular to the sample surface and propagate into the sample through the coupling agent water. When the ultrasonic wave encounters a defect during the propagation of the sample under test, a portion of the ultrasonic wave will be lost at the junction of the sample and the defect, i.e., between different media, and a certain range of masking will be formed behind the defect. According to the loss of ultrasonic energy, it is then possible to determine the presence and size of defects, including internal and external defects. The ultrasonic signal propagated through the coupling agent can be distinguished from the ultrasonic signal propagated through the solid sample. Because the speed of sound in a solid, such as a ferrite tile, is about 6700 m/s at room temperature of 25 \({}^{\circ}\)C, which is significantly higher than the speed of sound in a liquid, such as water, which is about 1500 m/s, the time of flight (TOF) for propagating the same distance of sound waves in a ferrite tile is only 1/4 of that in a water medium, resulting in the received signals showing a significant difference in the time domain. Based on the cleaning requirements after sample testing and the principle of acoustic impedance matching, the coupling agent was selected as water. Ultrasonic waves can be regarded as linear propagation in an isotropic ideal medium, following the geometric acoustic law. Let the initial sound pressure \(p_{0}\), the received sound pressure is \(P_{i}\), the ultrasonic angular frequency is \(\omega\), the imaginary number unit is \(j\) and the wave number be \(k=2\pi/\lambda\), respectively. The ultrasonic sound pressure \(p\) at the sound range \(r\) is  
\[\mathrm{p}_{i}(r,t)=\frac{p_{0}}{\mathrm{r}}e^{j(\omega t-kr)} \tag{1}\]  
The transmittance \(T\) of ultrasonic waves between a medium with acoustic impedances \(z_{1}\) and \(z_{2}\) is  
\[T=\left(\frac{4z_{1}z_{2}}{z_{1}+z_{2}}\right)^{2} \tag{2}\]  
The acoustic transmittance decreases with the increase of the acoustic impedance difference between the two sides of the dielectric interface. To ensure the sufficient intensity of acoustic transmission into the sample, it is necessary to match the acoustic impedance of the medium on both sides of the interface, so it is necessary to insert the liquid coupler with large acoustic impedance between the probe and the sample. As the thickness of the thin-walled sample is less than the diameter of the probe, the perforated circular hole defect is regarded as a rigid sphere with radius a. The direction of the incident sound wave is the positive direction of \(\theta\).