# Study on Molybdenum-Rhenium Alloy Ultrasonic Resonance Temperature Sensor
## 2 Materials and Methods  
### 2.1. Illtrasonic Guided Wave Resonance Measurement Principle  
When an ultrasonic wave is excited by a magnetostrictive [14] transducer and propagates along a waveguide rod to a resonant region, phenomena such as reflection, superposition, and attenuation occur, forming an ultrasonic resonance echo signal. The signal comprises two components: a reflected wave and an attenuated wave. When the excitation signal frequency aligns with the natural frequency of the resonator, a zero-crossing [15] appears in the echo signal, indicating that the resonator has entered a resonant state and thus is initiating resonance. When the incidence is complete, the energy stored in the resonator is reflected back through the waveguide rod at the intrinsic frequency of the resonator with the energy attenuation [16] equation shown below:  
\[A_{\mathrm{n}}=A_{0}e^{(-\frac{\pi}{Q})} \tag{1}\]  
where \(A_{0}\) represents the initial amplitude of the attenuated echo; \(A_{\mathrm{n}}\) is the amplitude at the n-th cycle; and \(Q\) is the quality factor of the resonator.  
When ultrasonic waves propagate in a waveguide rod, the change in temperature is actually detected by measuring the resonant frequency. The resonator, as the main temperature-sensitive component, is made of materials whose acoustic properties (e.g., speed of sound and modulus of elasticity) change with temperature, which in turn leads to a change in resonant frequency. These principles of ultrasonic guided wave resonance are illustrated in Figure 2.  
Figure 1: Schematic diagram of ultrasonic resonant temperature-sensing system design.  
The relationship between the natural frequency of the resonator and sound velocity can be expressed as follows.  
\[f=\frac{m}{2L}C \tag{2}\]  
where \(f\) represents the natural frequency of the resonator; \(m\) is the mode order of the resonator; \(L\) is the length of the resonator; and \(C\) indicates the propagation velocity of sound waves in solid material.  
When the length and mode of the resonator are constant, the natural frequency is directly related to the velocity at which sound waves travel through the material. There is a distinct functional relationship between the propagation speed of sound waves in a solid medium and the temperature of said medium.  
\[C=\sqrt{\frac{E(T)}{\rho(T)}} \tag{3}\]  
where \(E\) denotes Young's modulus; \(\rho\) represents the density of the solid medium; and \(T\) is the ambient temperature. As the measured ambient temperature increases, the Young's modulus of the waveguide material decreases, while the density experiences a slight reduction, leading to a change in sound velocity.  
Combining the above two equations, it can be seen that the change of ambient temperature causes the change of the Young's modulus and density of the material, which directly affects the propagation speed of the sound wave in the material, and further affects the intrinsic frequency of the resonator. Therefore, by determining the variation rule of resonance frequency at different temperatures, the operating temperature of the measured environment can be deduced. In order to ensure the reliability of the experimental data and the accuracy of the temperature measurement, and to reduce the testing error, it is necessary to carry out repetitive tests under the same experimental conditions, to obtain the variation curve of the resonance frequency with the ambient temperature at different temperatures.