## 2 Theoretical Approach
### Piezoelasticity under the Influence of Strain and Temperature Variations
This UT transforms the electrical signal pulse into an ultrasonic wave propagating through the materials. After passing through this entire region, the receiving UT will receive this ultrasonic wave and transform it into an electrical signal [10; 49]. The amount of energy transmitted between the transducers depends on several factors, such as the shape of the excitation signal and the excitation frequency [48; 49]. Energy transmission is improved as the transducer is excited close to its resonant frequency. In the case in question, the two transducers will be excited by consecutive sinusoidal pulses that sweep in frequency through a previously determined window in Hz, according to the resonant frequency of interest. The signals emitted by the transducers varying in frequency are called mirroring parameters S11 and S22, which, in the case of this research, are excited with an interval of 3 ms from one to the other. This research involves a complex analysis, allowing for examining the maximum energy through the valley of mirroring parameters S11 or S22 in power transmission and communication systems between UTs. The valley of curves S11 and S22 is where the highest energy transmission occurs. This value is also the value where the transition impedance is maximized. The analysis of parameters S21 and S12 allows for the analysis of the maximum energy transmitter through the valley of mirroring parameters S11 or S22 in power transmission and communication systems between UTs, allowing a quantification of the energy transmission through them. It is necessary to understand the communication between the transmitter and the receiver. This communication can be characterized by the parameter S21 when comparing the response of UT1 as a transmitter and UT2 as a receiver or by S12 when UT2 is the transmitter and UT1 is the receiver. For this reason, analyzing the maximum energy through the valley of mirroring parameters S11 or S22 in power transmission and communication systems between ultrasonic transducers is important. This transducer-receiver relationship is given by the logarithmic ratio between the Fourier transform of the voltage of the UT receiver (V2) and the Fourier transform of the voltage of the UT transmitter (V1). This relationship is expressed in Equation (2). \[S21=20\:log\left(\frac{\Re[V2(t)]}{\Re[V1(t)]}\right) \tag{2}\]  
Figure 1: Illustration of the propagation of acoustic waves through the pitch-catch technique for monolayers and multilayer of materials. (**a**) Conf.1, where the ultrasonic transducers (UTs) are on opposite sides of the same stainless-steel plate. In this configuration, the ultrasonic wave travels only through stainless-steel. (**b**) Conf.2, in this case, the pair of ultrasonic transducers are on different stainless-steel plates, but they are also aligned horizontally, and there is fluid between them. Signals S11, S22, S21, and S12 are provided by equipment called vector network analyzers (VNAs), whose properties and characteristics are described in Section 3. External factors such as temperature variation, mechanical deformation, and transducer alignment directly influence the ultrasonic wave. The temperature variation of the environment where the system is installed directly affects the speed of wave propagation [35]. It has emphasis because it changes on a large scale in open and inhospitable environments and significantly varies the wavelength [20]. Generally, the effect of temperature is measured and empirically removed. However, certain consequences of the temperature variations are not considered when doing this. For example, the UT can compress or lengthen the response of the signal or even change the appearance of the signal [50]. The advantage is that the longitudinal wave velocity can be related as expressed in Equation (3) [35].