# Role of Piezoelectric Elements in Finding the Mechanical Properties of Solid Industrial Materials
## Using Piezoelectricity to Calculate Solid-Material Properties
### Longitudinal Velocity of Sound  
Longitudinal waves in materials refer to the particles of the material that are moving in the same or maybe opposite in direction with respect to the exciting vibration wave, such as a compression wave. The popular expression of longitudinal wave velocity \(\mathrm{v}(C_{L})\) is \(C_{L}\ =\ \lambda\ \times\ f\) where \(\lambda\) is the wavelength and \(f\) is the frequency or \(C_{L}\ =\ \mathrm{w}\mathrm{k}\) where \(\mathrm{w}\) is the angular frequency and \(\mathrm{k}\) is the propagation constant. Considering that PE is often used as a generator of sonic and ultrasonic waves, many researchers have attempted to study the longitudinal velocity of sound in PE itself. Authors in [65] studied the effect of many different variables, such as thickness (t), volume percentage, density (\(\rho\)), and the modulus of elasticity (E), on the longitudinal velocity of sound (\(C_{L}\)) for ceramic materials with PE (1 MHz to 10 MHz). Lead zirconate titanate Pb[ZrxTi1-x]O\({}_{3}\) (PZT ) using the formula \(C_{L}\ =\ (E_{t}/\rho)^{0.5}\) where \(E_{t}\), as an elastic modulus, was regarded as a function of volume fraction of PZT. This function was calculated by [66; 67], who found the relationship between the volume percentage of PZT and the longitudinal velocity of sound as an inverse relationship. Then, authors in [68] studied the effect of lateral dimensions of un-poled and poled PZT-5H ceramics (15 MHz) and the longitudinal velocity of sound was generated from it. The same result was obtained (an inverse relationship between the dimensions of PE and the longitudinal velocity generated from it) in which the transducer's diameter (un-poled and poled PZT-5H) increased to 14%, and the longitudinal velocity of sound will increase to 1.4% and 0.9% respectively. In the same context, authors in [69] used picoseconds in ultrasonic devices to calculate the longitudinal velocity in PE (111) epilayers of In\({}_{\mathrm{x}}\)Ga(1-x)As. As a sensor, authors in [70] utilized a 300 kHz PE to calculate the longitudinal velocity for the following samples: polycrystalline metals (Al and Cu), single-crystalline alkali-halides (LiF and KCl), poly-crystalline Al\({}_{2}\)O\({}_{3}\), fused silica (SiO\({}_{2}\)), and an invar alloy. The samples were square-shaped plates with a common dimension of approximately 40 mm \(\times\) 40 mm \(\times\) 10 mm except for Cu, which was 35 mm \(\times\) 35 mm \(\times\) 10 mm. To evaluate the velocity of sound in the single crystal YBazu307 under the effect of temperature change, the authors in [71] needed a PE as a generator of short 10 ns to 20 ns pulses of ultrasonic waves (500 MHz bandwidth). Figure 6: Connection of the impact tup. The relationship between the relative velocity \(\Delta v/v\) of sound in a single crystal YBazCu307 to temperature is non-linear. Also, an inverse relationship was found. authors in [72] found a positive relationship between longitudinal and transverse velocity with temperature. The experiment was performed using two PEs, one as an emitter transducer and another as a receiver, for sending and receiving ultrasonic pulse waves (405 kHz), on silica aerogels 10-cm wide where, aerogels undoubtedly come under a class of fascinating materials having high porosity, low density, low thermal conductivity, large surface area, transparency in the visible range, and many more peculiar properties for scientific and industrial uses [68]. They succeeded obtaining a longitudinal wave by cutting piezoelectric crystal in the x-direction, and then they obtained a transverse wave by cutting piezoelectric crystals in the y-direction.