D it can be: c2 = 0 = c2 , T2 ux = 0.(14)In this case, there
D it really is: c2 = 0 = c2 , T2 ux = 0.(14)Within this case, there’s only a y- or z-axis disturbance, as well as the displacement resolution is: ux = uz = 0, uy = ux (x)eit , uy = Bexp(-ix or ux = uy = 0, uz = uz (x)eit , uz = Cexp(-ix ). (16) ), (15)To get a homogeneous, isotropic, and Compound 48/80 Protocol infinite elastic medium, the results show that the propagation mode of any plane wave is either a longitudinal wave (cL ) or perhaps a transverse wave (cT ). 2.3. Wave Equation in Viscoelastic Media Based on the correspondence principle of uncomplicated harmonic waves, the option of plane simple harmonics in infinite viscoelastic media might be obtained in the corresponding option in elastic media [34]. In other words, replacing the constant from the elastic medium in Equation (13) using the complex function on the viscoelastic medium, the longitudinal wave option is: uy = uz = 0, ux (x, t) = ux (x)eit , ux = Aexp(-ix ). (i ) two(i ) (17)Sensors 2021, 21,8 ofSimilarly, the transverse wave solution is: ux = uz = 0, uy (x,t) = uy (x)eit , uy = Bexp(-ix or ux = uy = 0, uz (x,t) = uz (x)eit , uz = Cexp(-ix ). (i ) (19) ), (i ) (18)For a remedy in viscoelastic media, although only the parameters and of the remedy for elastic media are replaced by (i ) and (i ), the propagation Tasisulam Autophagy traits of ultrasonic waves are significantly various. We receive the following: 1. The velocities of longitudinal and transverse waves in viscoelastic media are denoted as cLv and cTv , respectively. We receive: cLv = Re (i ) 2(i ) , cTv = Re (i ) ,(20)2.exactly where (i ), (i ), cLv , and cTv are functions from the angular frequency . For that reason, when ultrasonic waves propagate in viscoelastic media, frequency dispersion will occur. In a perfect elastic medium, the plane wave is not attenuated, even though in a viscoelastic medium, the plane wave attenuates as the propagation distance increases. The attenuation coefficients of the longitudinal and transverse wave are denoted as L and T , respectively, and we’ve: L = -Im (i ) 2(i ) , (21)T = -Im, (i ) exactly where Re represents the genuine part of the complicated quantity, and Im represents the imaginary a part of the complex number. The ultrasonic attenuation in viscoelastic media is higher than that in elastic media. The major lead to is that the solutions on the wave equation in viscoelastic media are functions with the angular frequency . To acquire the identical detection resolution as that in elastic media, a four-laminated ultrasonic transducer using a frequency of 1 MHz was proposed. two.four. Structure of Laminated Transducer The structure in the four-laminated transducer is presented in Figure 1. It’s composed of a case, epoxy potting, backing material, piezoelectric wafers, a matching layer, a copper sheet, a ground wire, a signal wire, along with a connector. The case protects the internal components in the transducer. The epoxy potting acts as insulation between the case and the internal components. The backing material not merely controls the vibration of the piezoelectric wafers but also absorbs the sound waves emitted by the piezoelectric wafersSensors 2021, 21,9 ofbackward as significantly as possible. With a thickness of half the wavelength, the piezoelectric wafers convert from electrical energy to acoustic power and vice versa. The matching layer is utilized to ease the impedance matching amongst the piezoelectric wafers along with the detected object, and also the thickness is really a quarter on the wavelength. The copper sheet may be the typical electrode in the piezoelectric wafers connected in series. To transmit el.