D it can be: c2 = 0 = c2 , T2 ux = 0.(14)Within this case, there
D it is: c2 = 0 = c2 , T2 ux = 0.(14)Within this case, there’s only a y- or z-axis disturbance, and the displacement answer 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 infinite elastic medium, the results show that the propagation mode of any plane wave is either a longitudinal wave (cL ) or even a transverse wave (cT ). two.three. Wave Equation in Viscoelastic Media Based on the correspondence principle of straightforward harmonic waves, the remedy of plane simple harmonics in infinite viscoelastic media might be obtained from the corresponding remedy in elastic media [34]. In other words, replacing the constant from the elastic medium in Equation (13) with all the complicated function on the viscoelastic medium, the longitudinal wave solution is: uy = uz = 0, ux (x, t) = ux (x)eit , ux = Aexp(-ix ). (i ) 2(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 option in viscoelastic media, despite the fact that only the parameters and of your remedy for elastic media are replaced by (i ) and (i ), the propagation qualities of ultrasonic waves are substantially different. We obtain the following: 1. The velocities of longitudinal and transverse waves in viscoelastic media are denoted as cLv and cTv , respectively. We obtain: cLv = Re (i ) two(i ) , cTv = Re (i ) ,(20)2.where (i ), (i ), cLv , and cTv are functions on the angular frequency . Consequently, when ultrasonic waves propagate in viscoelastic media, frequency dispersion will occur. In a perfect elastic medium, the plane wave isn’t attenuated, whilst in a viscoelastic medium, the plane wave attenuates as the propagation distance increases. The attenuation coefficients in the longitudinal and transverse wave are denoted as L and T , respectively, and we have: L = -Im (i ) 2(i ) , (21)T = -Im, (i ) exactly where Re represents the actual part of the complicated quantity, and Im represents the imaginary part of the complicated number. The ultrasonic attenuation in viscoelastic media is higher than that in elastic media. The key result in is that the solutions of the wave equation in viscoelastic media are functions in the angular frequency . To acquire precisely the same detection resolution as that in elastic media, a four-laminated ultrasonic transducer with a frequency of 1 MHz was proposed. two.four. Structure of Laminated Transducer The structure with the four-laminated transducer is presented in Icosabutate In Vitro Figure 1. It is actually Guretolimod Cancer composed of a case, epoxy potting, backing material, piezoelectric wafers, a matching layer, a copper sheet, a ground wire, a signal wire, in addition to a connector. The case protects the internal components on the transducer. The epoxy potting acts as insulation amongst the case plus the internal components. The backing material not simply controls the vibration on the piezoelectric wafers but in addition absorbs the sound waves emitted by the piezoelectric wafersSensors 2021, 21,9 ofbackward as a lot as you possibly can. Using a thickness of half the wavelength, the piezoelectric wafers convert from electrical power to acoustic power and vice versa. The matching layer is utilized to ease the impedance matching among the piezoelectric wafers as well as the detected object, and also the thickness is actually a quarter from the wavelength. The copper sheet is the typical electrode with the piezoelectric wafers connected in series. To transmit el.