# Dynamic Viscosity and Transverse Ultrasonic Attenuation of Engineering Materials
## 1 Introduction  
Dynamic viscosity and attenuation of elastic waves have been studied for a long time in geophysics, ultrasonics, and polymer physics [1; 2; 3; 4]. The frequency range of interest is below 10 Hz in seismology, above 20 kHz in ultrasonics, and from 20 mHz to 20 MHz in polymer physics. Dynamic mechanical analysis methods determine the storage (or elastic), loss (or viscous), and complex (E*) moduli, as a function of frequency, temperature, or time [5]. These are especially useful for polymers and fiber reinforced plastics. Geophysicists have used attenuation behavior as a tool to examine the interior of the earth and recently produced Q tomographs, globally mapping the areas of high attenuation [6]. Here, Q is a measure of attenuation and is called the quality factor with Q\({}^{-1}\) = \(\Delta\)E/2\(\pi\)E\({}_{\rm m}\), where \(\Delta\)E is energy loss and E\({}_{\rm m}\) is the maximum energy stored per cycle [1]. In ultrasonics, damping factor \(\eta\) (=Q\({}^{-1}\)) and attenuation coefficient \(\alpha\) (=\(\pi\eta\)/\(\Lambda\) with wavelength \(\lambda\)) are more commonly used. [PERSON] [1] showed that \(\eta\) is also related to the viscosity (\(\eta_{\rm KV}\)) term in the Kelvin-Voigt model of viscoelastic solids as \(\eta\) = 2\(\pi\)/\(\eta_{\rm KV}\)/E\({}^{\prime}\), with frequency \(f\) and the elastic modulus E\({}^{\prime}\). Thus, ultrasonic attenuation induced by damping can be attributed to dynamic viscosity. Ultrasonic waves are widely used as non-destructive evaluation (NDE) methods to achieve the quality assurance and failure prevention in many industries [3; 7; 8]. One approach is ultrasonic testing (UT) that uses active interrogation of target components and the other is acoustic emission (AE) that relies primarily on \"acoustic\" signals emitted from the target under stress. Both are major contributors to the field of structural health monitoring (SHM) [9; 10]. In UT and AE applications, attenuation parameters have more passive roles than in seismology, that is, high attenuation often produces poor signal-to-noise ratio when one seeks a high spatial resolution by using higher frequency testing. This requires the knowledge of ultrasonic attenuation coefficients in designing an optimal test strategy. In many SHM applications, guided ultrasonic waves are useful and their attenuation coefficients are related to those of longitudinal and transverse (or shear) waves [11]. However, past studies on experimental determination of attenuation coefficients were limited, especially for transverse wave attenuation [12]. The dearth of transverse wave attenuation is likely due to the lack of practical applications requiring direct transverse wave measurements in UT or AE NDE fields [7; 8]. In geophysics, various wave modes are often studied together and the attenuation coefficients for both the longitudinal and transverse modes can be found for rocks in the literature (wave modes being referred to as P and S for compressional and shear). This reflects the importance of seismic S waves since these produce the strongest shaking of the buildings and structures. With the dynamic mechanical analysis for polymers and FRP, the axial and torsional moduli correspond to longitudinal and transverse modes. For the longitudinal wave attenuation in the MHz range, a recent work measured and provided the attenuation coefficients from 348 tests [13]. The present work is to follow this previous study by first establishing a reliable method for getting transverse wave attenuation coefficients and to fill the extant gap by obtaining them for representative engineering materials. While the transvers \(\times\) mode testing was not conducted for all the samples used in [13], more than 270 test results will be presented below. Of the total, 50 tests were for additional longitudinal attenuation testing. The attenuation coefficients for longitudinal and transverse waves can be compared directly as identical samples were used for both tests.