# Dynamic Viscosity and Transverse Ultrasonic Attenuation of Engineering Materials
## 1 Introduction
For 61% of the tests, the comparison yielded the expected trend of higher transverse wave attenuation, but some materials and material groups exhibited lower (17% of total) or comparable (23%) transverse wave attenuation. The existence of the three trends for seismic wave attenuation was shown analytically in [14]. While the transverse wave attenuation is higher in some rocks, the opposite cases were also reported [1; 14; 15]. The observed attenuation behaviors reported below are discussed in terms of various damping and scattering mechanisms. As was the case in [13], the current understanding of attenuation mechanisms is found inadequate to explain many of the findings. In describing the attenuation behavior of materials, additional terms have to be defined. First, the attenuation coefficient is written for the longitudinal and transverse waves as \(\alpha\) and \(\alpha_{\text{t}}\). These are frequency dependent, and are usually separable between the linear term and power-law terms. The linear parts, often called damping term, of \(\alpha\) and \(\alpha_{\text{t}}\), are written as \(C_{\text{d}}f\) and \(C_{\text{dt}}f\), with frequency \(f\), while the power-law terms as \(C_{\text{n}}f^{\text{n}}\) and \(C_{\text{nt}}f^{\text{n}}\). When damping factors are independent of \(f\), \(C_{\text{d}}\) and \(C_{\text{dt}}\) are unchanged with frequency and we have:  
\[\alpha=C_{\text{d}}f \tag{1}\]  
and  
\[\alpha_{\text{t}}=C_{\text{dt}}f \tag{2}\]  
where \(C_{\text{d}}\) (or \(C_{\text{dt}}\)) is equal to \(\pi\
u\eta\) (or \(\pi\
u_{\text{t}}\eta_{\text{t}}\)) with \(\
u\) (or \(\
u_{\text{t}}\)) being the longitudinal (or transverse) wave velocity and \(\eta\) (or \(\eta_{\text{t}}\)) the longitudinal (or transverse) damping factor. For many materials in the usual ultrasonic frequency range of 20 kHz to 20 MHz, the linear relations are adequate. Of the power-law relations, n = 4 is the most often observed. This arises from Rayleigh scattering and \(\rm C_{4}=C_{R}\) was used in [13]. The corresponding term for the transverse mode is \(\rm C_{RI}\). A combination of the linear and Rayleigh terms becomes the following for the two wave modes:  
\[\alpha=C_{d}f+C_{R}f^{4} \tag{3}\]  
\[\alpha_{t}=C_{dt}f+C_{RI}f^{4} \tag{4}\]  
Equation (3) was called the Mason-McSkimin relation after its first use by [PERSON] and [PERSON] [16]. Here, the same name is also used for Equation (4). Additionally, frequency-squared (\(f^{2}\)) and frequency-cubed (\(f^{3}\)) dependences of \(\alpha\) were observed in [13] and were named as Datta-Kinra and Biwa relations, respectively [17; 18]. In the following, the experimental methods used are described along with the background information in Section 2. Section 3 presents test results of ferrous metals including cast iron, non-ferrous metals, polymers, fiber-reinforced plastics, and ceramics and rocks in five subsections. These are accompanied by discussions for various sub-groups, including comparisons between the longitudinal and transverse attenuation characteristics. Section 4 provides a discussion of the attenuation behavior encompassing different types of materials, followed by the summary and conclusions sections. An Appendix A gives the elastic moduli data of the materials tested in the present study.