# Review on Structural Health Evaluation with Acoustic Emission
## 2 Signal Attenuation
### Guided Wave Attenuation  
In using AE to examine structures, we typically deal with two-dimensional wave propagation since thick-walled structures--like nuclear pressure vessels and concrete dams--are exceptional cases. For theoretical developments and early experimental results, see [PERSON]'s book from 1967 [20] or numerous other books that followed dealing with wave propagation, e.g., [PERSON] [21]. In most instances, AE signals propagate along the surface as Rayleigh waves or through thin-walled structures as Lamb waves. These waves are collectively known as guided waves. When they spread on two-dimensional structures, signal loss occurs from geometrical spreading (with inverse square root dependence on travel distance, x, or \(1/\surd\)x dependence). Additionally, attenuation is caused from material absorption and scattering with exponential decay, or \(\exp(-a\alpha)\) with attenuation coefficient \(a\). Combined, signal amplitude decreases following \((1/\surd)\cdot\exp(-a\alpha)\). For dispersive Lamb waves, signal loss also arises from dispersion or frequency-dependent wave speed, which spreads vibration energy over a longer period, reducing signal amplitude [13; 20]. When the surfaces are covered with liquids or other damping matters, attenuation occurs from vibration energy leakage. These appear as additional \(a\) values. [PERSON] et al. [22] gave an extension of wave propagation theory to anisotropic plates with dissipation given in terms of quality factors, \(Q\). They presented several examples of attenuation in fiber reinforced composite plates. [PERSON] and [PERSON] [23] in 1957 gave theory and experimental confirmation for Rayleigh wave attenuation. Measured Rayleigh attenuation coefficients (\(a_{\mathrm{R}}\)) for PMMA were nearly linear with frequency and was 55 dB/m at 100 kHz. [PERSON] [20] provided a parametric equation for \(a_{\mathrm{R}}\) and listed three more \(a_{\mathrm{R}}\) values for aluminum (Dural or 2017 alloy), glass and polystyrene (15.4, 73.7, 422 dB/m at 1 MHz, respectively). In 1964, [PERSON] et al. [24] derived theoretical expressions for Lamb wave attenuation coefficients, \(a_{\mathrm{L}}\). These guided wave attenuation coefficients, \(a_{\mathrm{R}}\) and \(a_{\mathrm{L}}\), depend on the attenuation coefficients of longitudinal and transverse bulk waves, or \(a_{\mathrm{P}}\) and \(a_{\mathrm{I}}\). [PERSON] et al. [24] also measured \(a_{\mathrm{L}}\) values for the 0 th and 1 st symmetric and asymmetric modes, or S\({}_{0}\), S\({}_{1}\), A\({}_{0}\), and A\({}_{1}\) modes on low carbon steel plates (C = 0.15%). The measured \(a_{\mathrm{L}}\) values ranged from 3.8 to 5.7 dB/m at 1.3 MHz and were comparable to their longitudinal and shear wave attenuation coefficients, \(a_{\mathrm{P}}\) = 3.7 and \(a_{\mathrm{I}}\) = 3.9 dB/m. At zero thickness limit for S\({}_{0}\), \(a_{\mathrm{L}}\) = \(\surd\)2 \(a_{\mathrm{I}}\). Pressure-vessel and pipeline steels are known for their low bulk wave attenuation of \(a_{\mathrm{P}}\) = 1 to 10 dB/m at 2 MHz as tabulated in [PERSON]'s book [25], which are reduced further at lower frequencies. Such low attenuation values for Al and steels below 10 dB/m were reported by [PERSON] and [PERSON] [26], [PERSON] [27], [PERSON] [28], and [PERSON] [29; 30] among others at frequencies up to 20 MHz. These data were obtained using directly bonded, low-loss quartz transducers. Some of these tests also included \(a_{\mathrm{t}}\) measurements. Thus, guided waves are expected to propagate with low loss when materials possess low \(a_{\mathrm{P}}\) values. Wave attenuation is characterized using several different parameters. In the ultrasonic and AE fields, attenuation coefficient \(a\) is commonly used to represent an exponential decay. We use two units for \(a\); One is dB/m and the other Np/m with 8.686 dB = 1 Np. Np stands for Nepers, a non-dimensional unit, and is useful in numerical computation. This \(a\) is also related to damping (or loss) factor \(\eta\) by \(\eta\) = \(a\lambda/\pi\), where \(\lambda\) is the wave length.