## Finite difference time domain simulation

Additional acoustic properties are defined with respect to plane wavefronts, that is, perturbations in the particle displacement fields u(x,t)u(x,t) propagating with a speed of sound c along planes of constant phase defined by the wave normaln, and with a particle vibration direction defined by the polarizationp:equation(2)up=ppexp iω[t-nqxq/c]-αnqxq ,up=ppexp iω[t-nqxq/c]-αnqxq ,where hereafter summation is applied over repeated indices (Einstein’s notation). Given a direction n, c and p are determined by the elastic properties Cpq and ρ [18], and to a lesser degree by the transmitter frequency ω=2πfω=2πf [1]. The material attenuation coefficient αα describes the damping of the wavefront due to scattering and absorption, and is associated to the complex-valued viscoelastic matrix Cpq∗=Cpq(1+itanδpq), with tanδpqtanδpq the loss tangent tensor [31]. Rewriting Eq. (2) as ppexp iω[t-nqxq/Λ∗] ppexp iω[t-nqxq/Λ∗] and recalling C∗=ρ(Λ∗)2C∗=ρ(Λ∗)2 along the material axes L, R, T [18] it Veliparib follows that C∗≈ρc2(1+i2αc/ω)C∗≈ρc2(1+i2αc/ω) and:equation(3)α=π(f/c)tanδα=π(f/c)tanδMaterial attenuation values for wood are very scarce at ultrasonic frequencies, due to the practical difficulties to isolate αα from other transmission loss mechanisms (coupling, diffraction, interference, etc.). Table 3 compiles available αα and cc data, together with own ACU measurements, from which tanδtanδ are derived with Eq. (3). Despite the large anisotropy and variability of αα (2–25 dB cm−1), the loss tangent tanδtanδ is remarkably invariant (0.09–0.13) for longitudinal wave propagation in cross-grain (RT) planes, for the broad frequency range 100–1000 kHz and even for different wood species. Based on this empirical observation, a constant tanδtanδ was adopted for the modeling of timber cross-grain sections, with tanδpq≈tanδ=0.1tanδpq≈tanδ=0.1. This implies a constant attenuation per wavelength factor αλαλ, with λ=c/fλ=c/f, which is characteristic of hysteretic damping, and leads to linear frequency attenuation laws. Assuming a constant tanδtanδ significantly simplifies the modeling, since if the anisotropic αα is known in one material direction, values in every other direction can be calculated from Cpq and ρ.