Wave Shape

The three-dimensional propagation of waves in a solid is most easily visualised by the Huygens’ Principle (Christiaan Huygens 1629-1681). The Huygens’

Principle states that “all points on a wave front serve as point sources of spherical secondary wavelets. After time δt, the new position of the wavefront will be a surface tangent to the secondary wavelets”. This principle is illustrated in Figure 5.15.

Figure 5.15. Illustration of the Huygens’ Principle.

The illustration in Figure 5.15 is only meant as a visual aid and not as an accurate model to describe the wave propagation in a standing tree measurement; the anisotropy and inhomogeneity of material properties due to knots, compression wood grain angle, variation in moisture content, etc. will never allow for

spherical wave propagation. However, Huygens’ Principle does illustrate how a point-source impulse applied at the top edge of a cross section can spread to form a plane wavefront in a cylinder (neglecting boundary effects from the cylinder walls), e.g. Figure 5.16.

Figure 5.16. Spherical propagation of wavefront in a cylinder.

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0.5D D

Impulse location

t +δt Wavefront Wavelets

t

Zhang et al. (2011) performed a similar experiment to that presented in this chapter, except over a larger range of transit lengths and found that the stress should be evenly distributed after approximately 10 diameters from the impact point. Closer inspection of the stress wave contours in Zhang et al. (2011) near the impact point show a similar distribution to the slowness plots of ultrasound propagation at various angles to the fibre direction in Bucur and Berndt (2001), where slowness is the inverse of speed in s m^-1. The results from Bucur and Berndt (2001) show an elliptical variation in slowness between the longitudinal and tangential planes, with the tangential plane for Douglas fir approximately 3 times slower than the longitudinal. The stress wave contours in Zhang et al.

(2011) are approximately 2.5 times greater in the longitudinal than the radial direction after 100 µs for green (i.e. freshly felled) red pine. The difference in the transmission time between longitudinal and the tangential or radial directions, is due to the anisotropic nature of wood. Modelling the wave shape in a log based on these elliptical (rather than spherical) parameters estimated from Bucur and Berndt (2001) and Zhang et al. (2011), gives a reasonable approximation to the experimental findings in Zhang et al. (2011), shown in Figure 5.17. Figure 5.17 a uses the value obtained from Bucur and Berndt (2001) and Figure 5.17 b uses the value obtained from Zhang et al. (2011).

Figure 5.17. Model of wave shape using an elliptical rather than a spherical propagation shape. a. Value obtained from Bucur and Berndt (2001):

longitudinal velocity 3 times faster than tangential. b. Value obtained from Zhang et al. (2011): longitudinal velocity 2.5 times greater than radial.

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D

D

Figure 5.17a.

Figure 5.17b.

There are two aspects of the stress wave propagation in this experiment that are worth considering: Firstly, the outward propagation of the wave toward the opposite side of the cross section. The difference in size of the quasi-plane wave between the billets is shown in Table 5.5. This ‘flattened’ area of the wave extends more than twice as far into the large billet as the small, with the medium billet falling in between the two. This could be due to differences in stiffness between the billets, with the maximum dynamic MOE for each billet of 16.58 kN mm^-2, 14.51 kN mm^-2 and 13.75 kN mm^-2 for the small, medium and large

billets, respectively. Putting this in terms of wave propagation by elliptical wavelets, differences in MOE could result in variations in the eccentricity of the ellipse, with stiffer material being more elongated in shape and less stiff material more circular. This effect also raises some interesting questions about how quickly the wave would spread outward when it reaches the lower stiffness core;

minimum values of dynamic MOE ranged through 6.7 kN mm^-2, 7.51 kN mm^-2 and 5.05 kN mm^-2 for the small, medium and large billets respectively.

Importantly, the extent of the quasi-plane wave area changes with transit length.

Table 5.5 shows size of the quasi-plane wave for both transit lengths on each billet, with the 1.0 m transit lengths 2.2 – 3 times larger than the 0.5 m transit length. If a quasi-plane wave behaves in a similar manner to a plane wave in that its speed is controlled by the entire area it covers, then the properties of a tree measured during a standing tree TOF test are not simply the area 20 – 30 mm wide, directly between the probes as is generally assumed. In this case the area controlling the speed of the wave would extend approximately 40 mm, 64 mm and 90 mm in the radial and tangential directions for the small, medium and large logs respectively.

The variation in size of the quasi-plane wave combined with the radial variation

Andrews (2003) found a similar result; explaining that the stress wave was highly sensitive to the grain direction. Gerhards (1980) also showed that a stress wave will advance in the direction of the grain rather than normal to the long axis of the test specimen. The bias shown in this experiment is consistent with the left handed spiral grain growth pattern in Sitka spruce (Moore 2011). The implication of this is that the receiver probe in a standing tree TOF test is never aligned with the fastest arrival time as it is placed along the same azimuth as the starter probe. Therefore the accuracy of the instruments will be affected. If the grain angle were constant, this could be accounted for empirically. In this experiment the differences between minimum TOF and values that would be obtained in a standing tree TOF test were 4.6% 9.9% and 8.3% for the small, medium and large billets, respectively. However, even if the difference was consistent, the accurate placement of probes is essential for accuracy as

misplacement will either overestimate or underestimate the TOF. This property of stress waves does raise the potential capability of being able to measure spiral grain on standing trees by finding the minimum TOF time and measuring the angle relative to the impact location.