Using a high-speed video camera, we studied oblique collisions of lapilli-size pumice cylinders (with no rotation before impact) on flat pumice targets. Our results show that the rebound angle, the ratios of the components of velocities and the energy loss vary with the impact angle. In particular, in collisions with an average yaw angle approximately equal to zero, we observed relatively larger rebound angles at small and large impact angles and smaller values in between (the angles are measured from the horizontal surfaces of the targets). We observed also that the ratio of the normal components of velocities decreases and the ratio of the horizontal components increases when the impact angle increases. Furthermore, the ratio of the kinetic energy after to that before collisions, in general, decreases when the impact angle increases. Thus, our experiments reveal features that could be useful in modelling pumice-pumice collisions in geophysical flows.
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 In granular flows, the energy dissipated by particle impacts is important. For example, in collisions with a rough ground surface, the conversion of the horizontal translational momentum into the vertical fluctuation momentum affects the flow mobility [Iverson, 1997]. Collisions are important also in dense gas-fluidised beds, where the hydrodynamics is strongly dependent on the amount of energy dissipated in particle-particle encounters [Goldschmidt et al., 2001]. Volcanic multiphase flow models do account for particle interactions [Dobran, 2001] and geophysical flows with pumice fragments can occur, for example, as pyroclastic flows, volcanic conduit flows and eruption jets [Sparks et al., 1997].
 In nature, particles have irregular shapes and their collisions are not necessarily normal to surfaces. Thus, we present here the studies of oblique collisions of pumice cylinders on flat pumice surfaces (cylinders were used to increase the complexity of the particle shape compared to that of a sphere). The purpose of these experiments is to understand the effect of the impact angle on the rebound features and provide an example of their variability that can be useful in modelling pumice-pumice collisions in geophysical flows. The rebound features are the rebound angles, the ratios of the components of velocities and the energy losses. Our results reveal a complexity which suggests that the description of pumice-pumice impacts using a constant coefficient of restitution is an oversimplification.
 Here, we do not refer to the coefficient of restitution also because it applies to collisions where the energy losses are only due to plastic deformations and vibrations. In geophysical flows, the collisions are more complex because of, for example, erosion, shattering and comminution of the rock fragments. The coefficient of restitution e purports to describe the degree of plasticity in collisions of two bodies and is generally defined as the ratio of the final to initial components of the relative velocities normal to the contact surface [Goldsmith, 2001]. This coefficient can be measured by dropping a non-rotating particle onto the horizontal surface of a large rigid body. If the particle has been dropped from the height h1 and rebounds to the height h2, then e is equal to . However, the use of e can lead to paradoxical conclusions in eccentric collisions with friction [Stronge, 1991]. Moreover, e depends on several properties including nature of materials, masses, elastic moduli and radii of curvature at the contact point. It depends also on the vibrational energy of the bodies and consequently on their initial relative normal velocities and shapes [Goldsmith, 1952]. Examples of the effects of different shapes can be found in Chau et al. . In addition, temperature can have important effects in some materials such as pitch [Pochettino, 1914]. This coefficient can also depend on the ratio of the diameter of the impinging spheres to the thickness of the target [Sondergaard et al., 1990].
 The collisions of flat-ended pumice cylinders on flat pumice surfaces were studied using a high-speed video camera at 2000 frames per second. In our experiments we kept the range of impact velocities as small as possible (average 24.88 ± 0.39 m/s), whereas the impact angle varied from 18 to 74° from the horizontal surface of the targets. We used pumice cylinders with a length of 0.89 cm, a diameter of 0.55 cm and an average mass equal to 0.11 ± 0.004 g. Their bubble diameter mode is smaller than 1 mm. These cylinders were drilled from irregularly shaped pumice fragments that were collected in the pyroclastic deposits of the Medicine Lake Volcano in northern California, USA. We blew these cylinders at flat pumice targets using blowpipes (connected to a compressed air bottle) so that their flight trajectories were lying in a plane perpendicular to the sight direction of the high-speed video camera. These cylinders were shot at four flat pumice targets (10 cm long and 6 cm wide) embedded in a concrete slab with mass equal to 1.6 kg that was resting on a table during the experiments. The targets were obtained cutting a large piece of pumice perpendicularly to the stretching direction of the bubbles. Each target was used more than once aiming at unused portions of their surfaces. The impact points were not located on the line connecting the centres of gravity of cylinders and slab. The pumice used for cylinders and targets have the same characteristics.
 In the high-speed movies, the linear average velocities of the cylinders were determined by tracking the positions of their centre of mass in 1 ms time windows immediately before and immediately after the collisions. These movies show that, after impact, the cylinders rotate about a central diametral axis that is perpendicular to the surface of the images, thus the rotational inertia I of the cylinders is [Den Hartog, 1961]:
where m is the mass of the solid cylinder, L is its length and R is the radius of the base. The translational and rotational kinetic energies are equal to 0.5mV2 and 0.5Iω2 respectively, where V is the linear velocity and ω is the angular velocity. We computed also the ratio of the kinetic energy after impact to that before impact:
where K stands for kinetic energy, T for translational, R rotational, A after impact and B before impact. No target motion is discernible in the high-speed movies.
 The symbols for angles and velocity components are illustrated in Figure 1A. In this paper, we always refer to the absolute value of the components of velocities and the impact and rebound angles are always measured from the horizontal surfaces of the targets. The pumice cylinders were at room temperature, whereas the temperature of the pumice targets was higher than ambient (approximately 120°C) because of the lamps used for the high-speed video camera recording. We do not deal with impacts where the cylinders touch the target more than once, break into large pieces or slide visibly. Importantly, because we verified experimentally that the angle between the longitudinal axis of the cylinder and the velocity vector (i.e. the yaw angle at the time of impact) affects the rebound angle, in this paper we study only collisions with yaw angle approximately equal to zero (1.7 ± 0.4°). This means that the cylinders first hit the target with the rear edge of their base (Figure 1B). In our plots, we fitted the data with lines in a least square sense to provide a first approximation of the trends. In this paper, the numbers that follow the symbol ± are the standard errors of the means.
3. Results and Discussion
 The rebound angles are relatively larger at small and large impact angles with smaller values in between (Figure 2A). A similar trend is also obtained with larger cylinders (inset in Figure 2A). Thus, a general conclusion of these experiments is that the impact angle affects the rebound angle. The high-speed movies show that the larger rebound angles at small impact angles are related to the profile of the indentation carved by the cylinders into the targets. When the impact angle decreases with a zero yaw angle, the indentation in the target presents progressively steeper steps in front of the cylinders (Figure 1C) that force the rebounding cylinders increasingly upward. Conversely, at large impact angle, the relative increase of the normal components of velocities results in an increase of the rebound angles (ideally, with flat-ended cylinders and zero yaw angle, we expect β = 90° at α = 90°). We also observed a larger scatter in the data at small impact angles. The scatter in the data can be due to variability in the effect of the indentation profile, which depends on, for example, sharpness and shattering of the edges of the cylinders and inhomogeneous distributions of target properties such as hardness and bubble sizes. The data scatter is also affected by the yaw angle variability. The plot of α + β (Figure 2B) indicates that, in general, the right triangles of the components of the velocities (Figure 1A) are not geometrically similar (same shape but different size), which happens when α = β or α + β = 90° (similar triangles would provide a relationship between α and the components of velocities before and after impact). Although, we can expect that, in nature, the collisions of irregularly shaped clasts are more complex than the abstraction represented by cylinders, the effect of the impact craters on the rebound characteristics of particles with asperities should have general validity. The importance of the impact crater carved at the time of impingement in fragile material like pumice suggests that the rebound characteristics may be relatively independent of pre-collision roughness of the target surfaces. The high-speed movies reveal also that our pumice-pumice collisions produce fine ash. It has been shown that pumice-pumice impacts (for example in buoyant plumes) can generate electrically-charged ash particles which produce electrostatically-bound aggregates [James et al., 2002].
 In our experiments, VNA/VNB decreases and VHA/VHB increases as the impact angle increases (Figure 3). Both ratios present a large range of values and to evaluate their effect, we should consider that when the impact angle increases the normal component of a constant impact velocity increases and the horizontal one decreases. A dependence of the ratio of the normal components of velocity on the incidence angle has been reported also in plaster spheres impacting plaster plates [Chau et al., 2002]. Concerning the ratio of the kinetic energies, its maximum value at each α decreases when the impact angle increases (but larger ranges of values occur at smaller impact angles) showing the consequences of increasingly more normal impacts (Figure 4A). This is the result of a decrease of both the translational and rotational kinetic energies after impact when the impact angle increases. The cylinders did not rotate before the collisions but all of them acquired a clockwise (for configuration in Figure 1A) rotation afterwards. Here, we found that the cylinders tend to rotate slightly less rapidly when the impact angle increases (Figure 4B). In our collisions, less than 60% of the initial kinetic energy is conserved (Figure 4A). During impact, the kinetic energy can be dissipated in the form of frictional heat generation, particle deformation, solid electrification, surface erosion and particle breakage [Fan and Zhu, 1998]. Although we deal only with cylinders that hit the target with the rear edge of their base, other impact geometries are possible. In this case, the dissipation of energy is also affected by which portion of the particles first hits the target, because sharp edges or tips are more readily shattered than other parts [Wong et al., 2000]. Also the impacts of the larger cylinders of the inset in Figure 2A produce plots of the ratios of the velocity components and energy losses versus α with trends similar to those of the smaller cylinders described in this paper.
4. Implications for Volcanic Flows
 The study of pumice collisions can provide fundamental information on energy dissipation in pumice flows as well as on the evolution of particle sizes and shapes. For this reason, the paucity of these studies in the literature is surprising. For example, we suggest that in moving pyroclastic flows, the huge overriding co-ignimbrite ash clouds can be the result of particle collisions and subsequent comminution. The average impact velocity in our experiments (25 m/s) is in the lower portion of the range of possible pyroclastic flow speeds which vary from a few meters per second in the smallest observed events to estimated values of over 200 m/s in the most violent ones [Sparks et al., 1997]. Although the relative velocity between clasts is probably different from the speed of the flows with respect to the ground, at increasingly higher impact velocities, the comminution of pumice is expected to become more important.
 As explained in the introduction and demonstrated by our results, the use of the coefficient of restitution is not a good approximation. Thus, constitutive equations for volcanic flows may need to be developed to account for the appropriate collisional losses of kinetic energy (that, however, vary significantly for example with the impact angle as in Figure 4A) to provide a realistic evaluation of volcanic hazards (such as the run-out distances in the case of pyroclastic flows). Ideally, models should explore the sensitivity of results to the variability of the impact features in fragment collisions. The assumption of equal and constant rebound angles, velocity ratios and energy losses in all collisions of all particles may lead to incorrect descriptions of flow behaviour. The problem is that particle collisions are phenomena of great complexity that depend on a combination of numerous variables.
 Our paper focuses on the effect of the impact angle, but the effect of other properties (such as impact velocity, shape of the fragments, nature of material, temperature etc.) should also be investigated. For example, in silicate glasses, the temperature dependence of the elastic properties is small up to the glass transition, but the fracture strength depends upon thermal and handling history [Bansal and Doremus, 1986]. The shape of the fragments is also an important variable as shown by the effect of the edges of the cylinders and we can expect that the range of irregular shapes available in nature constitutes a significant increase in complexity with respect to experimental oversimplifications represented by cylinders or spheres. This complexity increases even further if we take into consideration, at the same time, the combined effect of all the variables that can affect rebounding rock fragments.
 This work was supported by NSF grant EAR-0207471. We thank Professor W. Goldsmith for his useful suggestions. Dr. S. Lane and Dr. L. Mastin are also thanked for their careful reviews of the paper.