Scaling and dynamics of washboard roads

Granular surfaces subjected to forces due to rolling wheels develop ripples above a critical speed. The resulting pattern, known as washboard or corrugated road, is common on dry unpaved roads. We investigated this phenomenon theoretically and experimentally using laboratory-scale apparatus and beds of dry sand. A thick layer of sand on a circular track was forced by a rolling wheel on an arm whose weight and moment of inertia could be varied. We compared the ripples made by the rolling wheel to those made using a simple inclined plow blade. We investigated the dependence of the critical speed on various parameters and described a scaling argument that leads to a dimensionless ratio, analogous to the hydrodynamic Froude number, which controls the instability. This represents the crossover between conservative dynamic forces and dissipative static forces. Above onset wheel-driven ripples move in the direction of motion of the wheel, but plow-driven ripples move in the reverse direction for a narrow range of Froude numbers.

Figure 1

The two versions of the experimental apparatus. The top photo shows the setup in Cambridge, where the table carrying the sand roadbed rotates beneath an arm holding a wheel or plow, which is stationary in the laboratory frame. The bottom photo shows the setup in Lyon, where the roadbed is stationary beneath an arm that rotates. In both experiments the moving parts rotate counterclockwise, as seen from above.

Figure 2

Schematic views of (a) the wheel and (b) the plow blade. (c) shows the geometry of the equivalent physical pendulum. The plow is rigidly fixed to the arm and makes an angle of attack $\alpha$ with respect to the horizontal.

Figure 3

(Color online) Typical space-time plots showing the development of ripples from a flat bed, with red indicating a higher arm position and blue indicating a lower arm position. Both parts show 1000 rotations and have the same scale. The driving direction is to the right. (a) A free wheel just above the critical velocity. (b) The plow with $v$ below the critical velocity for the first 300 rotations and well above critical for the next 700 rotations.

Figure 4

(Color online) Parts (a) and (b) show different segments of a ten day experiment during which the speed was decreased in steps of $0.0021\text{m}{\text{s}}^{-1}$ every 500 rotations. The driving direction is to the right. In part (a), the speed is increasing from 0.684 to $0.691\text{m}{\text{s}}^{-1}$ (Fr: 2.04–2.09), triggering a transition from flat bed to backward traveling ripples. In (b), the speed is decreasing from 0.694 to $0.687\text{m}{\text{s}}^{-1}$ (Fr: 2.10–2.06), triggering a transition from 21 to 20 ripples. Here $M_G=0.0785\text{kg}$, $w=0.120\text{m}$, $\rho =1201\text{kg}{\text{m}}^3$, and the angle of attack $\alpha$ was $36.8°$.

Figure 5

(Color online) The domain of existence of the ripple pattern for the plow as a function of the speed $v$ and the gravitational mass $M_G$. The solid line corresponds to a constant Froude number $\text{Fr}=2.58$. The inset shows $v^2/g$ vs ${\left(M_G/\rho w\right)}^{1/2}$, a scaling for which Eq. (6) predicts the boundary to be a straight line.

Figure 6

(Color online) The critical speed $v_c$ for the onset of the ripple pattern for wheels. Part (a) shows $v_c$ vs $M_G$ for blocked or nonrolling wheels, while (b) shows the same data for freely rolling wheels. The solid line corresponds to a constant Froude number, according to Eq. (11), i.e., to an $M_G^{1/6}$ power law dependence. It is evident that the predicted scaling is not obtained for rolling wheels.

Figure 7

The dependence of the critical velocity $v_c$ on the width $w$ of the plow. The Froude numbers of the three points shown were 1.84, 1.86, and 1.99, and hence the critical Froude number ${\text{Fr}}_{\text{c}}$ remained approximately constant while $w$ was varied. The solid line shows a satisfactory fit to the $w^{1/4}$ scaling expected from Eq. (7).

Figure 8

The dependence of the critical Froude number on the angle of attack of the plow. The angle $\alpha$ is measured with respect to the horizontal (see Fig. 2). At low angles, the transition was very sharp and the error bars lie within the symbols.

Figure 9

The dependence of the critical Froude number on the mean angle $\theta$ that varies with the height of the support.

Figure 10

(Color online) The domain of existence and transitions for forward and backward traveling ripples using the plow. The data and solid line show the transition between backward and forward traveling ripples at $\text{Fr}=3.80$. The dashed line shows the onset of backward ripples from the flat bed at $\text{Fr}=2.58$, as in Fig. 5.

Figure 11

Schematic mechanism for the formation of backward traveling ripples. The arrow shows the direction of motion of the plow. If the rate of erosion by the plow increases with the angle $\beta$ with the surface, then material is preferentially removed from the downward face of the ripple and deposited on the upward face of the following one, giving rise to a net backward ripple motion. During this process, the plow does not leave the surface.

Figure 12

(Color online) The zoo of states for the plow and wheel. Each figure contains 1000 revolutions at constant speed. The range of ripple heights is normalized for each figure, using the same color scale as in Fig. 4. (a) Plow at $0.737\text{m}{\text{s}}^{-1}$. (b) Plow at $0.827\text{m}{\text{s}}^{-1}$. (c) Plow at $0.964\text{m}{\text{s}}^{-1}$. (d) Plow at $1.001\text{m}{\text{s}}^{-1}$. (e) Wheel at $1.166\text{m}{\text{s}}^{-1}$. (f) Wheel at $1.325\text{m}{\text{s}}^{-1}$.