1 Introduction

Ocean beaches are littered with pebbles that are constantly exposed to the erosion forces of the surf waves and the sand. The first explanation for the smooth, rounded shape of the pebbles was given by Aristotle [1]. He proposed that the abrasion is more efficient at regions far from the centre of the stone where greater impulses can be more readily delivered. Aristotle himself claimed that spherical shapes are dominant. In the last century, some field and laboratory studies on pebble shapes have been published [2,3,4,5,6,7]. However, the experimental conditions of these studies do not correspond to the situation that forms a pebble in the surf wave. Also, a number of mathematical models for the evolution of pebbles were published [8,9,10,11,12]. But all these mathematical models ignore the physical basis for the formation of the elliptical shape: the frictional abrasions process arising from rotation about the axis of greatest moment of inertia as a consequence of the conservation of angular momentum. Indeed, an observer will immediately notice the rolling motion of the pebble in the water backflow on the beach. The importance of this process was investigated in Ref. [13] with a model and a detailed physical analysis. Unfortunately, in this paper an invalid force expression was used. As a consequence, the claimed convergence of the ellipticity \(\varepsilon ={\left\{1-{b}^{2}/{a}^{2}\right\}}^{1/2}\) to a common equilibrium value does not hold. In a recent paper [14], not only force and curvature but also the contact duration with the sand surface during rotation was taken into account. It was shown that the pebbles neither approach the same ellipticity nor become more spherical or disc-like, but rather the ellipticity \(\varepsilon\) increases.

2 Observation of pebble motion in the surf: from sliding to hopping

In Ref. [13], four different abrasion processes were identified. An incoming wave may lift up pebbles, and when it slows down on the inclined sandy beach, first the larger pebbles and then the smaller pebbles fall to the ground and roll up the beach. When the water flows back to the sea with increasing speed, four different scenarios may occur, which are shown schematically in Fig. 1.

  1. (i)

    In a weak backflow, a pebble with axes \(a>b>c\) laying on its flat side, slides seawards without rolling motion. The abrasion of the c-axis was shown to increase with decreasing \(b/a\)-ratio, and thus elongated pebbles are thinner on average than rounder pebbles—in good agreement with observation [13].

  2. (ii)

    A slightly stronger backflow can cause the pebble to initially rotate around the longer a-axis, since the potential energy for erection of the pebble for rotation around the a-axis, \({W}_{\mathrm{pot},a}=mg\left(b-c\right)\), is less than the potential energy for erecting it for rotation around the c-axis, \({W}_{\mathrm{pot},c}=mg\left(a-c\right)\).

  3. (iii)

    The most interesting case occurs for strong backflow. From the theory of gyroscopes, it follows that for the rotation of a symmetrical body about free axes only the rotation about the axis of the smallest (a-axis) and of the largest (c-axis) moment of inertia are stable. In a medium with friction, such as a water–sand mixture, only the rotation around the axis of the greatest moment of inertia is stable [15]. Therefore, the pebble quickly rises and performs a fairly stable rotation around the c-axis. In contrast to the brief rotation around the a-axis, often 10–20 rotations around the c-axis were observed, and the pebble rolled several metres. This rotation leads to a very effective abrasion process on a narrow surface strip perpendicular to the ab-plane. It was shown recently [14] that this process always leads to an increasing ellipticity \(\varepsilon\).

  4. (iv)

    With stronger surf, the rotation frequency of the pebble increases. When a limiting frequency is exceeded, the rolling movement around the c-axis changes into a hopping movement.

Fig. 1
figure 1

Sliding, rotation and hopping of an elliptical pebble (schematically) and the relevant energies for the transitions between different states

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In this work, the consequences of the hopping movement for the shape of the pebbles are examined and compared with empirical results on a large number of elliptical stones from the beaches of different coasts. It should be noted that the transitions from sliding to rolling around the a-axis, as well as the transition from rolling around the a-axis to rolling around the c-axis, depend on the mass of the pebble and hence on the density \(\rho\) of the mineral. In contrast, the transition from rolling to hopping depends only on the size of the pebble, but not on the density of the mineral. This makes it possible to investigate the influence of hopping on the material removal of the ab-plane, allowing the shape of pebbles from coasts with different surf strength to be compared with one another, even if these pebbles consist of different minerals.

3 Empirical basis: a collection of 1500 elliptic-like pebbles

Over a time of about ten years, about 1500 nearly ellipsoidal pebbles were collected on flat sandy beaches, showing low density of pebbles and thus negligible collisions between them. In this way, it is ensured that the frictional sliding and rotation of the elliptical pebble around the c-axis are the main abrasion process in the surf waves. Only pebbles appearing outwardly isotropic were included. They were collected mostly on the beaches of the Canary Islands, the Cap Verde Islands and along the Turkish south coast between Side and Alanya. A smaller selection was collected on the Baleares, Madeira, Porto Santo and on the North Sea island of Heligoland. In this work, only the pebbles from the Turkish south coast are to be compared with those from Canary and Cap Verde Islands. The reason for this selection is that the surf conditions in the Mediterranean and the Atlantic differ significantly. The Cap Verdes and the Canaries are of volcanic origin, and these pebbles consist of dark grey to black igneous basalt. The pebbles from the beaches of the southern Turkish coast mainly consist of light grey and yellow–brown marble (calcite, Ca[CO3]). Figure 2 shows twelve elliptical pebbles each from the Turkish south coast (calcite) and from the Canary and the Cap Verde islands (basalt), a common sea star and two deep sea urchins.

Fig. 2
figure 2

Twelve elliptical pebbles each from the Turkish south coast (calcite) and from the Canary and Cap Verde islands (basalt), a common sea star (Asterias rubens) and two shells of deep sea urchins (Coeloplereus maillardi)

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Figure 3 shows the number of elliptical pebbles of calcite and basalt, respectively, in the interval \(\Delta \left(b/a\right)=0.02\) as a function of the ratio \(b/a\). For comparison, since the total number of pebbles from calcite \(\left({N}_{\mathrm{calcite}}=750\right)\) is significantly larger than the number of pebbles of basalt \(\left({N}_{\mathrm{basalt}}=466\right)\), different ordinates have been chosen as shown in Fig. 3. The adaption of the frequency distributions with two sets of Gaussian distributions \(f\left(b/a\right)\) and \(g\left(b/a\right)\) shows significant differences for both minerals: the maxima of the Gaussian distributions of \(f\left(b/a\right)\) and \(g\left(b/a\right)\) have slightly different values (calcite: \({x}_{c}=0.760 \pm 0.0033;\) basalt: \({x}_{c}=0.739 \pm 0.0045\)). The difference in the position of the maxima \(\Delta {x}_{c}=0.021\) is about five times greater than the mean uncertainty \(\langle {\sigma }_{c}\rangle =0.0039\) of the Gaussian fits. The shift of the maximum for the basalt pebbles to smaller values of b/a is an indication of an additional erosion of the b-axis by hopping in the stronger surf of the Atlantic.

Fig. 3
figure 3

Frequency distribution for 750 elliptical pebbles of calcite and 466 pebbles of basalt as a function of the axes ratio \(\left(b/a\right)\). Plotted is the number N of pebbles in the interval with \(\Delta \left(b/a\right)=0.02.\) The data were fitted by Gaussian distributions \(f\left(b/a\right)\) for calcite and \(g\left(b/a\right)\) for basalt

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In addition, the half-widths of the distributions differ significantly: the basalt distribution with \({\mathrm{FWHM}}_{\mathrm{bas}}=0.230\) is about 24% broader than the calcite distribution with \({\mathrm{FWHM}}_{\mathrm{cal}}=0.175\). If there are two competing erosion mechanism, one would expect a broader frequency distribution, because in general both mechanisms involve frequency distributions whose maxima lie at different axial ratios. The larger half-widths of the distribution function for the basalt pebbles are an indication that in addition to the abrasion caused by rolling about the c-axis, abrasion due to hopping at higher rotation frequencies plays a greater role for basalt pebbles than for calcite pebbles.

In an earlier work [16], it was shown how the elliptical shape develops from a non-elliptical stone (ovoid) through abrasion during rotation around the c-axis. The formation time

$$\tau =\frac{3 {H}_{R}}{\pi \rho g {a}_{o}}$$

is proportional to the abrasion hardness \({H}_{R}\) and inversely proportional to the density \(\rho\) of the mineral. On comparing stones of basalt and calcite of the same size \({a}_{o}\), the ratio of the formation times depends only on the densities and the abrasion hardness:

$$\frac{{\tau }_{\mathrm{bas}}}{{\tau }_{\mathrm{cal}}}=\frac{{\rho }_{\mathrm{cal}} \cdot {H}_{R,\mathrm{ bas }}}{{\rho }_{\mathrm{bas}}\cdot {H}_{R,\mathrm{ cal }}}=11.9.$$

A rough estimate shows that the total time required for the formation of an elliptical stone from the edged fragment of a mineral, with a hardness such as basalt, is of the order of 1000 years.

This example shows that the formation times also depend on the size of the stones. The mean values of the axes of the 750 pebbles of calcite are \({\left(\langle a\rangle ,\langle b\rangle ,\langle c\rangle \right)}_{\mathrm{cal}}\)= (1.749, 1.316, 0.705) cm and of the 466 pebbles of basalt are \({\left(\langle a\rangle ,\langle b\rangle ,\langle c\rangle \right)}_{\mathrm{bas}}=\left(2.283, 1.629, 0.779\right)\) cm. The mean values of the masses \(m= \left(4\pi /3\right) \rho abc\) are \({\langle m\rangle }_{\mathrm{cal}}=18.7 \mathrm{g}\) and \({\langle m\rangle }_{\mathrm{bas}}=42.8 \mathrm{g}\). The average mass of the basalt stones is therefore more than twice as large as that of the calcite stones.

In a weak backflow of the water, the heavier basalt pebbles are less likely to rotate about the c-axis than the lighter calcite pebbles. They will more often lie on their flat side and glide seaward without rolling. During this grinding process, the c-axis is ground down, so that the c/a-ratio will be reduced. This can be seen by comparing the c/a-ratios of calcite and basalt: the c/a-value of basalt \({\langle c/a\rangle }_{\mathrm{bas}}=0.348\) is significantly smaller than the c/a-value of calcite \({\langle c/a\rangle }_{\mathrm{cal}}=0.403\).

With a strong backflow of the water, the transition from rotation around the c-axis to hopping occurs while maintaining this axis of rotation. Upon impact after hopping, a narrow area perpendicular to the ab-plane is eroded.

For the transition from the rotation around the c-axis to hopping, according to Eq. (2) in Sect. 4.1 the minimum angular velocity \({\omega }_{\mathrm{min}}\) depends only on the length of the a-axis and on the b/a-ratio, but not on the density of the pebble. The independence from the density of the mineral makes possible to compare the shape of elliptical stones from different coasts. For this comparison, the minimum angular velocities for hopping were calculated using Eq. (2) from the mean values of the length of the a-axis and from the mean values of the b/a-ratios for all calcite and basalt stones. The result is \({\langle {\omega }_{\mathrm{min}}\rangle }_{\mathrm{cal}}=35.9 {\mathrm{s}}^{-1}\) for calcite and \({\langle {\omega }_{\mathrm{min}}\rangle }_{\mathrm{bas}}=30.4 {\mathrm{s}}^{-1}\) for basalt. On average, the transition to hopping occurs for basalt stones at lower angular velocities than for calcite stones. The relative difference in the minimum hopping frequencies between calcite and basalt is \({\Delta \nu }_{\mathrm{min}}/\langle {\nu }_{\mathrm{min}}\rangle =16.7\%\).

This experimental finding of the smaller b/a-ratio of the basalt stones compared to the calcite stones is a strong indication for the reduction of the b/a-ratio that is caused by preferential ablation of the b-axis when hopping in the mean stronger surf on the beaches of the Atlantic islands. The theoretical justification is given in the following Sects. 4.1 and 4.2.

4 Abrasion due to hopping

4.1 Trajectories of the base point

The most effective mechanism for the formation of the ab-plane is the rotation around the c-axis. At this rotation, the force is not constant, because the centre of mass of the ellipsoid performs an up-and-down movement. When the stone touches the b-axis, the centre of mass must be accelerated upwards, and this acceleration has to be added to the acceleration due to gravity. If, on the other hand, the stone rolls over the a-axis, the acceleration is directed downwards, and this acceleration must be subtracted from the acceleration due to gravity. The acceleration on the a-axis is decisive for the hopping process and is given by [14]:

$${a}_{\mathrm{CM}}=- a{\omega }^{2}\left(1-{\left(\frac{b}{a}\right)}^{2}\right)$$

From this, one obtains for the force between the sand and the pebble at the a-axis

$${F}_{s,a}=\frac{4\pi }{3}\rho abc\cdot \left\{g-a{\omega }^{2}\left(1-{\left(\frac{b}{a}\right)}^{2}\right)\right\}$$
(1)

The transition from rolling to hopping of the pebble is given by \(g+{a}_{CM}=0\); this is achieved at a minimum angular velocity around the c-axis of

$${\omega }_{\mathrm{min}}=\sqrt{\frac{g/a}{1-{\left(b/a\right)}^{2}}}$$
(2)

This angular velocity depends only on the b/a-ratio and on the axis length a, but not on the density \(\rho\) of the mineral of the pebble.

For the further discussion, some relations are required, which are derived from Fig. 4a. The figure schematically shows the rolling of the pebble around the c-axis, with \(\varphi\) the angle between the a-axis and the radius \(r\) from the centre of mass to the supporting point PS and with \(\left(\omega t\right)\) the angle between the a-axis and the vertical distance \({r}_{\perp }\) from the centre of mass and the point \({P}_{\perp }\). Using the well-known tangent equation for an ellipse, one finds

Fig. 4
figure 4

a Rotation of an elliptical pebble around the c-axis, with \(\varphi\) the angle between the a-axis and \(r\) and \(\left(\omega t\right)\) the angle between the a-axis and \({r}_{\perp }\) (schematic). b Rotation of an elliptical basalt-pebble around the axis of the largest moment of inertia in the draining water at the beach of Sal (Cap Verde). The outline of the pebble, the direction of motion and the direction of rotation are indicated

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$$\mathrm{tan}\alpha ={\left(b/a\right)}^{2}\left(1/\mathrm{tan}\varphi \right) .$$
(3)

From Fig. 4a, the relation \(\mathrm{tan}\alpha \cdot \mathrm{tan}\left(\omega t\right)=1\) can be obtained. With Eq. (3), this gives a simple relation between the angle \(\varphi\) and the angular velocity \(\omega\) of the rotation of the elliptic pebble,

$$\mathrm{tan}\ \varphi = {\left(\frac{b}{a}\right)}^{2}\mathrm{tan}\left(\omega t\right)$$
(4)

which will be used in the following calculations for the hopping motions of the pebbles.

During the rotation about the c-axis, the circumferential velocity is

$${v}_{c}\left(\varphi \right)=r\left(\varphi \right) \omega =a \omega \cdot \frac{b/a}{\sqrt{{\left(b/a\right)}^{2}{cos}^{2}\varphi +{sin}^{2}\varphi }} .$$
(5)

The transition from rolling to hopping will be fulfilled first at the a-axis. There one has \({ v}_{c}=a\omega\), and the vertical component of the velocity of the centre of mass \({v}_{\mathrm{CM},\perp }=0\) vanishes. Without taking into account slipping at the supporting point PS, the horizontal component of this velocity is \({v}_{\mathrm{CM},\parallel }=a\omega\). This gives the parabolic path for the movement of the centre of mass after lifting:

$${\varvec{r}}={\varvec{a}}+\left({\varvec{a}}\times{\varvec{\omega}}\right) t+\frac{{\varvec{g}}}{2}{t}^{2}$$

With \(x\left(t\right)=a\omega \cdot t\) and \(y\left(t\right)=a- \left(g/2\right) {t}^{2},\) one obtains for the vertical component of the centre of mass in a dimensionless representation:

$$\frac{y\left(t\right)}{a}=1- \frac{g}{2 a {\omega }^{2}}\cdot {\left(\frac{x}{a}\right)}^{2}=1- \frac{g}{2 a {\omega }^{2}}\cdot {\left(\omega t\right)}^{2}.$$

For the impact point of the stone, one needs the path of the base point \({y}_{\mathrm{BP}}\left(\omega t\right)\) of the ellipsoid. This can be obtained by subtracting the vertical component of the radius, \({r}_{\perp }\left(\omega t, b/a\right)\):

$$\frac{{y}_{\mathrm{BP}}\left(\omega t\right)}{a}=1- \frac{g}{2 a {\omega }^{2}}\cdot {\left(\omega t\right)}^{2}-\frac{\sqrt{1+{\left(b/a\right)}^{2}{\mathrm{tan}}^{2}\left(\omega t\right)}}{\sqrt{1+{\mathrm{tan}}^{2}\left(\omega t\right)}}$$
(6)

The minimum angular velocity at which the pebble starts to hop can be obtained from the condition:

$$\begin{aligned}\frac{\partial }{\partial \left(\omega t\right)}\left(\frac{{y}_{\mathrm{BP}}}{a}\right)&=0 \text{ for } \left(\omega t\right)\Rightarrow 0\\ \frac{\partial }{\partial \left(\omega t\right)}\left(\frac{{y}_{\mathrm{BP}}}{a}\right)&= - \frac{g}{a{\omega }^{2}}\cdot \left(\omega t\right)-\frac{\left({\left(b/a\right)}^{2}-1\right)\cdot \mathrm{tan}\left(\omega t\right)}{\sqrt{\left(1+{\mathrm{tan}}^{2}\left(\omega t\right)\right)\cdot \left(1+{\left(b/a\right)}^{2}\cdot {\mathrm{tan}}^{2}\left(\omega t\right)\right)}}\equiv 0\\{\omega }_{\mathrm{min}}^{2}&=\frac{g}{a}\cdot {\mathrm{lim}}_{\left(\omega t\right)\Rightarrow 0}\left\{\frac{\sqrt{\left(1+{\mathrm{tan}}^{2}\left(\omega t\right)\right)\cdot \left(1+{\left(b/a\right)}^{2}\cdot {\mathrm{tan}}^{2}\left(\omega t\right)\right)}}{\left(1-{\left(b/a\right)}^{2}\right)\cdot \left(\mathrm{tan}\left(\omega t\right) / \left(\omega t\right)\right)}\right\}\\{ \omega }_{\mathrm{min}}&=\sqrt{g/a}\cdot \frac{1}{\sqrt{1-{\left(b/a\right)}^{2}}}\end{aligned}$$
(7)

The result agrees with Eq. (2). The lifting of an elliptical stone with \(b/a=0.76\) and \(a=2 \mathrm{cm}\) is achieved for an angular velocity \({\omega }_{\mathrm{min}}=34.1 {\mathrm{s}}^{-1}\) which corresponds to a frequency of \({\nu }_{\mathrm{min}}={\omega }_{\mathrm{min}}/2\pi =5.42 \mathrm{Hz}\).

In order to demonstrate the different paths of the centre of mass and the base point of the ellipsoid, these trajectories are shown in Fig. 5 for a typical stone with \(b/a=0.76\) and a frequency of \(\nu =8 \mathrm{Hz}\). At the same time, the location and position of the stone are drawn in four positions between the lift-off and the impact at \({\left(\omega t\right)}_{o}\cong 90^\circ\).

Fig. 5
figure 5

Trajectories of the centre of mass and of the base point for a typical pebble at a frequency of \(8 \mathrm{Hz}\). Location and position of the pebble at four positions between the lift-off and the impact are shown

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Figure 6a shows the trajectories of the base point in a 3D representation as a function of \(\left(\omega t\right)\) and of the frequency \(\left(5 \mathrm{Hz}< \nu <10 \mathrm{Hz}\right)\) for an elliptical pebble with \(a=2 \mathrm{cm}\) and \(b/a=0.76\). The angular and frequency range of hopping is determined by the condition \({y}_{\mathrm{BP}}/a\ge 0 .\) For the critical value of \({\nu }_{\mathrm{min}}=5.42 \mathrm{Hz}\), the trajectory shows an initial horizontal slope. The line of intersection of the \({y}_{\mathrm{BP}}\)-surface with the (0,0)-plane defines the hopping angle \({\left(\omega t\right)}_{o}\), which increases with increasing frequency up to about 107° at \(\nu =10\mathrm{ Hz}\).

Fig. 6
figure 6

a Trajectories of the base point in a 3D representation as a function of \(\left(\omega t\right)\) and of the frequency \(\nu\) for an elliptical pebble with \(b/a=0.76\) and \(a=2 \mathrm{cm}\). b Trajectories of the base point in a 3D representation as a function of \(\left(\omega t\right)\) and of the ratio \(\left(b/a\right)\) at a frequency of \(\nu =8 \mathrm{Hz}\)

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Figure 6b shows the trajectories of the base point in a 3D representation as a function of \(\left(\omega t\right)\) and of \(b/a\) at a frequency of \(\nu =8 \mathrm{Hz}\). For a slim pebble with \(b/a=0.5\), the hopping angle is largest, with \({\left(\omega t\right)}_{o}\cong 110^\circ\), and becomes narrower for rounder pebbles. Hopping is no longer possible for elliptical stones with \(b/a>0.86\) at the rotation frequency of 8 Hz.

To determine the jumping distance of the stones, the angle of impact \({\left(\omega t\right)}_{o}\) was calculated as a function of the rotation frequency. This results from the condition \({y}_{\mathrm{BP}}\left(\omega t\right)/a=0.\)

$$\begin{aligned}0&\equiv 1- \frac{g}{2a{\omega }^{2}}\cdot {\left(\omega t\right)}_{o}^{2}- \frac{\sqrt{1+{\left(b/a\right)}^{2}\cdot {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}}{\sqrt{1+ {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}}\quad \hbox{resp.}\\ \nu &=\frac{1}{2\pi }\sqrt{\frac{g}{2a}} \cdot {\left(\omega t\right)}_{o}\cdot \sqrt{\frac{\sqrt{1+{\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}}{\sqrt{1+{\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}-\sqrt{1+{\left(b/a\right)}^{2}\cdot {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}}}\end{aligned}$$
(8)

Figure 7 shows the hopping angle \({\left(\omega t\right)}_{o}\) as a function of the rotation frequency \(\nu\), obtained by inverting Eq. (8). For \(b/a=0.76\), the starting frequency of hopping is \({\nu }_{\mathrm{min}}=5.42 \mathrm{Hz}\). For the more elongated pebble with \(b/a=0.65\), the starting frequency \({\nu }_{\mathrm{min}}=4.63 \mathrm{Hz}\) is lower, while the rounder pebble with \(b/a=0.87\) requires a higher starting frequency of \({\nu }_{\mathrm{min}}=7.15 \mathrm{Hz}\).

Fig. 7
figure 7

Hopping angle \({\left(\omega t\right)}_{o}\) as a function of the rotation frequency \(\nu\) for three values of b/a

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In all cases, the hopping angle increases very steeply above \({\nu }_{\mathrm{min}}\) with a flatter slope at higher frequencies. This means that, contrary to the intuitive expectation, the hopping angle quickly reaches large values at frequencies just above \({\nu }_{\mathrm{min}}\). For example, for an elliptical pebble with \(b/a=0.76\), the hopping distance of \({\left(\omega t\right)}_{o}=\pi /2\) is already reached for \(\left(\nu -{\nu }_{\mathrm{min}}\right)/{\nu }_{\mathrm{min}}\cong 8.1 \%\). In the angle range \(0\le {\left(\omega t\right)}_{o}\le \pi /4\), the area around the a-axis is removed by the impact while in the angle range \(\pi /4\le {\left(\omega t\right)}_{o}\le 3\pi /4\) the area around the b-axis is removed. As Fig. 7 shows, the frequency range for the removal in the area around the a-axis with \(\Delta \nu =\nu \left(\pi /4\right)- {\nu }_{\mathrm{min}}\) is relatively narrow, so that a preferred removal in the area of the b-axis can also be expected when jumping.

4.2 Material removal on impact of the pebble

If the elliptical pebble rolls around the c-axis, the pressure at the contact surface and the slippage between the pebble and the sandy beach determine the amount of erosion. In the case of bouncing, the kinetic energy upon impact determines the amount of abrasion.

To calculate the material removal of the pebble, the vertical speed \({v}_{\mathrm{BP}}\) of the base point is derived from Eq. (6) by \(\mathrm{d}{y}_{\mathrm{BP}}/\mathrm{d}t\):

$$\frac{{v}_{\mathrm{BP}}}{a \omega }=-\frac{g}{a {\omega }^{2}}\cdot \left(\omega t\right)+ \frac{\left(1-{\left(b/a\right)}^{2}\right) \cdot \mathrm{tan}\left(\omega t\right)}{\sqrt{1+{\left(b/a\right)}^{2}{\mathrm{tan}}^{2}\left(\omega t\right)} \cdot \sqrt{1+{\mathrm{tan}}^{2}\left(\omega t\right)}}$$
(9)

Figure 8a shows the vertical speed \({v}_{BP}\) in a 3D representation as a function of \(\left(\omega t\right)\) and of \(b/a\) for an elliptical pebble with \(a=2 \mathrm{cm}\) at a rotation frequency \(\nu =8 \mathrm{Hz}\). The vertical speed \({v}_{BP}\) is positive in the ascending part of the trajectory of the base point \({y}_{\mathrm{BP}}\) and becomes zero at the maximum. Therefore, the line of intersection of the \({v}_{\mathrm{BP}}\)-surface with the (0,0)-plane defines the dependence of \({y}_{\mathrm{BP},\mathrm{max}}\left(\left(\omega t\right),b/a\right)\).

Fig. 8
figure 8

a Vertical velocity \({v}_{\mathrm{BP}}\) of the base point in a 3D representation as a function of \(\left(\omega t\right)\) and of \(b/a\) for an elliptical pebble with \(a=2 \mathrm{cm}\) at a rotation frequency of \(\nu =8 \mathrm{Hz}\). b The square of the impact velocity \({v}_{\mathrm{BP} o}\) in a 3D representation as a function of the impact angle \({\left(\omega t\right)}_{o}\) and of \(b/a\)

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For the kinetic energy of the impact \({W}_{\mathrm{BP}}{\left(\omega t\right)}_{o}= \left(m/2\right){ v}_{\mathrm{BP} o,}^{2}\) the velocity \({v}_{\mathrm{BP} o}\) at the angle of impact \({\left(\omega t\right)}_{o}\) must be calculated. With Eq. (8), one obtains for the impact velocity:

$$\frac{{v}_{\mathrm{BP} o}}{a\omega }=-\frac{2}{{\left(\omega t\right)}_{o}}\cdot \left\{ 1-\sqrt{\frac{1+{\left(b/a\right)}^{2} {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}{1+{\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}} \right\}+ \frac{\left(1-{\left(b/a\right)}^{2}\right) \mathrm{tan}{\left(\omega t\right)}_{o}}{\sqrt{1+{\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}\cdot \sqrt{1+{\left(b/a\right)}^{2}{ \mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}}$$
(10)

Figure 8b shows the square of the impact velocity in a 3D representation as a function of the impact angle \({\left(\omega t\right)}_{o}\) and of \(b/a\) for an elliptical pebble with \(a=2 \mathrm{cm}\). For small values of \(b/a\), this surface shows a pronounced maximum in the range of \({\left(\omega t\right)}_{o}\cong 110^\circ\), which suggest a strong material removal in the angular range near the b-axis.

For the material removal on impact, however, not only the kinetic energy, but also the size of the area of the stone with which it hits the underlying sand is important. When rotating around the c-axis, this area is dependent on the respective curvature at the point of impact. The Gaussian curvature in the ab-plane is \({K}_{a}={a}^{2}/{\left(bc\right)}^{2}\) on the a-axis and \({K}_{b}={b}^{2}/{\left(ac\right)}^{2}\) on the b-axis. The ratio \({K}_{a}/{K}_{b}=3\) is obtained for a typical stone, so that the curvature plays a significant role in the energy density of the impact. The generalization to \({K}_{ab}\left(\varphi \right)\) with

$${r}^{2}\left(\varphi \right)=\frac{{b}^{2}}{{\left(b/a\right)}^{2}\cdot {\mathrm{cos}}^{2}\varphi +{\mathrm{sin}}^{2}\varphi }$$

and with Eq. (4) leads for this curvature as a function of \(\left(\omega t\right)\) to:

$${K}_{ab}\left(\omega t\right)=\frac{1}{{c}^{2}}\cdot {\left(\frac{a}{b}\right)}^{2}\cdot \frac{1+{\left(b/a\right)}^{6}\cdot {\mathrm{tan}}^{2}\left(\omega t\right)}{1+{\left(b/a\right)}^{2}\cdot {\mathrm{tan}}^{2}\left(\omega t\right)}$$
(11)

Figure 9a shows the curvature \({K}_{ab}\) in a 3D representation as a function of \(\left(\omega t\right)\) and of \(b/a\). For small values of \(b/a\), a pronounced minimum occurs in the range of the b-axis. For example, for \(b/a=0.5,\) the ratio of the curvatures at the b- and a-axis is \({K}_{b}/{K}_{a}={\left(b/a\right)}^{4}= 1/16\).

Fig. 9
figure 9

a Curvature \({K}_{ab}\) in a 3D representation as a function of \(\left(\omega t\right)\) and of \(b/a\) for an elliptical pebble with \(a=2 \mathrm{cm}\). b The product \({v}_{BP}^{2}{\left(\omega t\right)}_{o}{K}_{ab}\) as a measure of the energy density at the impact in a 3D representation as a function of \(\left(\omega t\right)\) and of \(b/a\)

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The curvature \({K}_{ab}\) has the dimension of an inverse area. Therefore, the product of the curvature and the square of the impact velocity is a measure of the area density of the energy. This product \({v}_{\mathrm{BP}}^{2}{\left(\omega t\right)}_{o}{ K}_{ab}\) is shown in Fig. 9b in a 3D representation as a function of \({\left(\omega t\right)}_{o}\) and of \(b/a\). The minimum of the curvature at the b-axis causes a weak maximum of the energy density at \({\left(\omega t\right)}_{o}\cong 77^\circ\) and a large maximum at \({\left(\omega t\right)}_{o}\cong 125^\circ\). Both angles are closer to the b-axis than to the a-axis, so that the material removal in the range of the b-axis strongly predominates. For a pebble with \(b/a=0.76\), the impact area on the b-axis is about three times larger than on the a-axis. Nevertheless, because of the very strong \({\left(\omega t\right)}_{o}\)-dependence of \({v}_{\mathrm{BP}}^{2}{\left(\omega t\right)}_{o}\) (Fig. 8b), the energy density on the b-axis and thus the material removal is greater than on the a-axis.

4.3 Torques on jumping pebbles

If the elliptical stone hits the sandy beach in the angle range \(0\le \left(\omega t\right)\le \pi /2\) , a torque \({\varvec{T}}={\varvec{r}}\left(\varphi \right)\times {\varvec{F}}=\mathrm{d}{{\varvec{L}}}_{\omega }/\mathrm{d}t\) acts on it, which increases the angular momentum and the angular velocity, so that larger hop width will occur afterwards. If, however, the angle of incidence exceeds \(\pi /2\), the sign of the torque \({\varvec{T}}\) changes, so that it becomes \(\mathrm{d}{\varvec{L}}/\mathrm{d}t<0\) and the angular momentum decreases. With the force \({{\varvec{F}}}_{{\varvec{B}}{\varvec{P}}}\) at the impact point this torque \({{\varvec{T}}}_{{\varvec{a}}{\varvec{b}}}\) can be calculated:

$${{\varvec{T}}}_{{\varvec{a}}{\varvec{b}}}={\varvec{r}}\times {{\varvec{F}}}_{{\varvec{B}}{\varvec{P}}};\quad {T}_{ab}= r \cdot {F}_{\mathrm{BP}}\cdot \mathrm{sin}\left(\omega t- \varphi \right).$$

Eliminating the angle \(\varphi\) with Eq. (4) gives for the torque

$${T}_{ab}= a \frac{\left(1-{\left(b/a\right)}^{2}\right)\cdot \mathrm{sin}\left(\omega t\right)}{\sqrt{1+{\left(b/a\right)}^{2}{\cdot \mathrm{tan}}^{2}\left(\omega t\right)}}\cdot {F}_{\mathrm{BP}}$$
(12)

The force cannot simply be calculated as a time derivative of the momentum \(\mathrm{d}\left(m{v}_{\mathrm{BP}, o}\right)/\mathrm{d}t\) since the velocity at the point of impact varies discontinuously but the force is certainly proportional to \(\left|{v}_{\mathrm{BP},o}\right|\). Figure 10 shows the torque in dimensionless units \({T}_{ab}/\left(m {a}^{2}{\omega }^{2}\right)\) in a 3D representation as a function of \({\left(\omega t\right)}_{o}\) and of \(b/a\).

Fig. 10
figure 10

Torque \({T}_{ab}/\left(m {a}^{2}{\omega }^{2}\right)\) at the impact as a function of \({\left(\omega t\right)}_{o}\) and of \(b/a\)

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The torque is positive for \({\left(\omega t\right)}_{o}\le 90^\circ\), disappears at the b-axis and is negative for \({\left(\omega t\right)}_{o}\ge 90^\circ\). The asymmetry of the torque surface with respect to the b-axis results from the fact that \(\left|{v}_{\mathrm{BP},o}\right|\) for \({\left(\omega t\right)}_{o}\ge 90^\circ\) is larger than for \({\left(\omega t\right)}_{o}\le 90^\circ\) (Fig. 8b).

In this way, the pebble receives a (smaller) positive angular momentum \(\Delta {L}_{\omega }=\int {T}_{ab}\mathrm{d}t\) when it hits in the area \(0^\circ \le \left(\omega t\right)\le 90^\circ\) and a (larger) negative angular momentum \(- \Delta {L}_{\omega }=\int {T}_{ab}\mathrm{d}t\) when it hits in the area \(90^\circ \le \left(\omega t\right)\le 180^\circ\).

In the considerations so far, it has always been assumed that the hopping of the stone starts when it rolls over the a-axis, because the necessary cut-off frequency is lowest here (Eq. (2)). The hopping distance \({\left(\omega t\right)}_{o}\) depends on the b/a-ratio and on the frequency. In all subsequent impacts, torques should occur that cause changes in the angular momentum \(\Delta {{\varvec{L}}}_{{\varvec{\omega}}}=\int {{\varvec{T}}}_{{\varvec{a}}{\varvec{b}}}\mathrm{d}t\), which change the angular momentum \({{\varvec{L}}}_{{\varvec{\omega}}}=\boldsymbol{ }{I}_{{\varvec{c}}}\cdot{\varvec{\omega}}\) and thus also the angular velocity \(\omega\). However, the experimental study on hopping with a model stone described in the following paragraph shows that the relative changes in angular velocity are only of the order of \(\Delta \omega /\omega \cong 3\cdot {10}^{-2}\). The physical reason for this is that the angular velocity required for hopping and thus the angular momentum is already so large that the torques can only slightly change the angular momenta.

5 Experimental verification with a model stone

It is possible to observe the rotation of the elliptical stone around the c-axis on the sandy beach, but the view in the water–sand mixture is severely restricted (Fig. 4b). In particular, the transition from rolling to hopping can hardly be seen. Therefore, a model in the form of an elliptical cylinder was used for the verification, which rolls on an inclined plane with variable inclination and starts to hop at higher frequency. The cylindrical shape with \(a=3 \mathrm{cm}, b=2 \mathrm{cm}\) and a width of \(d=3 \mathrm{cm}\) guarantees a stable rotation around the d-axis. The material chosen was aluminium with a density \({\rho }_{\mathrm{Al}}=2.7 \mathrm{g}/{\mathrm{cm}}^{3}\) comparable to the density of calcite with \({\rho }_{\mathrm{Cal}}=2.8 \mathrm{g}/{\mathrm{cm}}^{3}\). To prevent slipping on the inclined plane, this was outfitted with a rubber coating.

The measurements were provided by photographing the rotating cylinder with stroboscopic lighting with pulse frequencies \({\nu }_{p}=540 \mathrm{Hz}\) and a pulse length of \(\Delta {\nu }_{p}\cong 15 \mathrm{\mu s}\). At the same time, the movement of the centre of mass is recorded by an LED attached to the axis of rotation.

Measurements of 10 to 15 values of the rotation frequency \({\nu }_{R}=\left(1/2\pi \right)\cdot \left(\Delta \varphi /\Delta {t}_{P}\right)\) per revolution were performed, from which the mean value \(\langle {\nu }_{R}\rangle\) and the standard deviation \({\sigma }_{R}=\sqrt{\langle {\nu }_{R}^{2}\rangle -{\langle {\nu }_{R}\rangle }^{2}}\) were calculated. In addition, in the angular range of hopping, the hopping width \({L}_{H}\) was determined as the distance between successive base points of the parabolic trajectories as a function of the angle \(\delta \left(^\circ \right)\) of inclination. Figure 11a shows the rolling of the elliptical cylinder at a slight incline and Fig. 11b shows the hopping with a greater incline of the plane. The path of the centre of mass recorded by the light trace of the LED shows a cycloid-like shape in the area of rolling and the expected parabolic shape in the area of hopping.

Fig. 11
figure 11

a Rolling of an elliptical cylinder on the inclined plane with an incline of \(\delta =4^\circ .\) b Hopping of an elliptical cylinder on the inclined plane with an incline of \(\delta =12^\circ\)

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Figure 12 shows the rotation frequency \(\langle {\nu }_{R}\rangle\) as a function of angle of inclination \(\delta \left(^\circ \right)\) of the inclined plane. In the rolling area, a linear increase of \(\langle {\nu }_{R}\rangle\) between \(2^\circ \le \delta \le 5.5^\circ\) is observed.

Fig. 12
figure 12

Rotation frequency \(\langle {\nu }_{R}\rangle\) and hopping width \({L}_{H}\) as a function of the angle of inclination

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At angles of inclination \(\delta \ge 5.5^\circ\), in the area of hopping, the frequency of rotation \({\nu }_{R}\cong 4.7 Hz\) is constant. In addition, Fig. 12 shows that the standard deviation \({\sigma }_{R}\) decreases with increasing angle of inclination of the plane—the approximation of constant angular velocity becomes better fulfilled for larger values of \(\delta\). The very small changes in the angular velocity of successive hopping processes show that the torque that occurs only plays a subordinate role. As a consequence of the almost constant angular velocity, the hopping width (\({L}_{H}\cong 8.5 \mathrm{cm})\) is almost independent of the angle of inclination.

The minimum frequency for the lifting of an elliptical stone with \(b/a=2/3\) and \(a=3 \mathrm{cm}\) is \({\nu }_{\mathrm{min}}=3.86 \mathrm{Hz}\) according to Eq. (2). The measured frequency for the elliptical cylinder with \(b/a=0.666\) and \(a=3\mathrm{ cm ~is~ }{ \nu }_{R}\cong 4.7 \mathrm{Hz}\), so it is about 22% larger.

This difference in the hopping frequencies results from the different shapes and their influence on the moments of inertia and the masses of the elliptical cylinder and ellipsoid: The moment of inertia of an elliptical cylinder with respect to the d-axis is

$${I}_{C}=\frac{\pi }{4}\cdot \rho \cdot abd\cdot \left({a}^{2}+{b}^{2}\right).$$

The moment of inertia of an ellipsoid for a rotation around the c-axis is

$${I}_{E}=\frac{4 \pi }{15}\cdot \rho \cdot abc\cdot \left({a}^{2}+{b}^{2}\right).$$

With the width \(d=2c\) the ratio of the moments of inertia \({I}_{C}/{I}_{E}= 15/8\), so the moment of inertia of the elliptical cylinder is significantly greater than that of the ellipsoid.

The potential energy of both bodies on the inclined plane depends on their masses: For the elliptic cylinder \({m}_{C}=\pi \rho abd\), and for the ellipsoid \({m}_{E}=\left(4\pi /3\right) \rho abc\). With \(d=2c\) the ratio of the masses is \({m}_{C}/{m}_{E}= 3/2\). The moments of inertia related to the respective masses

$$\left({I}_{C}/{m}_{C}\right)/\left({I}_{E}/{m}_{E}\right)=5/4=1.25.$$

The minimum frequency of rotation of the elliptical cylinder when hopping should be greater than that of the ellipsoid by this factor. The measured difference of 22% corresponds very closely to this value.

In contrast to the rolling of the cylinder, in which the frequency of the rotation increases linearly, the frequency of the rotation does not change when jumping, even with a steep incline of the plane. The residual very small changes in rotational frequency at successive hops show that the angular momentum transfer must be very small \(\left(\left|\Delta {{\varvec{L}}}_{{\varvec{\omega}}}\right|\ll \left|{{\varvec{L}}}_{{\varvec{\omega}}}\right|\right).\) The question arises as to how this behaviour is compatible with the principle of the conservation of energy.

Considering jumping of the elliptical cylinder, the total energy is made up of the rotational energy related to the momentary axis of rotation, the translational energy and the potential energy of the centre of mass. The axis of rotation is the axis through the centre of mass (d-axis) and the translational energy is determined by the horizontal velocity \({v}_{\parallel }\). The potential energy is given by the height difference \(\Delta h\) between successive points of impact of the parabolic path:

$${W}_{\mathrm{tot}}\left({\omega }_{R}{,v}_{\parallel },\Delta h\right)=\frac{1}{2}\cdot \frac{m}{4}\left({a}^{2}+{b}^{2}\right)\cdot {\omega }_{R}^{2}+ \frac{m}{2}\cdot {v}_{\parallel }^{2}+mg \Delta h .$$

Since, according to Fig. 12, the rotational frequency \({\nu }_{R}\) in the hopping area is independent of the inclination angle \(\delta\) of the inclined plane, the rotational energy is constant. The same applies to the translational energy, since the hopping width \({L}_{H}\) is almost independent of the angle \(\delta\). Therefore, the potential energy \({W}_{\mathrm{pot}}=mg \Delta h\) is completely converted into deformation energy (heat) of the rubber coating of the inclined plane with each impact. This applies to the examined angular range \(5.5^\circ \le \delta \le 10^\circ\), in which the potential energy changes almost by a factor of two.

It should be noted that the conditions on the sandy beach and in the experiment are comparable only to a limited extend. In the experiment, the movement of the elliptical cylinder results from its potential energy at the starting point. On the beach, the movement results from the force exerted on the elliptical stone by the backflow of the water. The weak slope of the sandy beach plays only a minor role compared to the force of the water backflow. The flow velocity \({v}_{w}\) of the water backflow and thus the driving force will generally not be constant. In addition, even with laminar flow of the water, the velocity has a gradient \(\delta {v}_{w}/\delta h\), which influences the forces on the stone. Despite these differences, the experiment on the inclined plane and the observation of the rotating pebble on the sandy beach largely agree on one point:

Due to the rubber pad used to avoid slipping of the pebble on the inclined plane, the impact of the elliptic cylinder is completely inelastic. Likewise of the deformable sand surface, the hopping process on the beach is also largely inelastic, so that a large part of the kinetic energy is converted into heat.

6 Consequences from the experimental observations on the model stone.

As shown in Fig. 11b the crest height \(h\) of the parabola when jumping is not very much greater than the length of the a-axis. When hitting the sandy beach, the potential energy \({W}_{\mathrm{pot}}=mg\left(h-{r}_{\perp }\right)\) is converted into kinetic energy. This energy is given by.

$${W}_{\mathrm{BP}}= mga \left\{ \frac{h}{a}- \sqrt{\frac{1+ {\left(b/a\right)}^{2}{\cdot \mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}{1+ {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}}\right\}.$$

The energy density per area A which causes the material to be removed at the point of impact of the pebble is given by the product of the energy \({W}_{\mathrm{BP}}\) and the curvature \({K}_{ab}\left(\omega t\right)\) (Eq. (11)):

$$\frac{{W}_{\mathrm{BP}}}{A}=\frac{mg a}{{c}^{2}}\cdot \left\{ \frac{h}{a}- \sqrt{\frac{1+ {\left(b/a\right)}^{2}{\cdot \mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}{1+ {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}} \right\}\cdot {\left(\frac{a}{b}\right)}^{2}\cdot \frac{1+{\left(b/a\right)}^{6}\cdot {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}{1+{\left(b/a\right)}^{2}\cdot {\mathrm{tan}}^{2}{\left(\omega t\right)}_{o}}.$$

Figure 13 shows the energy density in a 3D representation as a function of \({\left(\omega t\right)}_{o}\) and of \(h/a\) for an elliptic pebble with \(b/a=0.76.\) For small h/a-ratios, the energy density disappears in the range of the a-axis, because the jumping height is very small. The maxima of the energy densities for small \(h/a\)-ratios (\({W}_{\mathrm{BP}}/A=0.1 mga/{c}^{2})\) are not on the b-axis, but at \({\left(\omega t\right)}_{o}\cong {60}^{\mathrm{o}}\) and \({\left(\omega t\right)}_{o}\cong {120}^{\mathrm{o}}\). As expected, these maxima shift towards the a-axis as h/a increases. Since the impact of the pebbles on the sandy beach is essentially inelastic, the jumping heights are rather low. This favours material removal in a broad angular range around the b-axis at small hopping heights.

Fig. 13
figure 13

Energy density as a function of \({\left(\omega t\right)}_{o}\) and of \(h/a\) for a pebble with \(b/a=0.76\)

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7 Summary

The experimental investigations with the elliptical cylinder on the inclined plane show that the rotation frequency \({\nu }_{R}\) is constant during hopping. The frequency required for hopping to occur is already so high that the changes in the angular momentum (\(\Delta {L}_{\omega }=\int {T}_{ab}\mathrm{d}t)\) due to the torques that occur are very small, so that the angular momentum \({L}_{\omega }={I}_{c}{\omega }_{R}\) remains practically constant.

Overall, the above calculations show that when an elliptical stone jump after rotation, the angular ranges about the b-axis are preferred for impact. More frequent hopping in stronger surf therefore leads to additional erosion of the b-axis and thus to a reduction of the b/a-ratio. These theoretical considerations for the hopping of elliptical pebbles after a rotation around the c-axis confirm the observation that elliptical pebbles from the coasts of the Atlantic have on average a smaller b/a-ratio than elliptical pebbles from the coasts of the Mediterranean (Fig. 3). The surf on the Atlantic is on average stronger than on the Mediterranean and so hopping on the beaches of the Atlantic plays a stronger role in the erosion.

It should be emphasized that the above considerations no longer apply in the case of extremely strong water backflow, because e.g. the assumption of an almost constant rotation speed is certainly no longer valid. However, since these extreme situations are rare, they should only slightly affect the shape of the elliptical stones.