PROCEEDINGS
of the Tomsk Technological Institute
of Emperor Nicholas II.

Vol. 13. 1909. No. 1.

III.

V. A. Obruchev.

On the question of the mode of movement of coarser sediments along the shores of water basins.

With a table of drawings.

1–12.

On the Question

Of the Mode of Movement of Coarser Sediments along the Shores of Water Basins.

(With a table of drawings).

In geological literature up to now little attention has been paid to the very interesting, and in practical respects important, question of the mode of movement of boulders, pebbles, and coarse sand along the shores of seas and lakes. These products of marine abrasion and river erosion cannot remain suspended in standing water and consequently must be displaced by rolling along the bottom. Their movement back and forth under the influence of surf waves advancing parallel to the shoreline, i.e., acting perpendicular to this line, has been studied and described in detail; but the displacement of these coarser sediments along the shore under the influence of waves moving at an oblique or right angle to the shoreline has not been explained.

Let us take the new, expanded and corrected edition of Mushketov’s Physical Geology, which is the most complete of all manuals not only in Russian but also in foreign literature; on p. 576 of volume II we find that the author, speaking of the width of an accretion shore, indicates that it “depends chiefly on the relation between the direction of the advancing waves and the extent of the shore; if the waves run perpendicular to the shore, then the accretion strip becomes wider than in the case when they strike at some sharp angle. If, however, the waves go parallel to the shore, then accretion almost does not occur, and the mineral material moves along the shore until it meets a shore favorable for deposition. Since the extent of the shore and the direction of the waves vary diversely, especially on winding shores, it is obvious that the width of the accretion shore is very inconstant and changes over short distances. Assuming that the waves run onto the shore at an oblique angle, it is obvious that their force upon striking the shore is resolved into two, of which

the surf is perpendicular to the shore, and the other—parallel; the first produces alluviation, and the second moves the sediment along the shore); the more acute the angle, i.e. the closer the waves are to a parallel position, the less will be deposited in a given place and, conversely, the more will be transported, and vice versa.”

Further (p. 577) the author examines the formation of spits (bars, arrows) and points out that the latter may consist of gravel, pebbles, and even boulders.

In describing the deposits left by the sea (p. 594), the author speaks of the deposits of the upper coastal zone (the beach), saying that they are distinguished by the greatest petrographic diversity, depending on the geological composition of the shores) and then examines in more detail the mode of formation of these deposits, where it is clear from the text that here he has in mind the work of a wave running perpendicular to the shore.

From what has been cited it is evident that the author assumes the possibility of the movement even of coarse material along the shore, but the mode of this movement is set forth by him quite unclearly. How is one to understand the statement that the force of an oblique wave is divided into two components—one parallel and one perpendicular to the shore—and that the first produces displacement of material along the shore, and the second—toward the shore? One and the same stone cannot move simultaneously in two directions; therefore one must think that the author imagines the mode of movement in such a way that part of the material moves toward the shore under the influence of one component, while another part is displaced along the shore under the influence of the other component. The question of the distance to which particles can be carried parallel to the shore from the point of their original origin has not been considered by the author at all, and his indication that the composition of beach deposits depends on the petrographic composition of the shore suggests that material foreign to a given shore cannot play an essential role.

In manuals in foreign languages we find almost the same attitude toward the question that interests us. Thus Lapparent**), like Mushketov, speaks of a component parallel to the shore, obtained with an oblique wave and causing a gradual displacement of pebbles along the shore; he gives examples of such displacement from the shores of the Pas-de-Calais, but over comparatively short distances.

*) Italics are mine.

**) Traité de Géologie, Vol. I, p. 244, Paris 1900.

In the very newest course of geology by E. Haug*) it is likewise said of the littoral drift, parallel to the shore, which creates a current that causes the transport of material along the shore; on p. 476 the author indicates that coarse material is transported only a short distance, whereas sand is often moved for a very long time along the shore, carried by a current whose constant direction is due to the predominance of some particular wind.

But, evidently, not all geologists share this view of the possibility of transporting even sand alone over great distances, if one judges from the latest note by P. Bréon (Bréon) in the proceedings of the French Academy of Sciences**). In the vicinity of Berck, on the shore of the Pas-de-Calais, this author studied the composition of the shingle and sand and found that, alongside chalk shingle, there often occur boulders of crystalline schists and eruptive rocks, outcrops of which are absent from the shore for a distance of not less than 250–300 kilometers, all the way to Brittany. The beach sand at Berck likewise consists of minerals borrowed chiefly from analogous rocks.

In trying to determine how the products of the destruction of these rocks could have come to be near Berck, the author remarks that the boulders have often still preserved sharp edges and could have been brought from the shores of Brittany on floating ice; but the transport of the sand cannot be attributed to the latter, and the mode of its movement, in the author’s opinion, remains unexplained.

Meanwhile, this question is explained quite simply if one becomes more closely acquainted with the manner of movement of coarser materials along the shore, a matter so unclearly set forth in manuals of geology. This lack of clarity prompts me to publish observations made several years ago, and the theoretical considerations based upon them.

Being in Alushta in the summer of 1902 and in Yalta in the summer of 1904, I spent entire hours on the seashore, observing the surf and collecting products of the sea’s activity for the Geological Cabinet of the Tomsk Technological Institute. At the same time it struck me that, at both points, the shingle and boulders of the beach include very diverse rocks, among them all the known eruptive rocks of the southern coast of Crimea—dacites, andesites, their tuffs, porphyries, porphyrites, melaphyres, quartz-augite diorites—whereas the steep coastal cliffs that border the beach—

*) Traité de Géologie. Vol. I, p. 473, Paris, 1907.
**) Comptes Rendus 1907, CXLIV, p. 759–760. Galets et sables du Pas de Calais.

and in Yalta and in Alushta, for several versts on both sides, consist exclusively of mountain shales and shaly sandstones. The latter, of course, enter into the composition of the pebbles in considerable quantity, but nevertheless the large participation of the ejected rocks, including those whose outcrops lie tens of versts from Yalta and Alushta, cannot fail to attract attention.

At first I tried to explain to myself this presence of pebbles of non-local rocks partly by their being brought by rivers from the depths of the Crimean mountains, partly by their being transported on sea ice along the shore. But the study of surf processes showed that the question is resolved more simply, and that pebbles and boulders can move along the shore so rapidly that their traversing tens and even hundreds of versts presents nothing improbable and requires only a corresponding interval of time.

I became convinced of this movement by means of a simple experiment. Among the pebbles and small boulders of the beach at both points one quite often finds yellow and red brick, whose bright color against the general dark background of the stony beach makes it possible to follow well the movement of one and the same stone under the influence of the waves. Throwing such a brick pebble, about the size of a fist, onto the beach exposed at low tide to waves of a sufficiently strong surf, I noticed that with a direct wave, i.e. one advancing perpendicular to the shoreline, the stone rolls back and forth, approximately along one and the same path; but with an oblique wave it moves along the shore in a zigzag line with a speed varying according to the force of the individual waves. The mode of movement is as follows (fig. 1): let the crest of the wave MN form an angle \(a\) with the shoreline NO; the stone is thrown at point A at a distance \(n\) from the shore; the overturning wave picks it up and rolls it in the direction of its motion, i.e. at an angle \(90^\circ - a\) to the shore, to point B at a distance \(a\) from A. At B the stone remains lying for several moments, until the ebbing water of the broken wave again picks it up and rolls it, but now in the direction of the greatest slope of the beach, usually at an angle of \(90^\circ\) to the shore, which it itself follows at low tide back into the sea; let it bring the stone to point C at a distance \(b\) from point B. The angles ABC and MNO, as angles with mutually perpendicular sides, are equal; and since we denoted the angle MNO by \(a\), then \(ABC = a\). The side \(AB = a\), and the side AR, which we shall denote by \(m\), from the triangle ABR will be equal to \(a \cdot \sin a\). But \(m\) expresses to us the magnitude of the displacement of the stone parallel to the shore under the action of one wave and

this quantity proves to be equal to the product of the magnitude of the stone’s displacement toward the shore by the sine of the angle formed by the wave crest and the shoreline. Thus,

\[ m = a.\operatorname{Sin}\alpha. \]

The next wave again picks up the stone and carries it to point D, again in the direction of its motion, a distance \(a_1\) from C; during the ebb of the wave the stone rolls to point E a distance \(b_1\) from D, and so on. Since we take the direction of the waves relative to the shore to be a quantity constant during our experiment, it is clear that the angle CDE will likewise be \(= \alpha\); and we shall find the magnitude of the displacement parallel to the shore under the second wave, \(m_1\), by dropping a perpendicular from point C to the extension of line DE; and since \(CD = a_1\), then

\[ m_1 = a_1.\operatorname{Sin}\alpha. \]

Reasoning in the same way with respect to the subsequent waves, which successively move the stone to F, G, H, I, K, L, and so on, we shall see that the final magnitude of the stone’s displacement parallel to the shore, as it moves through all the points of the zigzag line from A to L, will be equal to the sum of the magnitudes of the displacement under each separate wave, i.e.

\[ \mathrm{M} = m + m_1 + m_2 + m_3 + m_4. \]

Substituting into this equation, in place of \(m, m_1\), etc., the corresponding quantities \(a.\operatorname{Sin}\alpha,\ a_1 \operatorname{Sin}\alpha\), etc., we obtain

\[ \mathrm{M} = a.\operatorname{Sin}\alpha + a_1 \operatorname{Sin}\alpha + a_2 \operatorname{Sin}\alpha + a_3 \operatorname{Sin}\alpha + a_4 \operatorname{Sin}\alpha \]

and hence

\[ \mathrm{M} = \operatorname{Sin}\alpha\,(a + a_1 + a_2 + a_3 + a_4). \]

From this equation it is evident that the magnitude of the stone’s displacement parallel to the shore depends only on two factors: first, on the magnitude of the angle \(\alpha\) formed by the wave crests with the line of the shore, i.e. on the direction of the wind; and, second, on the sum of the magnitudes of the stone’s displacement by the individual waves in the direction toward the shore; these same quantities \(a, a_1\), etc., depend, of course, on the force of the individual waves, i.e. on the force of the wind, assuming that there are no obstacles in the stone’s path to its free motion and that the slope of the beach is everywhere the same.

The quantities \(b, b_1, b_2\), etc., expressing the motion of the stone under the influence of the ebbing water of the waves in a direction perpendicular to

of the shoreline, do not enter into our equation and, consequently, as if do not affect the displacement parallel to the shore. But in reality they do have such an influence, since the magnitude \(a, a_1, a_2\), etc. depends on them to a certain degree; but this influence is difficult to take into account: thus, for example, the displacement of a stone under the action of a receding wave has often not yet ended when the stone is already picked up by a new advancing wave. In some cases this influence may prove very strong; suppose, for example (Fig. 2), that under the influence of two weak waves a stone has moved from A to E, where it is picked up by a stronger wave; the latter could, by its force, roll it to point F, but the coastal cliff NO prevents this, and the stone reaches only its foot at G, thus losing the distance GF and the corresponding displacement parallel to the shore.

With different slopes of different parts of the beach, one would have to introduce into our equation also the value of the angle of this slope \(\beta\), since the magnitude of the displacement \(a\) naturally depends on it; the greater \(\beta\) is, the smaller \(a\) will be, since the greater will be the weight of the stone resisting the force of the wave. But the introduction of this factor would considerably complicate our equation and does not enter into the task of our note. It is sufficient for us to establish the following: the displacement of boulders, pebbles, gravel, and coarse sand parallel to the shore:

1) is caused by waves impinging at an oblique angle;

2) takes place along a zigzag line, composed of segments alternately perpendicular to the wave crest and to the shoreline;

3) the magnitude of this displacement depends:

a) on the magnitude of the angle formed by the waves with the shoreline, i.e. on the direction of the wind;

b) on the force of the individual waves, i.e. on the force of the wind;

c) on the angle of inclination of the part of the beach along which this displacement occurs.

When the angle \(\alpha\), i.e. the direction of the wind, changes, our equation will take the general form:

\[ \mathrm{M}=\operatorname{Sin}\alpha\,(a+a_1+a_2+\ldots a_n)+\operatorname{Sin}\alpha_1\,(a^1+a_1^1+a_2^1+\ldots a_n^1)+ \]

\[ +\operatorname{Sin}\alpha_2\,(a^{11}+a_1^{11}+a_2^{11}+\ldots a_n^{11})\ldots+\operatorname{Sin}\alpha_x\,(a^x+a_1^x+a_2^x\ldots a_n^x). \]

The magnitudes of the individual terms of the right-hand side of this equation may be greater or smaller depending on the duration of the wind of one and the same direction and on its force; when the magnitude of the angle \(\alpha\) is very small or when the force of the wind is very weak, individual

... the terms may be close to zero, i.e., at times the movement of the material along the shore may almost cease.

How great, theoretically, this displacement may be is shown by a numerical example; let \(\angle \alpha = 30^\circ\), and let the quantities \(a, .. a_4\) be equal to 3, 5, 4, 6, and 7 arsh., then

\[ \mathrm{M}=\operatorname{Sin}30^\circ(3+5+4+6+7)=\operatorname{Sin}30^\circ\cdot25=\frac{1}{2}\cdot25=12^{1}/_{2}\ \text{arsh.}, \]

i.e. under the influence of just five waves the stone will move \(12^{1}/_{2}\) arsh. along the shore.

If the corresponding quantities \(b,..b_4\) are taken to be 5, 3, 5, 6, and 4 arsh., then the total path traversed by the stone under the influence of these five waves along a zigzag line will prove equal to \(25+23=48\) arsh., i.e. almost four times greater than its displacement parallel to the shore.

If the waves go at an angle of \(45^\circ\) to the shore, then, taking the previous values for \(a,..a_4\), we obtain

\[ \mathrm{M}=\operatorname{Sin}45^\circ\cdot25=\frac{\sqrt{2}}{2}\cdot25=0.71.25=17.75\ \text{arsh.} \]

For the value \(\alpha=60^\circ\), we obtain \(\mathrm{M}=21.75\) arsh., and finally, for the value \(\alpha=90^\circ\), i.e. if the waves move parallel to the shoreline, \(\mathrm{M}=25\) arsh.; i.e., theoretically, the displacement of the material along the shore, other conditions being equal, should be greatest in the case of waves whose crests are perpendicular to the shoreline.

In reality, however, waves coming in the open sea perpendicular to the shoreline always prove oblique on the beach, since, owing to the friction of the lower layers of water against the surface of the inundated part of the coastal zone, the portions of these waves nearest the shore lag behind in their motion, and the wave crest turns into a curve convex toward the shore (Fig. 3). Consequently, those parts of the waves which form the shore surf and condition the mechanical work on the beach will always be directed at an angle \(\alpha\) smaller than \(90^\circ\) and, probably, even smaller than \(60^\circ\).

Our theoretical considerations concerning the movement of material along the shore are, of course, subject to numerous practical corrections. The beach on which the waves break is not a level surface free of obstacles, but consists of material of varying coarseness—boulders, shingle, gravel; therefore the motion of each separate stone is far from free. And, indeed,

we can observe that, under the influence of the wave, by no means all the material of the given stretch of beach is moved, but only part of it; the movement of the remaining part is hindered by various obstacles. Moreover, the movement of each individual stone occurs with interruptions, often very prolonged ones; for example, after several movements backward and forward, a stone will roll into a small depression or become wedged between two other, larger ones, and so forth, so that whole tens of waves may pass over it without moving it from its place; then a stronger wave will run in and the stone will begin to move again.

Thus, for example, of the 15 brick pebbles which I followed for a quarter of an hour, three were carried out into the surf by the very first wave and left the area of observation; five moved during this time a short distance parallel to the shore, namely from 3 to 10 paces; four covered from 11 to 20 paces, and only three from 25 to 40 paces. The angle \(\alpha\) in these observations was about \(30^\circ\), the number of waves per minute on average 12, the value \(a\), with the free movement of the stone, 3–5 arshins, and consequently in 15 minutes the stones should have covered

\[ \frac{15.12.4}{2} = 360 \]

arshins (taking \(a\) on average as 4 arshins), whereas in reality a few (20%) of them covered 9–14 times less, and the rest 18–120 times less.

Nevertheless, over the course of a prolonged interval of time the displacement of material along the shore must undoubtedly reach large magnitudes and therefore have great importance for the formation of coastal deposits. It is therefore surprising that this question has hitherto been so little developed in the geological literature, and that all manuals merely repeat the phrase about the decomposition of the force of an oblique wave into two components. Even Richthofen, who devoted much attention to the question of the work of sea-surf waves, says only that, with an oblique wave, each stone receives, simultaneously with the blow tending to move it toward the shore, a second blow, a lateral one*); in this way he depicts a mode of movement of a stone that is wholly at variance with reality.

The speed and direction of the movement of material parallel to the shore, as well as the quantity of material being moved, of course depend on the predominance of one or another set of winds at the given shore. We can imagine the complete absence of such movement only on a shore where exclusively winds blow that are perpendi-

*) F. v. Richhofen. Führer für Forschungsreisende, p. 346.

perpendicular to the direction of the shore (or else winds from the landward side, which cannot produce much wave action at the shore and thus have no influence on one or another displacement of material). But if, even several times a year, winds blow obliquely or parallel to the shore, then a corresponding displacement of material must inevitably occur; the more such winds there are, and the stronger they are, the more considerable this displacement will be, and the greater the masses of material that will be displaced. At the same time, frequent but weaker oblique winds must exert less influence than rarer but stronger ones.

It remains to consider briefly what influence a change in the direction of the shoreline has on the displacement of material.

One may suppose that headlands projecting far out into the sea must completely stop the displacement. In reality this is by no means true in all cases. Let us imagine (Fig. 4) a shore with a strongly projecting headland BCD; let sea winds of three directions, indicated by the arrows \(a, b, c\), blow in this locality, succeeding one another. On the coastal stretch AB the oblique winds \(a\) and \(b\) will produce a displacement of material parallel to the shore from A to B, while the direct wind \(c\) will not take part in this work. But, beginning from B, on the stretch BC the role of the winds partly changes: \(b\) becomes direct and ceases to take part in the displacement of material along the shore; \(a\) remains oblique (or, rather, changes from parallel to oblique) and continues to move the material from B to C; \(c\) changes from direct to oblique, but since it blows from C toward B, the displacement produced by it must be opposite to the displacement caused by wind \(a\), i.e., the material moved by wind \(a\) from B to C must be displaced from C to B under the action of wind \(c\). If wind \(c\) is stronger than \(a\), or, with more or less equal strength to \(a\), blows more often than the latter, then in the final result no displacement of material from B to C will occur, and at B a large quantity of material will accumulate, advancing from one side from A and from the other from C; therefore the bay at B will gradually become filled in, and instead of a re-entrant angle there will soon be a curve (shown in Fig. 4 by a dotted line). If, however, wind \(a\) is predominant, then, despite the retarding influence of wind \(c\), the material will be displaced from B to C, but at a reduced rate and in dimin—

the amount; bay B in this case too will gradually be filled in, but more slowly.

At point C the role of the winds for the sector CD changes again; \(a\) and \(b\), as blowing here from land toward the sea, will not have any essential influence on the movement of the material; but \(c\), being oblique, will cause movement from C to D both of material that has rounded the tip of the cape and of material forming on the sector CD itself.

At D we find a new change of roles; \(c\) becomes direct and will only contribute to the accumulation of material at D; \(a\) and \(b\), although oblique, yet, as still blowing from the land, cannot have a noticeable influence at D, and as a result at D in any case there must be shallow bays and a transformation of the re-entrant angle into an arc. But, moving from D to E, we shall find that the role of winds \(a\) and \(b\) will become ever stronger, since the influence of the cape on the wave they generate will become ever weaker, so that, the closer to E, the greater the amount of material will be moved along the shore, until we obtain conditions identical with those of point A.

From this example it is clear that capes projecting into the sea must, depending on the predominance of one or another wind, either play the role of a barrier preventing the further movement of material along the shore and causing its accumulation in both bays at the base of the cape; or else only slow the movement of material along the shore, likewise causing the bay to be choked, but more slowly.

Let us consider also the opposite case, when the shoreline forms a deep bay BCD (Fig. 5). Assuming the presence of the same three winds \(a\), \(b\), \(c\), we shall find movement of material from A to B under the action of winds \(a\) and \(b\); on the sector BC \(a\) loses its significance, as blowing from land, while \(b\) and \(c\) will produce movement of material from B to C. At point C wind \(a\), as before, has no significance; \(c\) becomes direct, and \(b\) almost direct, and therefore accumulation of material and gradual silting of the bay must occur; on the sector CD \(b\) becomes direct, while \(a\), gradually, as it approaches D, strengthens its influence and moves an ever greater amount of material from C to D; but \(c\) counteracts it and, as blowing from the open sea (whereas \(a\) blows from land across the bay), produces a stronger wave, moving material from D to C. As a result, if only \(a\) does not significantly predominate over \(c\) in strength and frequency, then on the sector CD there must occur movement of material from D to C and silting of the bay from this side as well. Finally, on the sector DE movement occurs from D to E.

Thus a deep bay is, with the same combination of winds, a more serious barrier than a cape, and interrupts the movement of material along the shore, absorbing this material; but owing to this absorption its role is only temporary. When the bay becomes shallower and contracts (Fig. 6), it will lose its significance; the waves of the sea, meeting the calmer waters of the shallow bay, lose their force to a considerable degree, and the agitation in the bay will be weaker; therefore at point B, from the material transported from A to B, a spit will begin to be built up in the direction of D, gradually cutting the bay off from the sea; this process of the formation of spits (bars, tongues) across the mouths of shallow inlets has long been known and is discussed in all manuals, so we shall not dwell on it. In time this spit will reach D and completely cut off the bay, turning it into a lake; if some river flows into the bay, then the flow of water into the sea will struggle against the complete cutting off of the bay, constantly washing out the spit over some stretch; but even in this case the movement of material farther along the shore will not stop, but will only be replaced in part by the interruption of the spit.

Consequently, deep bays, in the final analysis, are only temporary obstacles to the movement of material along the shore and, having then turned into shallow ones, only somewhat slow this movement until they become definitively silted up.

Let us now try to apply what has been said above to the southern coast of Crimea, in order to determine in what direction the material must move. This coast from Alupka to Feodosia has, in general, a direction from southwest to northeast (Fig. 7). According to Hann’s data*) in Crimea we have the following distribution of winds by points of the compass, in percent:

Winds N, NW, and W for the southern coast of Crimea blow from the land and, consequently, play no role in the movement of material; the wind SO is directed, in general, perpendicular to the shoreline, so that it can condition the movement of material only on separate sections—in bays and at capes; therefore we may also exclude it from our consideration. Of the remaining winds—oblique and parallel to the shore—S and SW must produce a move-

) I. Hann. Handbuch der Klimatologie*. Bd. III, p. 206, Stuttgart, 1897.

the transport of material from the southwest to the northeast, i.e., from Alupka to Feodosia, while NO and O—conversely, from Feodosia to Alupka; but the first two winds account in summer for 20%, in winter for 16% of the total number of winds, whereas the second, in summer, account for 33%, and in winter even for 43%, i.e., NO and O occur in summer 1.65 times, and in winter 2.7 times, more often than S and SW, and together with this are distinguished by greater force. Consequently, the transport of material in the final result must take place from Feodosia toward Alupka, and therefore it is entirely natural that among the boulders of the beach at Alushta and Yalta we find crystalline rocks occurring farther east, for example: at Alushta—melaphyres, dacites, andesites and their tuffs, exposed in the coastal cliffs of Karadag (between Feodosia and Sudak), and the keratophyre of Kuchuk-Uzen; and at Yalta—also quartz-augite diorites and porphyries of Mounts Kastel and Ayu-Dag, and the keratophyre of Ai-Danil.

The present note had already been written when I found in the Comptes Rendus a short article by Thoulet*), considering the same question of the transport of material along the shore and serving as a response to the above-mentioned article by Bréon. The author likewise comes to the conclusion that, under the influence of oblique waves, material is transported parallel to the shore along a zigzag line resembling saw teeth. According to Thoulet’s calculations, on the coasts of northern France, with 12–18 waves per minute, the displacement parallel to the shore reaches no more than 1 cm for each wave; taking into consideration the alternation of winds, he finds that each grain of sand, in order to move any distance along the shore, must travel a path at least 8000 times longer in its rolling back and forth. The author says that in the English Channel the transport occurs from W to O and that sand from the material of the crystalline rocks of Brittany, the occurrence of which on the shore for Bréon remained enigmatic, was brought precisely from W, i.e., from Brittany, by gradual transport along the shore, notwithstanding all the bends of the latter.

But since in Thoulet’s article, only 2 pages in length, the mode of movement of material parallel to the shore is considered too briefly, I believe that my note, even after the appearance of this article, retains its significance for clarifying the question.

Tomsk,
19 February 1908.

) Sur la marche des sables le long des rivages*. C. R. CXLIV, 1907, pp. 938–940.

After the article by V. A. Obruchev.