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OF THE

ROYAL ASTRONOMICAL SOCIETY

GEOPHYSICAL SUPPLEMENT

Vol. 3 No. 5 1934 JANUARY

ON TIDAL BORES WHICH ASSUME THE FORM OF A GROUP OF SHORT WAVES.

Vaughan Cornish.

(Received 1933 June 26)

In certain rivers the first rise of the spring tides has a visible head of water, sometimes a solitary ridge with an overfalling .front, sometimes a group of waves. On the Severn this visible head of the tide is now known as the “bore,” but as late as the eighteenth century was called the “eagre,” the name which is still applied to the phenomenon on the Trent and other rivers of the Eastern Counties.

In France the most general term for the visible head of the tide is barre, but the word mascaret, which has always been the local term on the Garonne, is now often applied to the barre of the Seine.

In the customary terminology of geophysics the word “ bore ” denotes a solitary wave with overfalling front, whether occurring as the head of the tide or as the successor of the curling breaker on a flat seashore.

The definition of “eagre ” in the New English Dictionary is “a tidal wave of unusual height,” thus conforming to the physicist’s conception of the tidal bore as a solitary wave ; but in E. Littrt’s Dictionnaire de la langue Frangaise the word barre is treated as a “ noun of multitude ” :-

‘‘ La barre, les premidres lames que la made montante pousse dam un fleuve.”

The conception of the tidal bore which has become customary among English physicists, viz. a solitary ramp of water constituting the visible front of the whole tidal wave, must, I think, have been largely due to the impression produced upon the late Sir George Airy on the occasion when

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184 Dr. Vaughan Cornish, Tidal Bores which 3 9 5

he viewed the Severn bore from the churchyard of Newnham in Gloucester- shire. This is situated on a bluff commanding an extensive view of the river and estuary. I have myself seen the bore more than once from this vantage point, and it is indeed, as Sir George Airy wrote, “ a majestic phenomenon.” *

That the celebrated astronomer was profoundly moved by the sight is evident even in the restrained mode of expression characteristic of the scientific writing of his day. I think that some light will be thrown upon the subsequent treatment of the subject of tidal bores if we endeavour to enter into the mind of the observer on this historic occasion.

Sir George Airy had devoted himself to the problems of the Moon’s motion, and had investigated the tides of the world. His visit to Newnham was, apparently, rather for the purpose of witnessing the illustration of a phenomenon than in order to test a theory; and, as it seems to me, he was thrilled by the idea that what he saw was the advancing front of the great solitary wave whose path across the oceans he had recorded in his map of cotidal lines. Years later Sir George Darwin went down to the Severn on a similar errand, but experienced a disappointment.

“ I n September 1897,” he writes,t “I was on the banks of the Severn at spring tide; but there was no proper bore, and only a succession of waves upstream, and a rapid rise of water-level.”

Thus it is evident that Sir George Darwin was on the look-out for a solitary ramp of water, and that no other kind of wave was in his view properly described as a bore. Yet the sentences which follow in his description show that he was not easy in his mind as to the discrepancy between theory and observation. He writes that “the heading back of the sea water by the natural current of the river; and the progressive change of shape of a wave in shallow water combine to produce a rapid rise of the tide in rivers,” but adds that this “serves rather to explain a rapid rise than an absolutely sudden one.”

I have now to describe my own observations. The channel of the tidal Severn is crossed in-several places by broad

“benches” of sandstone rock which suddenly reduce the depth, and it is at these “ benches ” that sightseers mostly congregate to admire the spectacle of the bore. On 1900 April 30 I took station beside Denny Pool on the upstream side of the broad shoal called Denny Rock. Presently the bore appeared round the bend of the river travelling in shallow water. A steep wave stretching from bank to bank, in places curling over in a scroll, was almost the only visible disturbance from my point of sight near the water level. Being at this time thoroughly indoctrinated with the conception of the bore as the steepened front of the whole tide-wave, I expected that it would preserve the form of a bar or barrier of water, and could scarcely

* Sir George B. Airy, “Tides and Waves,” &cy. Metropolitana. Afterwards published as a volume. Many English writers on the subject refer to this work, which evidently exercised a strong predisposing influence.

t George Howard Darwin, The Tides and Kindred Phenomena in the Solar System. (John Murray, 1898, p. 65.)

1934 Jan. assume the form of a Group of Short Waves '85

believe my eyes when on entering the deeper water in front of me it was instantaneously transformed into a group of rounded swells.

On 1901 October 30, at Framilode, lower down the river, I noted other incidents accompanying the passage of the bore from shallower to deeper water, which clearly indicated that it could not be regarded as the steep part of a wave many miles in length.

I watched the approach of the bore, which presented a crested front with a few ridges visible behind, and the rate of advance, which was quite steady, became clearly impressed upon the eye. On reaching deep water the front lost its steepness, and, simultaneously, new rounded waves appeared at the rear, so that the group comprised many more individuals. Still more singular was the circumstance that the rate of movement seemed suddenly to be retarded. The speed of the individual waves must have been slightly increased, but the rate of advance of the procession as a whole was much reduced by the formation of new waves behind.

On another occasion, when waiting on the bank of the Severn at Newnham for the arrival of the bore, I observed a curious effect which helps to explain the suddenness of change from the condition in which the first rise of tide eludes the eye to that in which it is spectacular. On the opposite bank I saw a small wave running and breaking on the edge of the shelving sand, but in the deeper water on our side, the true right of the river, I did not see any disturbance. I was therefore startled when a wooden post which projected several inches above the water was suddenly submerged. Thus the head of the tide in deep water was an invisible wave several inches in height. In this connection it is instructive to compare an observation which I made on the Serpentine Water in Hyde Park during a fresh breeze. Although the height of the waves was only 2 inches, the roughness of the surface was conspicuous because, as I found by direct measurement, the wave-length was only 36 inches. Thus the steepness at the shoulder of the waves was not less than

180' x & = 10' * and the parts of the sky reflected from front and back of the wave were consequently widely separate and therefore, under most conditions, of notably different brightness. But the wave-length in the group of the Severn bore is of the order of 30 feet, so that there would be very little difference in the brightness of the sky reflected from the front and back of a wave of a few inches in height.

It is interesting to recall the fact that the tiny capillary ripples made by insects moving on the surface of a pond are very conspicuous on account of their strong curvature.

In this connection I may refer to the distinction drawn by Sir George Darwin between a rapid and an absolutely sudden rise of water. It does not appear that he made any observations to determine the critical angle at which the first rise of the tide would become a visible wave.

In 1922 I stayed from September 15 to 24 at Burton-upon-Stather, a * See W. H. White, A Manual of Naval Architecture, 1896, p. 300.

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I 86 Dr. Vaughan Cornish, Tidal Bores which 3, 5

few miles above the outfall of the Trent into the Humber estuary, and made observations of the eagre up to a point a little beyond Wildsworth ; and in 1928 I stayed at Gainsborough from August 16 to 19 and made observations from above Wildsworth to Torksey, which is nearly as far as the eagre runs. On the afternoon of September 19, two days before New Moon, I watched for the turn of the tide from the jetty at Burton- upon-Stather, nearly 3 miles above the outfall to the estuary. At 3.50 p.m. the level was still falling, but by 4.4 p.m. the fall had ceased and a rise of 2 or 3 inches had occurred. The spots of foam upon the water continued to drift seawards until 4.8 p.m., but at 4.9 p.m. were travelling upstream. Neither the first rise nor the first flow were attended by any visible inflexion of the water surface. On the 22nd, the day after New Moon, I took my station on Garthorpe jetty on the left bank facing the Burton-upon-Stather jetty. At 6 p.m. the water was absolutely slack, so that the bore which reached the jetty 12 minutes later was not headed back by any current. In very shallow water near the right bank there then appeared a foaming ridge about 8 inches high, a ramp between two water- levels, there being no depression behind. The tide also advanced over the mud flats under the left bank in the same manner. No sooner had the bore passed than the water was seen to be running upstream.

Getting back into my motor-car I outstripped the progress of the bore and took my station at Keadby Bridge, 6Q miles farther up, before its arrival. Here there are no mud flats. The appearance of the bore was now very different, a long train of steep, rounded waves of which the first rose about I foot above the level of the river. The jointed masonry of the bridge piers provided a means of observing the rise of the water and of estimating its amount. One hundred well-marked waves passed the pier in 3 minutes, by which time I judged that the mean level had risen I foot. Having measured the speed of the bore, from Flixborough, which was 11 feet per second, and determined the period of the waves, 1.8 seconds, I was able to calculate the wave-length, which came out at 20 feet. The ordinary formula for calculating the maximum slope of deep-sea waves, namely,

height 180' x - length

would make the maximum slope of the first wave go. Actually it would be considerably more, since the crest was advanced far beyond the middle point.

The general slope of the water in the group of IOO waves was, however, only I in 2000-that is to say, an angle of less than oo 2'.

I made the following determination of rate of advance of the eagre and the period of its constituent waves :-

Speed Period (m.p.h.) (sea.) Locality

Mere Dyke 9.0 2.3 Keadby Drain 8.0 2'0 Keadby Bridge 7.5 1.8 West Butterwick 9.9 2.0

Average 8.6 m.p.h. 2 sea.

1934 Jan. assume the form of a Group of Short Waves '87

The average length from crest to crest was therefore 25 feet. At West Butterwick the height of the first ten waves was estimated at

4 feet, and at Wildsworth, where the river is narrower, the height of several waves was judged to be 5 feet, the highest which I have observed on the Trent. Here the light was failing, so I did not pursue the eagre further.

This place is 19 miles by water from the Humber.

In 1928 I stayed at Gainsborough from August 16 to 19 and followed the eagre from a point a little above Wildsworth, for another 16 miles, to Torksey which, as I have said, is near the end of its course.

On the afternoon of August 16, the day after New Moon, I judged the height of the waves at Ravensfleet, above Wildsworth, to be 3 feet. At Gainsborough Toll Bridge the height had increased to at least 34 feet.

On the morning of 18th I judged the height at Bowling Green Road, just below the town of Gainsborough, to be 34 feet, at Knaith one-half of this, and at Torksey somewhat less. At each of these places the bore was a group of many waves, the height of which from trough to crest diminished from front to rear. On the days when the bore was small the diminution was very gradual, but when high the gradation was more rapid, the most spectacular part of the disturbance being the first ten or fifteen waves. This character I had also noticed in 1922 at, and below, Wildsworth.

At Torksey, on the morning of August 18, an observation was made which confirmed and extended the lesson learned at Keadby Bridge in 1922. At Torksey a canal joins the Trent at right angles on the right bank. Leaving Knaith, where I had seen the bore arrive, I outstripped the tide in a motor-car and stationed myself at Torksey at the angle between canal and river on the upstream side of the canal. I stood looking north- wards across the canal, the river flowing by me on the left on a straight reach. Presently the bore appeared half a mile away, at first as a broad, bright band advancing with a smooth and stately motion, then revealed as a train of rounded waves 14 feet or less in height. When the bore passed the promontory between the river and canal (opposite to my post of observation) the procession of waves passed on up the river. I did not notice any such waves swing round the rectangular bend into the canal. The entry of the tide into the canal was, however, very spectacular. The water gave one great heave, so slow that it seemed to have more in common with a long ocean swell than with the progressive waves of rivers. This slow rise I take to have been the true profile of the whole bore, freed from the usual corrugations. The great mound of water at the entrance of the very shallow canal ranged itself with a definite transverse front, and then three or four rounded swells extending from bank to bank, I think 20 feet at most from crest to crest, quickly formed, and travelled up the canal. After a run of about 350 yards these burst against the lock-gates which barred the way. The resurgence also took the form of a group of waves athwart the canal, but although two or three front ridges were similar to those which had travelled up the canal, the rearward part of the pattern was quite different, being composed of diagonal, not transverse, ridges.

I 88 Dr. Vaughan Cornish, Tidal Bores which 3, 5

The difference is no doubt due to the fact that the channel was being partially emptied, not filled as on a rising tide.

When the resurgent waves reached the exit of the canal the whole of the bore had passed on upstream, although still within sight until it rounded a bend of the river about 37 miles above the junction of the Trent with the Humber.

On August 19, the last day of my visit to Gainsborough (four days after Full Moon), I motored to Ravensfleet early in the morning to await the bore at the same place where I had first seen it on the day of my arrival. Since then the tide had decreased considerably and there was a great change in the pictorial character of the group of waves. Although the speed re- mained the same, the effect on the first day after Full Moon was that of a destructive charge, on the fourth day of a stately procession. On the 16th the front wave, 3 feet high, was succeeded by eleven waves nearly as large, followed in their turn by many others considerably smaller ; and the whole group passed in I minute 45 seconds.

On the 19th the first wave, 14 feet in height, was followed by thirty-nine others, of which the first thirty were so nearly equal in height that the grada- tion of size was scarcely noticeable. The whole disturbance passed in I minute 40 seconds, so that its length must have been almost exactly the same as on the 16th. Getting on ahead of the bore I measured its rate of advance in the next reach, which was I 3 feet per second. From the character of the channel we are justified in assuming that this was very nearly the same as the speed at Ravensfleet itself, from which it follows that the length of the wave-procession there was 1365 feet, and since the period of the waves was 2.625 seconds, the wave-length was 34 feet.

On the morning of August 19 I saw a disturbance of the reflection some distance in front of the first wave-crest. On August 16, when the front wave was twice as high, there was no indication of a rise of water at such a distance in front of the crest, and I judge therefore that when the tides fall off, the front wave becomes more nearly symmetrical, the crest receding more and more towards the midway point. This circumstance must increase the rapidity of change from the spectacular to the invisible condition of the “first rise.”

I now have to consider the question whether the whole corrugated slope of some 1300 to 2000 feet, which constitutes the “head” of the spring tide in the Trent below Gainsborough, is the comparatively steep rise from the front trough of a wave whose next trough is many miles away.

A peculiarity which I observed in the bore of the lower part of the tidal Trent indicates that this slope may, on the contrary, be but the front of one of those surges in the tide of which the existence is revealed by the notches in the curves automatically drawn by a tide-gauge.

On 1922 September 21 and 22 there was a second bore following the first. The phenomenon is familiar to the dwellers by the river and is called the “ second shove.”

The waves of the second bore were decidedly lower than those in the first, but I think they often exceeded 2 feet in height. The wave-length

1934 Jan. assume the form of a Group of Short Waves 189

appeared to be the same, and where the second bore was well developed the number of waves was about the same as in the first bore.

On 1922 September 22, the day after New Moon, the time between the arrival of the first and second bore was

at Mere Dyke 3.5 minutes ,, Keadby Drain 5'0 9 9

,, West Butterwick 3-5 9 ,

On 1922 September 21, at West Butterwick, the interval between the arrival of the first and second bore was 3 minutes. I noticed that in the course of its progress from Butterwick to Wildsworth the second bore diminished whilst the first increased. The ferryman at Stockwith, higher up, told me that the second bore sometimes reached this place, but a resident of Gainsborough informed me that it did not come as far as the latter town.

The investigation of the conditions which produce the second eagre in the river Trent is a very promising line of research for the improvement of our knowledge of tidal bores in general, and is a piece of work which lies ready to our hand.

Is the second surge an incident of the progress of the tide above the out- fall of the Trent, or does it originate in the Humber?

The Ordnance map shows a feature in the Humber estuary just below the outfall of the Trent which may possibly account for the duplication of the Sore. This is Whitton Sand, a drying shoal 2 miles long by a mile wide dividing the estuary at low water into the broad Whitton Channel on the right, or Lincolnshire, shore, adjacent to the Trent, and the narrower channel on the left, or Yorkshire, shore by way of Faxfleet. There are several ways in which it might conceivably cause a second surge. This might follow on retardation in a shallower channel, or result from a freer flow after submergence of the sandbank, or be due to a wave reflected from the Yorkshire shore. The problem of the second surge cannot be solved without further data.

In conclusion I wish to urge upon students of geophysics the importance of a further investigation of the bore of the Trent, and in particular of the second surge, by that kind of team work which has now become fashionable in scientific, as in other, spheres of activity. The high spring tides of the Autumnal Equinox, which are specially favourable to the investigation, occur during the Long Vacation of our Universities, and a fortnight in September would be sufficient for the work. The prime essential is the co-operation of a number of workers adequately equipped with instrumental means, who could make simultaneous observations at different points. Tide-gauges and current-meters would be required both in the Trent, and in the Humber estuary between Trent Falls and Whitton Ness.

At points selected for kinematograph pictures of the bore in the Trent, stakes with altitudes marked in bands should be set up, so that the oscilla- tions of level during the passage of the short waves would be recorded on the film. Soundings for depth should be made before and after each tide at the points selected for observation. These observations would provide

These, however, are little more than guesses.

I90 MY. K. E. Bullen, The Errors in Calnclations 3, 5

the data hitherto lacking for an adequate mathematical investigation of the type of tidal bore most characteristic of English and French rivers, that in which the steep slope of the head of the tide breaks up into a group of short waves.

The phenomena are so thrilling to watch, and rural Lincolnshire has such charm in the harvest season, that a party of young men from one of the Universities would find this scientific expedition a very enjoyable episode in the Long Vacation.

ON THE ERRORS I N CALCULATIONS OF EPICENTRAL DISTANCES IN EARTHQUAKES.

K. E. Bullen, M.A., B S c . (Received 1933 August 16)

I. With a view to facilitating work on near earthquakes the author recently computed * the constants of 365 seismological observatories to four decimal places. In near earthquake work distances are required to I or 2 km., which corresponds to about 0'-01. This accuracy cannot be attained when three- figure values of the constants are used ; but it seemed likely that the yse of four-figure constants would be adequate for the purpose. It was later noticed (by Comrie) that for A > 20°, the formula cos A = aA + bB + cC would work to o O . 1 when used with four-figure constants.

It has therefore been thought desirable to give a complete analysis of the errors that can occur in the determination of epicentral distances of observing stations. Except in one instance to be mentioned later (where six-figure tables are used), it is presumed that four-figure tables and constants are used throughout a calculation. It has also been thought of value to expose the nature and extent of the actual circumstances that may conspire to give the maximum error.

2 . The constants a, b and c of the stations have been computed from the equations

a =sin 0 cos t$ ; where 0, t$ are a station's co-latitude and east longitude respectively and are known to the nearest minute.

It is assumed in what follows that A , B and C are similar constants for the epicentre of any particular earthquake, and that these will have been likewise calculated correct to four decimal places.

Two formula: are then available for determining the epicentral distance A of a given station :

b =sin 0 sin t$ ; c =cos 0 ;

cos A =aA+bB+cC ; (1)

(2) 2 vers A =(a - A)2 + ( b - B)2+ (c - C)z.

* British Association, Gray-Milne Trust, 1933.