424 



NATURE 



[Sept. 13, 1877 



laterally under lee of the barrier, suffers a reduction of 

 its height. In the second edition of my book on Har- 

 bours, I expressed regiet that no attempt had been made, 

 so far as I was aware, to obtain any numerical value of this 

 reduction of height derived either from theory or expe- 

 riment, although the extent of shelter which is to be 

 gained by the erection of our great national breakwaters 

 depends entirely upon its amount. 



From a few observations taken in the sea under lee of 

 the breakwater in Wick Bay, and from some experiments 

 made in a large brewer's cooling vat, it appeared that after 

 passing round an obstruction thi: reduction in tlu hciirht 

 0/ ivavcs varied as tJu- square root of the angle of deflec- 

 tion. The approximate formula given in my book was — 



X = I 00 — '06 sj a 

 where x represents the ratio of the reduced to the unre- 

 duced wave, and a the angle of deflection. 



On a recent visit to North Berwick, the finely-curved 

 storm and tide marks traced out on the sandy beach under 

 lee of the promontory at the harbour, reminded me of some 

 observations I had made many years ago at other parts 

 of the Firth of Forth. These observations, which were, 

 however, very imperfect, had for their object the deter- 

 mination of the reduction of the waves by ascertaining 

 the positions in reference to the centre of divergence of dif- 

 ferent pai ts of the line of high water mark where, of course, 

 all the wave forces become ni/. If a beach throughout its 

 whole extent consist of easily moved materials such as 

 sand or gravel, the incursion made at any one place by 

 the sea will obviously depend upon the force of the waves 

 which reach the shore at that place, providing the mate- 

 rials of the beach are homogeneous. In other words, the 

 heavier the waves at any part of the shore the farther 

 inland will the high-water margin retire beyond the tide 

 mark of more sheltered places. And where the waves 

 vary in height owing to some local cause, as, for example, 

 the existence of a sheltering promontory, the high water 

 mark instead of being straight and parallel to the prevalent 

 waves will assume a curved outline. 



At North Berwick, the projecting promontory at the 

 harbour, shelters a small Ijay or rather dia/it from the 

 heaviest waves that fall on that part of the coast. The 

 waves, therefore, are deflected at the pierhead, from which 

 point as a centre, each section of every w.ave taking its 

 own divergent direction, runs its course till its energy is 

 expended at high wattr mark. The maximum effect on 

 the beach will consequently be in the line of direction of 

 the undeflected swell, and the minimum effect will be in 

 the direction of the landward end of the promontory 

 where the waves are most deflected from their natural 

 course. Under these conditions, supposing the particles 

 of sand to be of uniform size and of the same specific 

 gravity, the high water margin must assume, as it does at 

 Noith Berwick, a curved outline owing to the inequality 

 produced by deflection on the height of the waves. 



If the distance between the pierhead and the high- 

 water mark measured parallel to the usual direction of the 

 undt fleeted swell (shown by the arrow in the diagi-am) be 

 assumed as unity, that distance may be regarded ?s the 

 measure of the amount of work that the undeflected part 

 of the wave has betn able to do, inasmuch as its force has 

 been wholly expended within that distance in driving the 

 beach landwards. The varying lesser distances betv/een 

 the same point and other parts of the high watermark, may 

 in like manner be regarded as representing the work that 

 has been done by the varying lesser forces exerted by the 

 different parts of the wave after being deflected. It is, no 

 doubt, true that the undeflected wave has the full force of 

 the wind to help it, while the deflected has not ; but in so 

 far as relates to the engineering aspect of the question, 

 this effect, even though it had been much greater than it 

 IS, would be of no importance, as the same conditions 

 hold true with an artificial as with a natural breakwater. 

 I may menticn in corroboration of the views that 



have been expressed that in the course of my practice 

 as an engineer I have, at difl'erent exposed parts of the 

 coast, had occasion to fill up a [small creek with soft 

 materials pioduced by works of excavation at an adjoin- 

 ing part of the shore. In the course of time the whole of 

 these artificial deposits have, in every instance, been 

 removed by the waves, and the former line of high- water 

 been restored. By analogy, therefore, we must believe 

 that if the bay at North Berwick were in like manner 

 filled up artificially with sand as far seawards as the pier- 

 head we should find, after a certain number of storms had 

 occurred, that the whole of the sand had been washed out 

 and the former line of high-water reproduced. If this be 

 true, then the different distances between the pier-head 

 and the high- water mark at North Berwick may justly 

 be regarded as the measures of the varying forces of the 



different sections of the deflected wave under lee of the 

 promontory. 



The first column in the accompanying table shows the 

 angles of deflection, while the second gives the measure- 

 ments from the pier-head to the high-water mark as taken 

 from the Ordnance map of North Berwick. The direc- 

 tions in which these measurements were taken are repre- 

 sented by dotted lines on the accompanying woodcut. 

 The third column shows the ratios of those measurements 

 to unity. The fourth column gives the ratios of the 

 heights of the deflected wave calculated by the formula 

 r = I - '06^/ a, and the last the p/us and minus differ- 

 ences. Though the employment of the square-root of tlic 

 angle may perhaps be regarded as somewhat unusual, 

 the formula as given is nevertheless more convenient for 

 use than a logarithmic spiral formula, which might give 

 nearly the same results. 



Although it is possible that the agreement of the 

 measurements with the results calculated by the formula 



