On the Laws of Individual Tides at Southampton and at Ipswich

VI. On the Laws of Individual Tides at Southampton and at Ipswich. By G. B. Airy, Esq., M.A., F.R.S., Astronomer Royal. Received February 16,—Read March 2, 1843. With the view of verifying the reported peculiarity in the tides at Southampton, I had proposed in the month of February 1842 to proceed thither for the purpose of examining, with my own eyes, the rise and fall of the water during one or more tides. As soon, however, as my purpose was made known to Colonel Colby, R.E., Director of the Trigonometrical Survey, and to Lieutenant Yolland, R.E., the Resident Officer at the Ordnance Map Office, Southampton, I received from those gentlemen the offer of placing at my service, for these observations, non-commissioned officers and privates of the corps of Royal Sappers and Miners, as well as of preparing and fixing the vertical scale of feet and inches, and of keeping a watch upon the general accuracy of the observed times. I was extremely glad to avail myself of this offer, for I believe that a more intelligent and faithful body of men does not exist than the Sappers employed in the Trigonometrical Survey; and I knew well the advantage of employing, upon a tedious business like this, a set of regular-service men stationed on the spot. A vertical scale of deal laths, upon which the divisions and figures were branded, was fixed near the end of the pier at Southampton, very near to the landing-stairs on the north side of the pier, so that the divisions could at all stages of the tide (with the assistance of a lantern at night) be easily read by a person standing on the stairs. The order of the graduations of the scale was increasing from the bottom upwards, and the zero of the scale was found, by levelling, to bear the following relation to certain fixed marks: [The mark in each case, except that of St. Paul's Church, is a horizontal cut, indicated by the point of an arrow. The arrow is on one side of the mark at Holyrood Church, and below each of the others.] The zero of the scale is, Below a mark cut on the metal of the coat of arms on the north side of the pedestal of Chamberlayne's column, Watergate Quay. . . . . . . . . . . . . . . . . . \{25\cdot765 Below the ground at the same place . . . . . . . . . . . . . . . . . . . . . . 21\cdot990 Below a mark cut in the stone jamb on the south side of the centre door of Holyrood Church. . . . . . . . . . . . . . . . . . . . . . . . . . \{36\cdot310 Below the floor of the portico . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\cdot370 Below a mark on the coin stone at the south-west angle of All Saints' Church. Below the flags of the footpath. Below a mark on the north side of the pedestal which supports the east lion at the north side of the Bar. Below the ground there. Below a mark in the bricks of the wall at the south-west side of the door of the toll-house, at the junction of the London and Salisbury roads. Below the ground there. Below the lower sill of the centre door of St. Paul's Church. Below the flag-stones. Below a mark on the south front of the west angle of the east wing of the buildings at the Ordnance Map Office. Below the ground there. The height of the water upon this scale was observed at every five minutes, from February 23, 15h 10m (astronomical reckoning), to February 27, 6h 20m. The watch by which the times were taken was a pocket chronometer belonging to Lieutenant Yolland, of which the error was small. The observations at February 24, 6h 5m and 6h 10m, were omitted from inadvertence (darkness coming on before proper preparations were made); those of February 24, 6h 35m and 6h 40m, were excluded from calculation, because the water was disturbed by a boat; and those of February 25, 2h 0m, 2h 5m, and 2h 10m, because they were evidently irregular, although no reason was assigned for their irregularity. When the seven complete tides embraced by these observations were laid down in graphical projection, each of them presented a curve similar in its general form to the extraordinary figure represented in the diagram, fig. 1, in Plate II.; four of the curves having three maxima of elevation in each tide. In the first two tides, the first maximum, as in the diagram fig. 1, was rather a stand-still in the rise than a rise and fall; and in the last tide (the highest of all), the first and third maxima were both of this character. In all the others, the three maxima to each tide were well marked. On February 24, from 15h to 17h, the wind blew strong from W. and S.W. On February 25, from 18h to 19h, it blew a gale from S.W. and W. From February 25, 21h, to February 26, 3h, the wind was strong from S.W. and W. And from February 26, 20h to 23h, there was a strong W.S.W. wind. At other times the air was almost perfectly calm. These changes in the state of the atmosphere do not appear to have produced any sensible alteration in the character of the individual tides. The epochs and magnitudes of the tides appear, however, to have been sensibly altered by these or other disturbing causes. Thus, if we examine the times and intervals of high water, they are as follows:— Curve representing the law of rise and fall of the tide at Southampton, as obtained from observation of seven tides. From 1842, Feb. 23rd, 15 h. 27 m. to Feb. 7th, 6 h. 5 m. AT SOUTHAMPTON AND AT IPSWICH. February 23. . . 22 5 Interval . . 12 30 24. . . 10 35 . . . . . . . . . . 12 20 24. . . 22 55 . . . . . . . . . . 12 30 25. . . 11 25 . . . . . . . . . . 12 10 25. . . 23 35 . . . . . . . . . . 12 25 26. . . 12 0 . . . . . . . . . . 12 15 27. . . 0 15 These changes may possibly be due to diurnal tide. The times and intervals of low water are the following: February 23. . . 15 25 Interval . . 12 30 24. . . 3 55 . . . . . . . . . . 12 5 24. . . 16 0 . . . . . . . . . . 12 30 25. . . 4 30 . . . . . . . . . . 13 20 25. . . 17 50 . . . . . . . . . . 11 25 26. . . 5 15 . . . . . . . . . . 12 25 26. . . 17 40 . . . . . . . . . . 12 25 27. . . 6 5 These changes are utterly lawless. The heights of high and low water, (taking for high water, in all cases, the second of the three maxima), and the ranges, are as follows: | High Water | Low Water | Range | |------------|-----------|-------| | ft. | ft. | ft. | | in. | in. | in. | | 16 | 8 1/2 | 12 | | | | 1 1/8 | | 17 | 0 | 13 | | | | 4 | | 16 | 11 1/2 | 14 | | | | 0 | | 16 | 11 1/2 | 14 | | | | 10 | The irregularities here are considerable, and, taken in conjunction with those of the times of low water, they seem to show that some powerful disturbing cause has affected the low water at February 25, 17th 50m. Nevertheless, the tidal curves for the tides on both sides of that low water are undistinguishable, as viewed by the eye, from the rest; and in the grouping for the reductions, to be shortly explained, the numbers furnished by them do not differ sensibly from those given by the others. Considering it then as established that the peculiarities in these tidal curves are the representatives of real peculiarities in the tides at Southampton (such as may always be expected in every individual tide of nearly the same magnitude as those observed on this occasion), and in no degree due to the accidental circumstances of weather; and considering also that these tides exhibit no certain trace of diurnal tide; I shall proceed to explain the method by which they have been reduced into mathematical form. The method is, in many respects, similar to that which I used for the Deptford tides, in a paper published in the Philosophical Transactions for 1842. The first step was, to divide the whole series of observations into groups, each group representing one tide. This was done by taking the interval from low water to low water as one tide, and supposing it to correspond to $360^\circ$ of phase, and converting the interval between every observation in that tide and its commencement into phase, by that proportion. The next step was to reduce the different observations of height to one uniform scale. This was done by assuming every range of tide, from high water to low water or from low water to high water, to be represented by 2, and converting the depression of the water at every observation below the high water of that tide into abstract number, by that proportion. In this manner every observation gave a phase expressed in degrees, and a converted depression expressed in abstract number. The next step was to collect from all the tides the phases between $0^\circ$ and $5^\circ$ and their corresponding converted depressions, and to take their mean; then those between 5° and 10°; and so on. The phases thus obtained differed little from 2°·5, 7°·5, &c.; and the correction of converted depression for these small differences being easily found from the preceding and following numbers, the converted depressions for 2°·5, 7°·5, &c. were found. In this manner the following Table was formed: Corresponding phases and converted depressions in the Southampton Tides; the phase commencing with low water and increasing by 360° in one tide; and the depression being measured from high water, the whole range being called 2·000. | Phase | Converted depression | Phase | Converted depression | Phase | Converted depression | Phase | Converted depression | |-------|----------------------|-------|----------------------|-------|----------------------|-------|----------------------| | 2°·5 | 1·996 | 9°·5 | 1·194 | 18°·5 | 0·064 | 27°·5 | 0·217 | | 7°·5 | 1·979 | 9°·5 | 1·185 | 18°·5 | 0·029 | 27°·5 | 0·251 | | 12°·5 | 1·939 | 10°·5 | 1·182 | 19°·5 | 0·018 | 28°·5 | 0·316 | | 17°·5 | 1·890 | 10°·5 | 1·181 | 19°·5 | 0·027 | 28°·5 | 0·384 | | 22°·5 | 1·838 | 11°·5 | 1·163 | 20°·5 | 0·051 | 29°·5 | 0·508 | | 27°·5 | 1·770 | 11°·5 | 1·128 | 20°·5 | 0·079 | 29°·5 | 0·598 | | 32°·5 | 1·698 | 12°·5 | 1·080 | 21°·5 | 0·104 | 30°·5 | 0·755 | | 37°·5 | 1·639 | 12°·5 | 1·018 | 21°·5 | 0·132 | 30°·5 | 0·890 | | 42°·5 | 1·574 | 13°·5 | 0·941 | 22°·5 | 0·145 | 31°·5 | 1·021 | | 47°·5 | 1·519 | 13°·5 | 0·846 | 22°·5 | 0·149 | 31°·5 | 1·201 | | 52°·5 | 1·479 | 14°·5 | 0·748 | 23°·5 | 0·152 | 32°·5 | 1·347 | | 57°·5 | 1·422 | 14°·5 | 0·651 | 23°·5 | 0·144 | 32°·5 | 1·491 | | 62°·5 | 1·382 | 15°·5 | 0·563 | 24°·5 | 0·139 | 33°·5 | 1·641 | | 67°·5 | 1·329 | 15°·5 | 0·441 | 24°·5 | 0·125 | 33°·5 | 1·769 | | 72°·5 | 1·292 | 16°·5 | 0·350 | 25°·5 | 0·129 | 34°·5 | 1·864 | | 77°·5 | 1·256 | 16°·5 | 0·262 | 25°·5 | 0·132 | 34°·5 | 1·932 | | 82°·5 | 1·225 | 17°·5 | 0·185 | 26°·5 | 0·143 | 35°·5 | 1·976 | | 87°·5 | 1·209 | 17°·5 | 0·119 | 26°·5 | 0·174 | 35°·5 | 1·996 | By means of these numbers the curve fig. 1 in Plate II. has been constructed. To express these numbers in a mathematical form by means of a periodic function, it was assumed that the converted depression could be represented by the formula \[ A + B \cdot \sin \text{ phase} + C \cdot \sin 2 \text{ phase} + D \cdot \sin 3 \text{ phase} + E \cdot \sin 4 \text{ phase}, \] \[ + b \cdot \cos \text{ phase} + c \cdot \cos 2 \text{ phase} + d \cdot \cos 3 \text{ phase} + e \cdot \cos 4 \text{ phase}. \] Then \( A \) is the mean of all the converted depressions. \( B \) is found by multiplying every converted depression by its value of \( \sin \text{ phase} \), and dividing the sum of all the products by 36. \( b \) is found in the same way, using the values of \( \cos \text{ phase} \) as factors. \( C \) is found by using \( \sin 2 \text{ phase} \) as factor, &c. In this manner the following expres- sion was found for the converted depression: \[ 0·900 + 0·469 \cdot \sin \text{ phase} - 0·085 \cdot \sin 2 \text{ phase} - 0·057 \cdot \sin 3 \text{ phase} - 0·017 \cdot \sin 4 \text{ phase} \] \[ + 0·766 \cdot \cos \text{ phase} + 0·174 \cdot \cos 2 \text{ phase} + 0·182 \cdot \cos 3 \text{ phase} - 0·029 \cdot \cos 4 \text{ phase}; \] or \[ 0·900 + 0·898 \cdot \sin (\text{ phase} + 58° 30') \] \[ + 0·194 \cdot \sin (2 \text{ phase} + 116°) \] \[ + 0·191 \cdot \sin (3 \text{ phase} + 107° 22') \] \[ - 0·034 \cdot \sin (4 \text{ phase} + 59° 43'). \] If we make phase \(+58^\circ30'\) = \(p\), this becomes \[0.900 + 0.898 \cdot \sin p + 0.194 \cdot \sin (2p - 1^\circ) + 0.191 \cdot (\sin 3p - 68^\circ 8')\] \[-0.034 \cdot \sin (4p - 174^\circ 17').\] The theory of waves, so far as it has yet gone, fails to explain the form of this expression. It is not reconcileable with that obtained on the supposition that the extent of vertical oscillation bears a sensible proportion to the depth, either when the length of the channel is indefinite, or when the channel is interrupted by a barrier*. The latter supposition is that which would seem to represent most exactly the circumstances of Southampton. It is to be remarked, however, that the investigations which I have cited suppose the section of the channel to be rectangular; a supposition which accords little with the state of the channel of the Southampton water, as shown by the extensive banks of mud discovered at low water there. The investigations also exclude friction. The following consideration is suggested by the form of the expression at which we have arrived for the converted depression. If the tide were such as we suppose the sea-tide to be (that is, if its depression were expressed by a single sine or cosine), and if (as above) the phase were measured from low water, then the converted depression would be expressed by \[A' + B' \cos \text{phase},\] or \[A' + B' \sin (\text{phase} + 90^\circ).\] From this we might be led to conceive that in all cases the argument of the principal term expressing the depression would be \((\text{phase} + 90^\circ)\), the phase being a quantity which commences from low water. But we find that it does in this instance depend on \((\text{phase} + 58^\circ 30')\), or on \((\text{phase} + 90^\circ - 31^\circ 30')\), differing from the former by \(31^\circ 30'\), which corresponds to one hour of time nearly. Now there seems to be no reason (except the convenience of mariners) why cotidal lines and speculations on the progress of the tide should be connected with high water rather than with low water, or any other phase of tide; the only thing with which, in a scientific view, they can properly be connected (as it appears to me), is the epoch of the argument of the principal term in the formula for depression. But, in the instance of Southampton, the time of actual low water differs from the time of greatest depression given by that term, by one hour; the times of high water differ about forty minutes. If then the peculiarity of the Southampton tide has been created in part by the shallowness of the English channel, and exists on the coast (we have no accurate information whether this is true or not), then the position of the cotidal line determined by the high water is erroneous, with reference to the views above mentioned, to a considerable extent. Another consideration suggested by the same expression is this. The mean of the depressions at high-water and at low-water is 1.000. But the constant term which enters into the general formula for the depression is 0.900. Now I conceive the latter to be the number which truly represents the mean level of the water. In all cases, * See Philosophical Transactions, 1842, p. 6. therefore, in which the mean level of the sea in different places is compared, I conceive it is not sufficient to take the mean of high and low water, but it is necessary to express the depression by the formula $A + B \sin \text{phase} + b \cos \text{phase} + C \sin 2 \text{phase} + \&c.$, and to adopt the constant $A$ as determining the mean level. The difference, at Southampton, is about nine inches, by which quantity the mean level, as defined above, is higher than the mean of high and low waters. The singularity of the results obtained at Southampton induced me to take measures for observing the tides in another estuary; and for this purpose I selected Ipswich as a good station. The positions of these places are not exactly similar: Southampton is nearly at the head of its gulf; Ipswich may be considered as being absolutely at the head of its gulf. The length of the Orwell does not differ materially from that of the Southampton water; and in depth, breadth, magnitude of tide, &c., it resembles it closely. Nevertheless, it will appear from the following results that the laws of the tides at these two places are extremely different. An application made by me to Charles May, Esq. of Ipswich, was by him laid before the Dock Committee; and I received from that body a most courteous offer of every assistance which could be required for the observation of the tides. The immediate superintendence of the observations was undertaken by James Jones, Esq., Dockmaster; and to the talent and spirit with which this gentleman undertook the troublesome business, I am entirely indebted for the opportunity of presenting the Royal Society with trustworthy results. The observations commenced on 1842, May 24, 23h 40m (astronomical reckoning), and finished on May 27, 3h 35m; embracing four complete tides. The height of the water was observed at every five minutes. The zero of the scale is the Lock Sill. Great attention was given to the determination of time. No irregularity appears on the face of the observations except about May 25, 19h, after a heavy squall from the S.S.E. with violent rain. The times and intervals of high water are as follows: | Date | Time | |------|------| | May 25 | 1h 5m | | Interval | 12h 15m | | 25 | 13h 20m | | 26 | 1h 50m | | 26 | 14h 0m | | 27 | 2h 34m | The times and intervals of low water are, | Date | Time | |------|------| | May 25 | 6h 55m | | Interval | 11h 40m | | 25 | 18h 35m | | 26 | 7h 30m | | 26 | 19h 45m | And the heights and ranges are,— | High Water | Low Water | Range | |------------|-----------|-------| | ft. in. | ft. in. | ft. in. | | 15 10 1/4 | | 12 0 3/4 | | | 3 9 1/2 | 12 0 1/2 | | 15 10 | | 11 3 1/4 | | | 4 6 3/4 | 12 3 3/4 | | 16 10 1/2 | | 11 11 1/2 | | | 4 11 | 12 0 1/2 | | 16 11 1/2 | | 11 8 | | | 5 3 1/2 | 11 0 1/4 | | 16 3 3/4 | | | The observations were reduced in exactly the same way as those made at Southampton, with this single difference, that the phase here began from high water (the first and last conspicuous phenomenon in these observations having been high water). And the values thus found for the converted depression are the following. Corresponding phases and converted depressions in the Ipswich tides: the phase commencing with high water and increasing by 360° in one tide; and the depression being measured from high water, the whole range being called 2·000. | Phase | Converted depression | Phase | Converted depression | Phase | Converted depression | Phase | Converted depression | |-------|----------------------|-------|----------------------|-------|----------------------|-------|----------------------| | 2·5 | 0·012 | 92·5 | 1·197 | 182·5 | 1·894 | 272·5 | 0·984 | | 7·5 | 0·053 | 97·5 | 1·276 | 187·5 | 1·869 | 277·5 | 0·940 | | 12·5 | 0·120 | 102·5 | 1·362 | 192·5 | 1·822 | 282·5 | 0·891 | | 17·5 | 0·184 | 107·5 | 1·449 | 197·5 | 1·779 | 287·5 | 0·841 | | 22·5 | 0·251 | 112·5 | 1·527 | 202·5 | 1·723 | 292·5 | 0·791 | | 27·5 | 0·322 | 117·5 | 1·599 | 207·5 | 1·658 | 297·5 | 0·736 | | 32·5 | 0·389 | 122·5 | 1·674 | 212·5 | 1·586 | 302·5 | 0·668 | | 37·5 | 0·449 | 127·5 | 1·731 | 217·5 | 1·516 | 307·5 | 0·625 | | 42·5 | 0·513 | 132·5 | 1·780 | 222·5 | 1·449 | 312·5 | 0·564 | | 47·5 | 0·570 | 137·5 | 1·828 | 227·5 | 1·381 | 317·5 | 0·497 | | 52·5 | 0·631 | 142·5 | 1·874 | 232·5 | 1·315 | 322·5 | 0·437 | | 57·5 | 0·689 | 147·5 | 1·910 | 237·5 | 1·258 | 327·5 | 0·367 | | 62·5 | 0·757 | 152·5 | 1·940 | 242·5 | 1·211 | 332·5 | 0·297 | | 67·5 | 0·822 | 157·5 | 1·964 | 247·5 | 1·179 | 337·5 | 0·231 | | 72·5 | 0·890 | 162·5 | 1·978 | 252·5 | 1·144 | 342·5 | 0·168 | | 77·5 | 0·961 | 167·5 | 1·985 | 257·5 | 1·107 | 347·5 | 0·100 | | 82·5 | 1·043 | 172·5 | 1·974 | 262·5 | 1·068 | 352·5 | 0·043 | | 87·5 | 1·117 | 177·5 | 1·941 | 267·5 | 1·030 | 357·5 | 0·011 | By means of these numbers the curve fig. 2 in Plate III. has been constructed. To make the phase commence at low water, its numerical value must be diminished by about 166°. curve representing the law of rise and fall of the tide at Ipswich, as obtained from observation of four tides. From May 25th, 1792, to May 27th, 1794. Treating these numbers in the same way as those for Southampton, and still conceiving the phase to commence at high water, we obtain the following formula for the converted depression: \[1.055 + 0.112 \cdot \sin \text{phase} - 0.096 \cdot \sin 2\text{phase} + 0.030 \cdot \sin 3\text{phase} + 0.012 \cdot \sin 4\text{phase}\] \[= 0.851 \cdot \cos \text{phase} - 0.041 \cdot \cos 2\text{phase} - 0.076 \cdot \cos 3\text{phase} - 0.023 \cdot \cos 4\text{phase};\] or \[1.055 + 0.858 \cdot \sin (\text{phase} - 82^\circ 31') - 0.104 \cdot \sin (2\text{phase} + 22^\circ 59')\] \[+ 0.082 \cdot \sin (3\text{phase} - 68^\circ 23') + 0.025 \cdot \sin (4\text{phase} - 63^\circ 2').\] Let phase \(= 82^\circ 31' = p\); this becomes \[1.055 + 0.858 \cdot \sin p + 0.104 \cdot \sin (2p + 8^\circ 1') - 0.082 \cdot \sin (3p - 0^\circ 50')\] \[= 0.025 \cdot \sin (4p + 87^\circ 2').\] The first three terms of this expression are extremely similar to those in the expression for the Southampton tide: the principal difference between the two expressions is in the fourth term, or that depending on \(3p\). It is very remarkable that the algebraical difference between the formulæ for tides which are so strikingly different in general character, depends entirely on a term of the third order of the fraction expressing the proportion of the vertical oscillation to the depth of the water; an order to which (so far as I am aware) theory has reached only in one instance, namely that of a wave travelling along an unlimited canal, and then only on the restricted suppositions of rectangular section and absence of friction*. It is also worthy of remark, that the terms of the second order do not agree with those given by the restricted theory to which I have alluded, and that they differ materially from those given by the Deptford tides†. From a consideration of the discordance between the observations and the present state of theory in regard to the form of these terms, as well as from remarking the influence which they have upon cotidal lines and mean levels, I am inclined to fix upon the circumstances of waves in canals as more deserving of notice at the present time, both in theory and in observation, than almost any other branch of the theory of tides. I take this opportunity of correcting an error in my paper on the Deptford tides‡. I have there stated, on the authority of Mr. Wheewell§, that the age of the tide as inferred from the heights is greater than its age as inferred from the times. This, however, is incorrect. The age inferred from the heights is, in every instance that has been properly examined, less than that inferred from the times‖. I trust that, in a subject which is at first examination very confusing, it will be regarded as a venial fault to have erred in such company as that of Mr. Wheewell. In a theoretical view, this correction is very important. The cause assigned for the * See Philosophical Transactions, 1842, p. 6, and Encyclopædia Metropolitana, Tides and Waves, Article 210. † Philosophical Transactions, 1842. ‡ Ibid. p. 8. § Ibid. 1838, p. 236. ‖ Encyclopædia Metropolitana, Tides and Waves, Article 543. greater age inferred from the heights* is certain, as it is one given by observations which (here according with theory) show that the rise of tide occupies a shorter time at springs than at neaps. Some other cause must therefore exist, of such a kind as to make the tide later in springs than in neaps, and of such a magnitude as completely to overrule that already assigned. I conceive that this cause may be the increased range of the tide in horizontal extent up the river, at springs; by which the tide is changed from a gulf-tide to a continuous-canal-tide, which travels more slowly. But without more rigorous theory, I dare not pronounce upon this point. Royal Observatory, Greenwich, February 10th, 1843. * Philosophical Transactions, 1842, p. 8.