# Gilbert T Walker - Papers and Reviews

We give below extracts from a few of Gilbert Walker's papers as well as extracts from reviews of a few of his publications. They exhibit the range of his interests: applied mathematics; meteorology; boomerangs; and the flight of birds.

Click on a link below to go to the information about that paper.

On Boomerangs.

The Physics of Sport.

The Meteorology of India.

Theory of Electromagnetism.

Some Problems of Indian Meteorology.

Meteorology and the Non-Flapping Flight of Tropical Birds.

The Flapping Flight of Birds.

1. Gilbert T Walker, On Boomerangs.
1.1. Gilbert T Walker, On Boomerangs [Abstract], Proceedings of the Royal Society of London 61 (1897), 239-240.

A typical returning boomerang resembles in general outline a symmetrical arc of a hyperbola, and is about 80 cm. in length measured along the curve. At the centre, where the dimensions of the cross section are greatest, the width is about 7 cm., and the thickness 1 cm.

Of the two faces, one is distinctly more rounded than the other; in addition the arms are twisted through about 4º, in the same manner as the blades of a right-handed screw propeller.

Such an implement, if thrown with its plane vertical, will describe a circular path of 40 or 50 metres in diameter, rising to a height of from 7 to 12 metres, and falling to the ground with its plane of rotation horizontal at a point somewhere near the thrower's feet.

The flight may be regarded as a case of steady motion, of which the circumstances gradually vary. In the more complicated, as well as the simpler, paths, observation makes it clear that everything depends on the changes in direction and inclination of the plane of the boomerang, and that the character of these changes is always the same; if they can be explained theoretically, the peculiarities of the motion may be accounted for.

Since the effects of the different forces at work are conflicting, it is necessary to adopt quantitative methods, even if the degree of accuracy attainable is not high; accordingly ratios comparable with a tenth are treated as small, and their squares neglected.

If we regard the boomerang as a thin, slightly distorted lamina, and integrate over it the forces indicated in S P Langley's paper on "Experiments in Aerodynamics," [Smithsonian Contributions to Knowledge, 1891] we can obtain equations of motion. From these, treating the motion as steady (to the first approximation), we may deduce the values of the angular velocities on which the direction of the axis of rotation depends. Five cases are worked out numerically, and the various effects of the "rounding" and "twisting" agree in character with the experimental facts; the discrepancies in actual magnitude are not larger than might, from the nature of the case, have been anticipated.

The theoretical results may be further tested by applying them to determine the conditions favourable to the production of other flights in which, after the first circle, a loop is described, either in front of or behind the thrower; in each of these cases success has been attained. An explanation is also afforded of the returning, of a boomerang without "twist," made by Mr O Eckenstein, and of the wonderfully long, straight trajectories of some of the native non-returning implements.

1.2. Gilbert T Walker, On Boomerangs, Philosophical Transactions of the Royal Society of London 190 (1897), 23-41.

The attempts that have hitherto been made to explain the flight of a boomerang have in general been of a somewhat fanciful nature.

Exception must be made in the case of such papers as those of Werner Stille, "Versuche und Rechnungen zur Bestimmung der Bahnen des Bumerangs " (Poggendorff, 'Annalen der Physik,' Bd. 147, 1872), and of Edmund Gerlach, "Ableitung gewisser Bewegungsformen geworfener Scheiben aus dem Luftwiderstandsgesetze"('Zeitschrift des Deutschen Vereins zur Förderung der Luftschifffahrt,' Heft 3, 1886). In the latter, which is the most noticeable contribution to the subject with which I am acquainted, the author gives an explanation in general terms of some of the effects of the air-resistance upon a symmetrical boomerang: he introduces, however, no analytical treatment of the dynamics of the rotating body and neglects entirely all consequences of the important deviations from symmetry which I have subsequently described as "twisting" and "rounding." Without one of these a return flight is, I believe, impossible.

For an account of the native Australian weapons, and in particular those of Victoria, reference should be made to the very complete descriptions given in Brough Smyth's book, 'The Aborigines of Victoria,' vol. 1, pp. 311-318; shorter notices are to be found in books of travel, such as that of Karl Lumholtz, 'Among Cannibals,' p.50.

Boomerangs may at the outset be divided into two classes - returning and non- returning; it is rather on weapons of the latter of these types that the natives of Australia rely when engaged in war or the chase. A typical returning boomerang resembles in general outline an arc of a hyperbola, and is about 80 centimetres in length measured along the curve. At the centre, where the dimensions of the cross section are greatest, the width is about 7 centimetres, and the thickness 1 centimetre; these dimensions become smaller as the ends are approached.

As a rule two properties are present. In the first place, the transverse section at any point would show that one surface possesses distinctly greater curvature than the other; secondly, the arms of the implement must be slightly twisted (from coincidence with the plane through each of them) after the fashion of the blades of a screw propeller or a windmill. The direction of the twist is such that rotation about a normal to the plane tends to set up linear velocity of the boomerang in the direction of the vector representing that rotation. These two peculiarities will in future be referred to as the "rounding" and the "twisting."

A weapon of this type is thrown in a horizontal direction in such a way as to impart considerable rotation in the vertical plane containing its initial direction of motion; the more convex surface is towards the thrower. The plane of rotation leans slowly over to the right (i.e. the vector representing the spin begins to point slightly upwards) and the path curls to the left. The projectile proceeds to describe a loop whose longer diameter is about fifty yards; it gradually rises until it reaches a height which is usually about thirty feet from the ground, travels horizontally for a time, and then gradually sinks to the earth.

The change in the angular motion has throughout the flight continued unaltered in character; the inclination of the plane of rotation to the horizon has steadily diminished from a right angle to zero, and the axis of the spin has veered continually to the left (as seen from above) in such a manner that as long as the linear velocity remains large, the angle between the direction of motion and the plane of rotation is small.

1.3. Gilbert T Walker, Boomerangs, Nature 64 (1901), 338-340.

Boomerangs may be studied for their anthropological interest as examples of primitive art, or for the manner in which they illustrate dynamical principles. But there is extraordinary fascination in making and throwing them, and in watching the remarkable and always graceful curves described in their flight; accordingly, my chief object in the following paper has been to diminish the practical difficulties of the subject by giving some of the results of ten years' experimental acquaintance with it.

1.4. Gilbert T Walker, Note on the Indian Boomerangs, Journal of the Asiatic Society of Bengal 20 (1925), 205.

The existence in the Madura district of projectiles resembling small boomerangs is well known: they are made of wood, bone or ivory, and have a knob on one end while the other end does not end in a point but is cut off in a straight line nearly perpendicular to the curve. I have not seen them thrown but from their shape I infer that the right forefinger is hooked round the knob while the straight edge of the other end is pressed gently against the chest: a flick with the wrist will then give considerable spin, and without this the implement will not travel far. Some at least of those that I have seen have a twist in their plane and might perhaps, without much modification, be capable of describing part, or even the whole, of a circle if thrown with great force; but I have not seen enough examples to know whether the twist, which resembles that of the arms of a windmill or of a screw propeller, is deliberate or accidental.
2. Gilbert T Walker, The Physics of Sport.
2.1. Gilbert T Walker, The Physics of Sport, The Mathematical Gazette 20 (239) (1936), 172-177.

A paper at the Annual Meeting of the Mathematical Association, 2 January 1936

If anything were calculated to make me anxious to do justice to my theme tonight it would be the association with your society of the men to whom I owe my earliest introduction to dynamics - at St Paul's School to Mr Pendlebury, your Secretary, and at Trinity College, Cambridge, to Professor Forsyth, your incoming President. The interest that they implanted has survived for half a century; and the applications to sport that I propose to describe are the immediate outcome of that interest. The dynamical effect which is most widely known in ball games and is perfectly familiar to all of you is the curling of the path of a rotating sphere when moving through the air. It is seen in the slice of a golf ball, the cut on a tennis ball and the swerve in cricket; and the explanation, often attributed to Tait, had been previously given by Rayleigh as well as, in general terms, by Newton.

But a racket court provides another anomaly; for there the ball is heavily undercut by the server and yet rebounds in the downward direction off the back wall instead of upwards as we should expect: obviously the direction of rotation must be reversed on impact on the front wall. Now one of the most conspicuous features in the court is the large size of the marks left by the ball flattening out on the walls: so that the idea of taking moments about the point of impact of a rigid sphere and thereby retaining the direction of rotation is impossible. The more natural supposition is that the ball flattens out to the moment of greatest compression and then rebounds with the velocities of all its parts reversed, bearing a constant ratio, say e, to their values on impact: and this idea, which gives a reversed spin of e times the original spin, explains our paradox.
...

A well-known tool of the stone age is the hatchet head or "celt" still in widespread use: and under favourable conditions this has curious properties. When placed on a fixed plate of clean glass many of them will spin in one direction and not in the other. The celt is oblong and at the point of contact with the plate the lines of curvature are not, in general, precisely parallel with the dynamical axes. There is rotational asymmetry, and this shows itself when the celt is spun. The theory brought out another paradox: when tapped at one end rotation is set up, and the direction of this may be reversed merely by raising the centre of gravity.

My last example shall be taken from living beings. When doing a high jump of six feet a man approaches the bar in a vertical attitude and when he clears it he is nearly horizontal; so, unless precautions are taken, he will during his descent, for a second time, turn through nearly two right angles; and his alighting would be most unpleasant but for violent contortions executed on the way down, which take some time to learn. Perhaps the best exponent of this art is the cat who, if suspended by the paws with her back only a few inches above a table, and released, will fall on her feet. I believe that the cat at once holds out its long hind-paws at right angles to its body and draws in its fore-paws, so that by a rapid contortion which rotates its forward portion by about 180º the hind-part, with its bigger moment of inertia, will rotate through only say 60º in the opposite direction; the contortion is then reversed with the fore-paws extended and the hind-paws withdrawn. They are completely spun round and catch the table, so that the forward part of the body can be quickly spun round and the cat is on its feet. I will illustrate this by getting on to a platform mounted so as to rotate without appreciable friction and, by repeated use of my arms, turn my body completely round as often as I wish, although at no instant is there the slightest angular momentum about the vertical.
3. Gilbert T Walker, The Meteorology of India.
3.1. Introduction to The Meteorology of India.

Gilbert T Walker, The Meteorology of India, Journal of the Royal Society of Arts 73 (3793) (1925), 838-855.

The Chairman [Sir Reginald Arthur Mant], in introducing the reader of the paper, said that Sir Gilbert Walker, after a brilliant career at Cambridge, where he came out as Senior Wrangler, was elected a Fellow of Trinity and appointed mathematical tutor, so that he might have looked forward to spending the rest of his days in peaceful contemplation of the higher mathematics. But "some far-reaching Nemesis steered him" from his home on the banks of the Cam and he fell under the spell of that most coy and capricious of all mistresses, the weather. In 1902 he joined the Meteorological Department in India, and two years later was appointed Director-General. For upwards of twenty years he studied the habits of his wayward mistress and learned to anticipate her moods. His annual forecasts of the monsoon rains both winter and summer were always anxiously awaited, because the amount and distribution of rainfall in India affected the happiness of countless millions of people. He perfected a system of storm signals by which he saved many ships and their crews from destruction. In recent years the Air Force had learned to rely on his forecasts of the winds before setting out on raids and expeditions. It seemed strange that a man sitting in a secluded office in Simla should be able to keep his finger on the pulse of the spirits of the air, but that was what Sir Gilbert did, and we would ask him now to give the Society some idea as to how it was done.

3.2. The Meteorology of India, by Sir Gilbert T Walker, C.S.I. , M.A., Sc.D., Ph.D., F.R.A.S., F.R.S., Professor of Meteorology, Imperial College of Science and Technology, and late Director of Indian Observatories.

The climate of India is of special interest, not merely as that of the greatest tropical region in the British Empire, but also because it seems to have been designed by nature with the object of demonstrating physical processes on a huge scale. The area involved includes the driest of deserts and the steamiest of swamps ; part of it is in great measure isolated from external influence to the north by the most complete mountain barrier in the world, so that many of nature's experiments are not disturbed by interference from outside; and in Ladak, which is a western continuation of Tibet, we can observe minimum temperatures which are not only relatively, but actually, arctic.

The explanation of the great temperature differences lies in the extreme transparency of the air to heat radiated from the ground during the very dry winter periods. At the Punjab hill stations on clear nights in December the warmth of the ground radiates fast into space; and the air in contact with the ground is chilled, becomes heavy and runs away downhill, its place being taken by unchilled air at 38º or 40º F., of which there is an unlimited supply : so the temperature at the hill station cannot fall below about 38º. On the other hand, at Rawalpindi, at the foot of the hills, the air, after being chilled, can find no escape and the minimum of 37º so produced is, night after night, lower than that of the hill stations. If we want a low minimum at a high elevation, we must find a place like an air tank without an outlet: accordingly at Annandale, 800 feet below the Simla ridge, where the air is largely enclosed, the night temperature is about 180 lower than on the ridge. An obvious application of the principle was to use a low canvas screen to check the escape of air from the Simla skating rink ; and while in the gentler frosts of early winter the ice was improved, the screen became unpopular in January, because the ice was too hard to be skated on with pleasure. Even in the daytime the chilling by radiation at Simla is so powerful that it is customary to flood the rink at 10.30 in the morning, for the water will be good ice for skating by 4.30 p.m., though the temperature in the shade four feet above the ice varies between 43º and 50º F. At Dras, in Ladak, the conditions in a broad valley at 10,000 feet are so favourable for radiation that 72 Fahrenheit degrees of frost are occasionally measured at our observatory. The familiar "cold weather line" of the Punjab hill stations marks the boundary over the plains between the chilled air below and the warmer air above; the evening's smoke and dust will rise through the cold air as soon as they are warm enough to do so, but on reaching the boundary these will be colder than the heated and light air above the boundary; further ascent will be impossible and the smoke must spread out in a horizontal sheet.
4. Theory of Electromagnetism.
4.1. E P Adams, Review: Outlines of the Theory of Electromagnetism, by Gilbert T Walker, Science 34 (876) (1911), 493-494.

This volume contains a series of lectures delivered before the Calcutta University on some of the more important developments of electromagnetic theory. The chief novelty of it, compared with other English books on mathematical physics, lies in the consistent use of vector methods, and their advantage is shown by the large amount of material condensed into fifty-two pages. While little of the material is new, the book will be of great assistance to those who wish to familiarise themselves with the present condition of the theory, as well as to those who wish to obtain a working knowledge of vector methods applied to physical problems. For the latter object no better course could be devised than a careful study of this book, with frequent transformations of the vector formulae into their more familiar Cartesian equivalents.

The first chapter gives an outline of vector analysis, including the vector expressions for the more important analytical theorems of constant use. In the second chapter vector methods are applied to magnetostatics, and here the advantages of these methods are most clearly brought out. The third chapter gives a statement of the Hertzian form of Maxwell's equations. In Chapter IV Hertz's theory for moving media is discussed and it is shown how experimental results prove its inadequacy.

The motion of a single charge moving with uniform velocity through the ether is considered in Chapter V, and in the next chapter the electron theory of Lorentz is applied to stationary media. The treatment of stresses within a material medium is not satisfactory; no account is taken of the variation of specific inductive capacity with the state of strain, and therefore the stress system obtained is that of Maxwell, which we know is not capable of experimental verification. In the last chapter Lorentz's theory is applied to moving bodies, ending with a brief account of aberration. The interpretation of the Lorentz transformation in terms of the theory of relativity is not touched upon.

There are many other matters that might properly come within the scope of this work, but it does not profess to be exhaustive, and as an outline it may be commended most highly.

4.2. G W O Howe, Review: Outlines of the Theory of Electromagnetism, by Gilbert T Walker, Science Progress in the Twentieth Century (1906-1916) 6 (24) (1912), 700.

These lectures were primarily intended to be of use to the lecturers in the outlying colleges and also to the more advanced students in Calcutta. After a brief outline has been given of the methods of vector analysis, these are applied to various fundamental problems of magnetostatics and the electro-magnetic field from the Maxwell-Hertz point of view; the electron theory of Lorentz, both for fixed and moving media, is then developed by the same methods. The matter is necessarily highly condensed.
5. Some Problems of Indian Meteorology.
5.1. W E W, Review: Some Problems of Indian Meteorology by Gilbert T Walker, Geography 15 (4) (1929), 312.

This small booklet gives a sufficient number of diagrams to allow the reader to follow the arguments of the lecturer. Certain aspects of dynamical meteorology were discussed, and new pieces of research in several branches were described, particularly with reference to polygonal cloud formation, river lines in the general circulation of the atmosphere, tropical cyclones, and the long period forecast. The lecturer proposed the term "foreshadow" for the last phrase, and established very high percentages of correlation between meteorological phenomena which are widely separated both in time and place.
6. Meteorology and the Non-Flapping Flight of Tropical Birds.
6.1. R C Miller, Review: Meteorology and the Non-Flapping Flight of Tropical Birds, by Gilbert T Walker, The Condor 26 (2) (1924), 80-81.

Review of Meteorology and the non-flapping flight of tropical birds, by Gilbert T. Walker, Proceedings of the Cambridge Philosophical Society 21 (1923), 363-375.

Walker on Soaring Flight.- This paper is of considerable interest to students of flight, as it is written primarily as a criticism of the unusual views advanced by E H Hankin in his volume entitled "Animal Flight" (London, Iliffe), and in various articles published elsewhere. Hankin's not very hopeful conclusion, from a great mass of data on the soaring flight of Indian kites and vultures, is that the phenomenon is shrouded in complete mystery. Walker, whose observations cover the same territory and are for that reason the more valuable, finds in air currents a sufficient explanation of this type of flight.

In the early morning, in the vicinity of Agra, atmospheric conditions as affected by temperature are extremely stable up to a height of 1 kilometre, and moderately stable between 1 and 2 kilometres. In the afternoon, however, there may be a difference of 22º C. between the temperature of the ground surface and that of the atmosphere 1.2 meters above, and the temperature may drop further 17º up to an altitude of 1 km. This causes conditions of great instability, creating currents which are sufficiently powerful in April and May "to reverse the direction of the ground winds over the whole region represented by Bareilly, Lucknow and Benares." Over rocky or sandy soil, aviators report an "upward bump" to a height of several thousand feet, and a "downward bump" on passing over green vegetation or water. At Simla within $1\large\frac{1}{2}\normalsize$ hours of (after?) sunrise currents of 6 to 10 feet a second are common at heights of only 20 feet above tree-clad slopes facing the sun." At Agra a rather crude recording instrument, set on a tower 45 feet above the ground, on sunny days "indicated ascending currents beginning shortly before the upward gliding of birds and ending shortly after this had ceased."

It is stated that the "alula" type of wing has been successfully introduced in airplane design, its tendency being to reduce the angle of descent in gliding. This point, in the opinion of the reviewer, the author has insufficiently developed. Contrary to some theorists, a steady horizontal wind is as useless in gliding as a perfect calm. But winds are constantly changing in velocity and direction. Calculations are introduced to show that the requisite energy for soaring may be derived from successive gusts of wind. This is regarded as a sufficient explanation of what Hankin calls "wind soarability" in the absence of sunshine. Also Rayleigh's theory of energy derived from a progressively increasing wind velocity with gain of altitude may occasionally suffice to explain gliding; at least it is a "useful auxiliary." In the case of gulls circling about the stern of a steamer, calculations are introduced to show that the differential wind velocity astern affords an adequate source of energy for this type of gliding.

In general, however, ascending currents are regarded as the source of energy of soaring flight. The author states: "During the past 7 years I have not seen a bird gliding upwards in a region where, from physical causes, descending currents could be expected; and in most cases ascending air has been strongly indicated." The paper is concluded with a review of certain inaccuracies and discrepancies in Hankin's work, from the point of view of physics and mechanics, and mention of certain items of observation in which Walker cannot concur. Most students of flight have felt that Hankin took insufficient account of known physical laws in arriving at his conclusions, and will appreciate this timely criticism, which is at the same time a contribution to our knowledge of soaring flight, clearly and concisely expressed.
7. The Flapping Flight of Birds.
7.1. Gilbert T Walker, The Flapping Flight of Birds, The Aeronautical Journal 29 (179) (1925), 590-594.

In connection with gliding flight enough measurements of "lift" and "drag" have been made to enable us to calculate the conditions for success of an aeroplane fitted with wings of standard sections; but no attempt has been made, as far as the present writer is aware, to ascertain what would happen if a flying machine were fitted with wings of standard section and these were flapped in a rhythmical manner. Would it support and propel itself? Several authors, including M F Fitzgerald and Colonel J D Fullerton in the previous issue of this journal, have discussed various portions of this problem; but instead of appealing to wind-tunnel determinations the latter has used such expressions as $pSUV$ or $pSUV^{2}$ for the pressure on a wing of area $S$, where $p$ is the density of the air and $V, U$ are the component velocities along and at right angles to its surface.

7.2. Gilbert T Walker, The Flapping Flight of Birds II, The Aeronautical Journal 31 (196) (1927), 337-342.

In a paper in the number of this journal for November, 1925, it was shown that if a flying machine were fitted with wings of standard section and these were flapped in a rhythmical manner, the machine would be supported and propelled, its weight, dimensions and velocity being those of a typical bird; the degree of accuracy attempted in the analysis did not exceed five per cent.

A subsequent examination of the power involved established a high efficiency; but this was due to a large amount of negative work during up-beats; and although such conditions might be maintained in a mechanical model they were unlikely in a bird, which has a large muscle, the pectoralis minor, for lifting its wings. In a bird efficiency seemed to require that the angle of incidence of the outer portion during an up-beat should be negative; the angle was however made zero in order to secure adequate lift, and it is this feature which gives rise to the negative work.

Last Updated June 2021