Waves and ripples in water, air, and æther — A Reader’s Guide

A visit to the seaside—What is a wave?—Wave-motion on water—Definition of a wave—Sea waves—Various forms of wave-motion—Wave length, velocity, and frequency—Atlantic waves—Rules for speed of sea waves—Illustrations of wave-motion—A stone falling on water—Production of a wave-train—Wave-energy—Conditions for the production of wave-motion—Distinction between wave-velocity and wave-train velocity—Why a wave breaks—Waves in canals—Rule for speed of a canal wave—Falling bodies—A “bore”—Tidal waves—Ripples—Distinction between waves and ripples—Surface tension on liquids—A needle floating on water—Experimental production of ripples—Reflection and refraction of ripples and waves—Interference of waves and ripples—Photography of waves and ripples 1

WAVES AND RIPPLES MADE BY SHIPS.

Ship-waves—The viscosity of liquids—How it is demonstrated—Rotational and irrotational motion in fluids—Eddies and whirls—Smoke rings—Vortex motion—Professor Hele-Shaw’s experiments—Irrotational or stream-line motion in water—The motion of water round a ship—The motion of water along a pipe—Flow in uniform pipes and non-uniform pipes—Relation between fluid velocity and pressure—Skin resistance and wave-making resistance—The movement of a fish—Motion through a perfect fluid—The waves made by moving objects—Waves made by ducks and swans—Echelon waves—Ship bow waves—The form of ship-waves—Mr. Froude’s experiments—Ship-models and experimental tanks—How a ship is designed—Froude’s laws—Testing ship-models—The design of a racing-yacht—Comparison of British and American yachts—The Cup race—Scott Russell’s experiments on canal-boats 57

WAVES AND RIPPLES IN THE AIR.

Air necessary for the production of sound—A sounding body is in vibration—Harmonic motion—The difference between noise and music—The nature of an air wave—The physical qualities of air—Longitudinal or compressional waves—Wave-models to illustrate the nature of sound waves—Quality of a sound—Velocity of an air wave—An illustration on a gigantic scale—The voice of a volcano heard round the world—The effect of temperature on air-wave velocity—Comparison of theory and experiment—Circumstances affecting distance at which sounds can be heard—Funeral guns—Fog-signals and sirens—Effect of wind and density—Sensitive flames as sound-detectors—Inaudible sounds—The reflection and refraction of sound waves—A sound-lens and sound-prism—The interference of sounds—Two sounds producing silence—The phonograph—A soap-bubble film set in vibration by air waves 103

The difference between sounds and musical tones—The natural period of vibration of an elastic body—The effect of accumulated impulses—Free and forced vibrations—Breaking down a bridge with a pea-shooter—The vibration of a stretched string—Stationary waves—A string vibrating in segments—Acoustic resonance—Nodes and anti-nodes—The musical scale or gamut—Musical intervals—The natural gamuts and the scale of equal temperament—Concords and discords—Musical beats—Helmholtz’s theory of discords—Musical instruments—Pipes—Strings and plates—A pan-pipe—An organ-pipe—Open and closed organ-pipes—The distribution of air pressure and velocity in a sounding organ-pipe—Singing flames—Stringed instruments—The violin—The Stroh violin—The structure of the ear—The ear a wonderful air-wave detector and analyzer 147

ELECTRIC OSCILLATIONS AND ELECTRIC WAVES.

The conception of an æther—The phenomena of light require the assumption of an æther—The velocity of light—Interference of light—Two rays of light can produce darkness—An electric current—The phenomena of electricity require the assumption of an electro-magnetic medium—Properties and powers of an electric current—Alternating and continuous electric currents—Electromotive force and electric strain—A Leyden jar—The oscillatory discharge of a condenser—Oscillatory sparks—Transformation of electric oscillations—Hertz oscillator—Production of a wave of electric displacement—Detection of electric waves—Metallic filings detectors—The coherer—Inductance and capacity of circuits—Electro-static and electro-magnetic energy—An induction coil—Electric oscillations give rise to electric waves—The electron theory of electricity 185

WAVES AND RIPPLES IN THE ÆTHER.

The experiments of Heinrich Hertz—Electric radiation—Lecture apparatus for producing and detecting electric radiation—Electric transparency and opacity—Why this difference—The reflection of electric radiation—The refraction of electric rays—An electric prism and an electric lens—The electric refractive index—Interference of electric rays—The velocity of electric radiations identical with that of light—Dark heat rays—Actinic or photographic rays—The cause of colour—The frequency of light waves—The classification of electric or æther waves—The gamut of æther waves—The eye an æther-wave detector of limited power—The electro-magnetic theory of light—Artificial production of light—Use of Hertz waves in wireless telegraphy—Marconi’s methods—Marconi’s aerial and wave-detector—The Morse alphabet—How a wireless message is sent—The tuning of wireless stations—Communication between ships and shore—The velocity of wireless waves—Conclusion 232

One statute mile is 5280 feet. One nautical mile is 6086 feet = 1¹⁄₆ statute mile. A knot is a speed of 1 nautical mile per hour.

Hence the following rules:—

To convert Knots to miles per hour—multiply by 1¹⁄₆. Miles per hour to knots—multiply by ⁶⁄₇. Feet per second to miles per hour—multiply by ²⁄₃. ⁄ Feet per second to knots—multiply by ⁶⁄₁₀. Knots to feet per minute—multiply by 100.

WAVES AND RIPPLES IN WATER, AIR, AND ÆTHER.

WATER WAVES AND WATER RIPPLES.

We have all stood many times by the seashore, watching the _waves_, crested with white foam, roll in and break upon the rocks or beach. Every one has more than once cast a stone upon still water in a lake or pond, and noticed the expanding rings of _ripples_; and some have voyaged over stormy seas, whereon great ships are tossed by mighty _billows_ with no more seeming effort than the rocking of a cradle. In all these things we have been spectators of a _wave-motion_, as it is called, taking place upon a water surface. Perhaps it did not occur to us at the time that the _sound_ of the splash or thunder of these breaking waves was conveyed to our ears as a wave-motion of another sort in the air we breathe, nay, even that the _light_ by which we see these beautiful objects is also a wave-motion of a more recondite description, produced in a medium called the _æther_, which fills all space.

A progressive study of Nature has shown us that we are surrounded on all sides by wave-motions of various descriptions—waves in water, waves in air, and waves in æther—and that our most precious senses, our eyes and ears, are really wave-detectors of a very special form. The examination of these waves and their properties and powers has led us to see that waves in water, air, and æther, though differing greatly in detail, have much in common; and many things about them that are difficult to understand become more intelligible when we compare these various wave-motions together. In these lectures, therefore, I shall make use of your familiar experiences concerning sea and water waves to assist you to understand some of the properties of air waves to which we owe our sensations of sound and music; and, as far as possible, attempt an explanation of the nature of æther waves, created in the all-pervading æther, to which are due not only light and sight, but also many electrical effects, including such modern wonders as wireless telegraphy. In all departments of natural science we find ourselves confronted by the phenomena of wave-motion. In the study of earthquakes and tides, telegraphs and telephones, as well as terrestrial temperature, no less than in the examination of water waves and ripples, sound, music, or light and heat, we are bound to consider waves of some particular kind.

Fastening our attention for the moment on surface water waves, the first question we shall ask ourselves is—What is a wave? If we take our station on a high cliff looking down on the sea, on some clear day, when the wind is fresh, we see the waves on its surface like green rounded ridges racing forward, and it appears at first sight as if these elevations were themselves moving masses of water. If, however, we look instead at some patch of seaweed, or floating cork, or seagull, as each wave passes over it, we shall notice that this object is merely lifted up and let down again, or, at most, has a small movement to and fro. We are led, therefore, to infer that, even when agitated by waves, each particle of water never moves far from its position when at rest, and that the real movement of the water is something very different from its apparent motion. If we place on the surface of water a number of corks or pieces of paper, and then watch them as a wave passes over them, we shall notice that the corks or bits of paper rise and fall successively, that is, one after the other, and not all together. A little more careful scrutiny will show us that, in the case of sea waves in deep water, the motion of the floating object as the wave passes over it is a circular one, that is to say, it is first lifted up, then pushed forward, next let down, and, lastly, pulled back; and so it repeats a round-and-round motion, with the plane of the circle in the direction in which the wave is progressing. This may be illustrated by the diagram in Fig. 1, where the circular dotted lines represent the paths described by corks floating on the sea-surface when waves are travelling over it.

Accordingly, we conclude that we have to distinguish clearly between the actual individual motion of each water particle and that general motion called the wave-motion. We may define the latter by saying that to produce a wave-motion, each separate particle of a medium, be it water, or air, or any other fluid, must execute a movement which is repeated again and again, and the several particles along any line must perform this same motion one after the other, that is, lagging behind each other, and not simultaneously. We might illustrate this performance by supposing a row of fifty boys to stand in a line in a play-ground, and each boy _in turn_ to lift up his arm and let it down again, and to continue to perform this action. If all the boys lifted up their arms together, that would not produce a wave-motion; but if each boy did it one after the other in order, along the rank, it would constitute a _wave-motion_ travelling along the line of boys. In more learned language, we may define a wave-motion by saying that _a wave-motion exists in any medium when the separate portions of it along any line execute in order any kind of cyclical or repeated motion, the particles along this line performing the movement one after the other, and with a certain assigned delay between each adjacent particle as regards their stage in the movement_.

It will be evident, therefore, that there can be many different kinds of waves, depending upon the sort of repeated motion the several parts perform.

Some of the numerous forms of wave-motion can be illustrated by mechanical models as follows:—

A board has fastened to it a series of wooden wheels, and on the edge of each wheel is fixed a white knob. The wheels are connected together by endless bands, so that on turning one wheel round they all revolve in the same direction. If the knobs are so arranged to begin with, that each one is a little in advance of its neighbour on the way round the wheel, then when the wheels are standing still the knobs will be arranged along a wavy line (see Fig. 2). On turning round the first wheel, each knob will move in a circle, but every knob will be lagging a little behind its neighbour on one side, and a little in advance of its neighbour on the other side. The result will be to produce a wave-motion, and, looking at the general effect of the moving knobs, we shall see that it resembles a hump moving along, just as in the case of a water wave.

The motion of the particles of the water in a deep-sea wave resembles that of the white knobs in the model described. Those who swim will recall to mind their sensations as a sea wave surges over them. The wave lifts up the swimmer, then pushes him a little forward, then lets him down, and, lastly, drags him back. It is this dragging-back action which is so dangerous to persons who cannot swim, when they are bathing on a steep coast where strong waves are rolling in towards the shore.

Two other kinds of wave-motion may be illustrated by the model shown in Fig. 3. In this appliance there are a number of eccentric wheels fixed to a shaft. Each wheel is embraced by a band carrying a long rod which ends in a white ball. The wheels are so placed on the shaft that, when at rest, the balls are arranged in a wavy line. Then, on turning round the shaft, each ball rises and falls in a vertical line, and executes a periodic motion, lagging behind that of its neighbour on one side. The result is to produce a wave-motion along the line of balls. By slightly altering the model, each ball can be made to describe a circle in a direction at right angles to the line of the balls, and then we have a sort of corkscrew wave-motion propagated along the line of balls.

Again, another form of wave-motion may be illustrated by the model shown in Fig. 4. In this case a number of golf-balls are hung up by strings, and spiral brass springs are interposed between each ball. On giving a slight tap to the end ball, we notice that its to-and-fro motion is handed on from ball to ball, and we have a wave-motion in which the individual movement of the balls is _in the direction_ of the wave-movement, and not across it.

The kind of wave illustrated by the model in Fig. 3 is called a _transverse_ wave, and that shown in Fig. 4 is called a _longitudinal_ wave.

At this stage it may be well to define the meaning of some other expressions which will be much used in these lectures. We have seen that in a wave-motion each part of the medium makes some kind of movement over and over again; and of its neighbours on either side, one is a little ahead of it in its performance, and the other a little in arrear. If we look along the line, we shall see that we can select portions of it which are exactly in the same stage of movement—that is, are moving in the same way at the same time. The distance between these portions is called _one wave-length_. Thus, in the case of sea waves, the distance between two adjacent crests, or humps, is one wave-length.

When we use the expression, _a long wave_, we do not mean a wave which is of great length _in the direction_ of the ridge, but waves in which the crests, or humps, are separated far apart, measuring from crest to crest _across_ the ridges.

Strictly speaking, the wave-length may be defined as the shortest distance from crest to crest, or hollow to hollow, or from one particle to the next one which is in the same stage of its movement at the same time.

Another way of illustrating the same thing would be to pleat or pucker a sheet of paper into parallel ridges. If we make these pleats very narrow, they would represent what we call _short waves_; but if we make these pleats very far apart, they would represent _long waves_.

Another phrase much used is the term _wave-velocity_. Suppose that a seagull were to fly along over a set of sea waves so as to keep always above one particular hump, or wave-crest; the speed of the gull, reckoned in miles per hour or feet per minute, would be called the speed of the waves. This is something very different from the actual speed of each particle of water.

A third and constantly used expression is the term _wave-frequency_. If we watch a cork floating on a wave-tossed sea, we observe that it bobs up and down so many times in a minute. The number of times per second or per minute that each particle of the medium performs its cycle of motion is called the wave-frequency, or simply _the frequency_.

Again, we employ the term _amplitude_ to denote the extreme distance that each individual particle of the medium moves from its mean position, or position of rest. In speaking of sea waves, we generally call the vertical distance between the crest and the hollow the _height_ of the wave, and this is twice the amplitude. With regard to the height of sea waves, there is generally much exaggeration. Voyagers are in the habit of speaking of “waves running mountains high,” yet a sea wave which exceeds 40 feet in height is a rare sight. Waves have been measured on the Southern Indian Ocean, between the Cape of Good Hope and the Island of St. Paul, and of thirty waves observed the average height was found to be just under 30 feet. The highest was only 37¹⁄₂ feet in height. On the other hand, waves of 16 to 20 feet are not uncommon. Travellers who have crossed the Atlantic Ocean in stormy weather will often recount experiences of waves said to be 100 feet high; but these are exceedingly rare, if even ever met with, and unless wave-heights are obtained by some accurate method of measurement, the eye of the inexperienced voyager is apt to be deceived.

In all cases of wave-motion there is a very close connection between the wave-velocity, or speed, the wave-length, and the wave-frequency. This connection is expressed by the numerical law that the velocity is equal to the product of the length and the frequency.

Thus, supposing we consider the case of Atlantic waves 300 feet from crest to crest, which are travelling at the rate of 27 miles an hour, it is required to calculate the frequency or number of times per minute or per second that any floating object, say a boat, will be lifted up as these waves pass over it.

We must first transform a speed of 27 miles per hour into its equivalent in feet per second. Since one mile is 5280 feet, 27 miles per hour is equal to 2376 feet per minute. Accordingly, it is easy to see that the wave-frequency must be 7·92, or nearly 8, because 7·92 times 300 is 2376. The answer to the question is, then, that the floating object will rise and fall eight times a minute. This rule may be embodied in a compact form, which it is desirable to hold firmly in the memory, viz.—

_Wave-velocity_ = _wave-length_ × _wave-frequency_.

This relation, which we shall have frequent occasion to recall, may be stated in another manner. We call the _period_ of a wave the time taken to make one complete movement. The periodic time is therefore inversely proportional to the frequency. Hence we can say that the _wave-length_, divided by the _periodic time_, gives us the _wave-velocity_.

In the case of water waves and ripples, the wave-velocity is determined by the wave-length. This is not the case, as we shall see, with waves in air or waves in æther. In these latter cases, as far as we know, waves of all wave-lengths travel at the same rate. Long sea waves, however, on deep water travel faster than short ones.

A formal and exact proof of the law connecting speed and wave-length for deep-sea waves requires mathematical reasoning of an advanced character; but its results may be expressed in a very simple statement, by saying that, in the case of waves on deep water, the speed with which the waves travel, reckoned in miles per hour, is equal to the square root of 2¹⁄₄ times the wave-length measured in feet. Thus, for instance, if we notice waves on a deep sea which are 100 feet from crest to crest, then the speed with which those waves are travelling, reckoned in miles per hour, is a number obtained by taking the square root of 2¹⁄₄ times 100, viz. 225. Since 15 is the square root of 225 (because 15 times 15 is 225), the speed of these waves is therefore 15 miles an hour.

In the same way it can be found that Atlantic waves 300 feet long would travel at the rate of 26 miles an hour, or as fast as a slow railway train, and much faster than any ordinary ship.[1]