H2 Wave motion

Composition of two transverse oscillations

Just as a longitudinal and a transverse wave propagate simultaneously along a row of points, so do transverse waves. In Optics, a wave composed of two transverse waves has a special role. In order to be able to to relate to special directions of motion arising in this context, we relate them to an eye in the row of points, which looks along it (Figs. 290/291/292).

If the first point of the row receives an impulse perpendicularly to the row so that its points oscillate up and down, a transverse wave arises, the plane of which is vertical, that is, image and position of the row of points are represented by Fig. 283 (5), if you place the page vertically. In contrast, if the point is also pushed vertically to the row of points, but in such a manner that it oscillates horizontally, you obtain a transverse wave, the plane of which lies horizontally, that is, image and position of the row of points are then again represented by Fig. 283 (5), if you place the page horizontally.

These two transverse waves are now to propagate simultaneously along the row of points. Let the first point receive at the same instant a push upwards and towards the right side (the directions relating to the eye observing within the row); for the sake of simplicity, let both impulses be equally strong. The point then describes a straight line (inclined towards the right upwards and back towards the left downwards) which follows from the parallelogram of motions. The two straight oscillations combine into a single straight oscillation. You obtain a transverse wave, the type of which coincides with each of the transverse component waves, which both impulses would have generated on their own. Only the position of the plane of the oscillation as well as the amplitude of the combined wave differ. However, what happens if not both waves start simultaneously?

Especially important (for an interpretation of certain optical phenomena) is when the point receives the second impulse - let it again be horizontal - only after it has completed 1/4 of its vertical oscillation, that is, when it is at the return point above, has the velocity 0 and is just about to return downwards. Under these conditions, a quarter of the vertical transverse wave has formed, as the horizontal one is only about to start (one says: The two waves have a phase difference of 1/4 of the wave length). At the instant when the oscillating point turns around, it has two velocities - one vertically downwards, the other horizontally towards the right hand side. The first is zero, the second has the maximum starting value. Both velocities change: The vertical one increases from 0 to its maximum value, the other one decreases simultaneously from the same maximum value to zero. Due to the one velocity component, the point approaches its initial position of rest, while the other one distances it by the same amount from it, so that it always stays at the same distance from it, that is, it remains always at the same distance from it and moves from that return point along a circle around the shown directions, the point covers (for the eye observing in the row of points) the circle in the clockwise direction. Every point of the row does so, but starting each somewhat later than the preceding point.

The circles lie on a cylinder, the axis of which is the straight line, along which the points were located initially and the cross-section of which at right angle to the axis is a circle determined by the trajectories of the points. The image, which the row of points offer at the instant, at which the first point has completed its first circle, is a screw line around the cylinder (Fig. 292). To the eye observing from the side, the moved row of points is like a snake which creeps along the cylinder. (you obtain an instantaneous view of the forming wave from that of the water waves (Fig. 285), if you hold the figure vertically, so that they lie on top of each other like the coins in a horizontally oriented row and look along it. The difference in the mode of generation of the two waves is only that the horizontal push - again related to the eye in the row of points - in the first case is directed along the row, in the second case towards its right hand side. In the first case, there arise circles which rotate in the direction of the view, in the second case, circles which rotate about it.

What means "polarized"

There exists along the row of points only one direction of oscillation; however, there exists an infinity of them perpendicular to it. This last fact leads to a fundamental difference between transverse and longitudinal waves (which explains several phenomena of Light and Sound). We will discuss this aspect already here, although it is only required when we will dicuss polarization of Light. So, what is the meaning of the word polarized?

We will start with longitudinal waves (Fig. 290). Consider a row of molecules, which oscillate longitudinally. The longitudinal wave propagates in the direction of the arrow. If you look in this direction of the row of molecules, you see that none of the molecules leave the direction of the arrow. The projection of all of the oscillating molecules on to a plane, perpendicular to the direction of propagation of the wave, is at each instant a point.

This situation differs radically in the case of a transverse wave (Fig.291). There exist in a plane perpendicular to the direction of propagation around the direction of viewing an infinity of directions. If we take the direction of viewing as an axis, perpendicular through the face of a clock, towards which you look, every possible position of its hand can yields such a direction, for each of them is perpendicular to the direction of viewing. In the transversally oscillating row of molecules, everyone of these possible directions of oscillation is present - how then does the eye see at a given instant the arrangement of the molecules? Understand that at each instant each direction of motion is present and each distance from the line of viewing, which a molecule can reach within the amplitude. Moreover, there exist infinitely many molecules, which oscillate simultaneously and which the eye sees simultaneously projected on to the face of the clock. Hence you will now understand that at each instant the eye sees the totality of molecules projected on to a circular disk with the radius of the amplitude of oscillation and every point of the disc occupied by a molecule. You have to envisage (according to the wave hypothesis), as has been described here, the oscillations in an ordinary - one says natural ray of light. The ray is the straight line, along which the wave advances, whence it would be more correct to talk of a natural and polarized wave. But, as a rule, this is not done.

Next, consider a straight line polarized ray of light. Consider just a single one (one position of the hand of a clock) of the infinitely many possible directions of oscillation. All of them then occur all the time parallel to the same diameter of the circular disk, and therefore lie in a single plane. The cross projection of the wave is then reduced to a single diameter of the circle. You can then talk of sides of the ray - an observation which does not make sense in the case of a longitudinal wave. You see then that the are no oscillations towards certain sides. The oscillations are held fixed on a certain straight line, just like a magnet needle with two poles. This analogue is the reason for the term polarized. - The wave, described by Fig. 283, then belongs to a straight line polarized ray of light. Its plane of polarization is the plane of the drawing.

We can explain the difference between a natural (ordinary) and straight line polarized ray also as follows: Imagine a compass, a ray of light through its centre (point of suspension) of the compass needle and molecules oscillating always to and fro only along the needle. If the needle is at rest, the molecules only oscillate in one direction and its opposite direction, for example, only from North to South and vice versa. As a result, there arises a straight line polarized ray of light with its plane of oscillation a vertical plane in the North-South direction. (You can then talk of the East- and West-sides of the plane.) If you turn the needle of the compass, also the plane of oscillation rotates. If the needle turns all the time and so fast, that it covers the compass card one million times per second, also the direction of oscillation changes correspondingly fast, and the molecules oscillate in this time in all possible directions of the compass card perpendicularly to the ray - just as we imagine it to be in an ordinary ray of light. Heinrich Wilhelm Dove 1803-1879 has justified this concept by a corresponding experiment (with sufficiently fast rotating calcite).

The molecules in a polarized ray do not always oscillate on a straight line, so that here the term straight line is required. The points can also describe circles (Fig. 292). The side view is then a circular line consisting of the points which circulate around the direction of propagation (Fig. 293). The ray is then said to be circularly polarized - right circular or left circular - depending on the direction of travel. The circular trajectory then arises by two straight line polarized transverse waves propagating simultaneously along the row of molecules at the phases which differ by a quarter of the wave length. If this phase difference lies between zero and 1/4 wave length, you obtain an ellipse, and the ray is then said to be elliptically polarized. Then the particles circulating about the initial straight line form together a screw line about an invisible cylinder with an elliptic cross-section, around which they wind in the direction of propagation of the wave. The eye which looks along the initial positions of rest of the molecules sees then an ellipse formed by the particles circulating around their positions of rest. We do not know of otherwise polarized light.

Another wave, which is similar but special, is that of Light which propagates according to the electro-magnetic theory of Maxwell ( used in wireless transmission of text and language ) It too involves as components two cross waves, which propagate simultaneously along the same straight line and oscillated in mutually perpendicular planes. But each of them exists in its form uninfluenced by the other one - hitherto the forms of the components combined into a new wave. In this case, each wave represents physically something else: The one a pendulum like oscillation of electrical forces, the other one pendulum like oscillations of magnetic forces; they form together, at the same phase, an electro-magnetic wave. If the stretched out left forefinger is pointed in the direction of propagation of the wave, the thumb E indicates the direction of the electrical force anywhere in the wave, the central finger M indicates the direction of the magnetic force.

Superposition of oscillations

Hitherto, in compositions of waves, the impulses have formed an angle: One of them was horizontal, the other vertical. Things simplify greatly when the impulses have the same or mutually opposite directions! If the first point of the row is driven downwards, an action which generates the cross wave of Fig. 283, and if it is then 1/2 or 3/2 or 5/2. etc. periods later, when it passes again at its starting velocity through its initial position upwards, given the same push downwards, it obviously comes to rest, and so does eventually every point a little later than the preceding one along the entire row. In fact, the second impulse forms Wave II of Fig. 295, when every particle is given a push, equally strong in the direction opposite to that in Wave I.

If the second push had been directed upwards as the point passed through the position of rest upwards, it would add to the first and generate a wave, which gave every particle a push equally large and equally directed as that it had received from Wave I. In Fig. 296, Curve II would then lie in the same position as Wave I and the result would be Wave III. Both waves have the same length, amplitude and phase. The purpose of Fig. 353 is now readily understood. The two lower curves are wave components, the top curve the corresponding wave resultant. Distances from the position of rest on the same side are added, those on opposite sides subtracted.

Superposition of oscillations has technically most important significance in wireless transmission of voices. The oscillation of the sound source is converted in the microphone into an alternating current. In the ordinary telephone, a wire transmits this current to the person at the other end. In wireless transmission, the alternating current from the microphone is superimposed on an alternating current (generated by a special generator) radiated by the aerial. In other words: The electrical oscillation of the converted sound is forced upon the electrical oscillation of the antenna - this process is called frequency modulation - and the oscillation of the aerial becomes the carrier of the alternating current of the microphone (that is, indirectly the transmitter of the sound). In this form, the oscillation coming from the aerial cannot be understood by the listener, because, if it were converted directly into sound, if would yield a tone beyond the ear's frequency range. The process of making it audible is discussed under the heading beats.

It is logically clear that the motion of a particle, which is acted upon by several waves, must arise from the impulses it receives. But we still lack the means of visualizing the process. We only do so, if we can follow by eye two waves like those of Fig. 297 which run on the surface of water. The concentric circles about A and B represent wave rings (say, generated by a simultaneous disturbance at A and B). If the 1., 3., 5., 7., ··· circles about A and B correspond at given instants to wave crests, then 2., 4., 6., ··· correspond at the same instant to wave troughs. The points, at which a circle about A intersects a circle about B, correspond to points, where simultaneously both waves act. At the points, where odd (even) numbered circles intersect, elevations (depressions) coincide; at these points equally directed drives generate higher crests and lower troughs, as every system of waves causes singly. At points, where even numbered and odd numbered circles intersect, there meet depressions and elevations and annul each other, whence the surface is at rest and, indeed, you see there a system of lines joining these points.

This interaction of two wave systems is called interference. The wave systems are not changed. Each wave travels through the other one without being perturbed. The wave systems are superimposed. You can observe this phenomenon frequently on water surfaces. For example, follow by eye the waves behind a boat which cross those generated by another boat: Even while they cross each other, you can spot them individually, as they proceed without any change.

Standing waves. Nodes and bulges of oscillations

Interference also generates waves of a special kind which you encounter frequently: Standing waves. It arises at interaction of two waves with equal amplitudes, equal wave length, but opposite direction of propagation. (In the interferences in Fig. 296, both waves appear at the same end of the row of points and propagate in the same direction.)

If you intersect vertically a water surface (Fig. 297) through the line joining A and B, you see along AB two propagating waves, one from A to B, another from B to A. Both start at the same time and, as they propagate under the same conditions, they meet at the centre C. If the distance AB is covered by two wave lengths, the row of points - for the sake of visibility we return to this concept - when the waves meet, looks like those in Fig. 298a.

The wave radiating from B on its own would give the row of points the appearance of the dotted curve I, the one from A that of the dotted curve II. The following curves (b, c, d, e) show the row of points each 1/4 period later than the preceding one, that is, each of two waves has been advanced by 1/4 wave length in its direction. (The quarter wave lengths in Fig. 298 have been subdivided by vertical parallels, whence in each of the curves b, c, d, e the motion has been advanced by the distance of two consecutive parallels.) You can follow singly both waves, the dotted and the dashed ones, and identify their interference by the solid curve, the resulting wave.

The curves a to e of Fig. 298 demonstrate:
1.
Certain points remain
always at rest, namely those which are half wave lengths away from O, P, Q, R (on the line K), the nodes. The two interfering waves pass simultaneously at mutually opposite phases, whence always two equally large and oppositely directed motions cancel each other.
2. Figs. 298 c and e display after half periods all points in a straight line, whence these points pass simultaneously through the position of rest, whence they also return a quarter period later simultaneously - simultaneously! not successively as in the waves described hitherto. Figs. 298 b, d show the points one quarter period after the passage through the position of rest, that is, at their largest distances from it, that is, that all points of the same section, for example of OP, move in the same direction, those of two adjacent sections in opposite directions. They prove also that the amplitudes of the individual points differ (in the advancing waves, described hitherto, all points had the same amplitude!): At the centre between two nodes, the amplitudes are largest, they are smaller towards the nodes and vanish at the nodes. The amplitude at the centre is equal to the sum of the amplitudes of the two interfering waves, because the two waves meet there, and only there always (as the curves of Fig. 298 show) with the same phase. The points O', P', Q' , which are half wave lengths apart, are called oscillation bulges; they rise and fall always along the same line at the centre between the nodes (along the line B'). Since individual points have different trajectories, but cover them all in the same time, their velocities differ: They are largest at the bulges, decrease towards the nodes and vanish at the nodes.

The present wave differs essentially from those already treated: Its characteristic is that certain points of the row are fixed, namely the nodes, while the others pass simultaneously through the position of rest and turn around simultaneously. Pairs of neighbouring nodes subdivide the row of points into sections, every section oscillates like a unit about the position of rest, thus creating the wave form (similar to the oscillating string of Fig. 304), adjacent sections oscillating in opposite directions. Since you see at the same location always the same form of motion, the wave apparently stays put, whence it is called a standing wave.

There exist also standing longitudinal waves. They arise when transverse waves are replaced by longitudinal waves. Also here arise nodes, at which the motion vanishes, and bulges, at which it is always strongest. However, the nodes and bulges of the standing longitudinal waves do not always lie at the same locations, at which the nodes and bulges of the standing transverse waves occur. If longitudinal waves start from A and B of the row of points simultaneously under the same conditions, they also meet at the centre. But they apply their mutually opposite impulses ® ¨ ¬ , whence there too occurs a node. (Two interfering transverse waves apply at the point equally directed drives, whence there occurs a bulge. If you pursue this process further, you discover that the nodes of standing longitudinal waves lie at the positions of the bulges of standing transverse waves.

If you look at the figures and realize that the compactions and dilutions of the longitudinal waves correspond to those points, at which at equal wave length the transverse wave curve intersects the straight line, you will understand that the nodes of the longitudinal waves are those points, at which there occur alternate compactions and rarefactions and which are at rest, while the bulges are the points with constant average density and strongest motion.

From a geometrical point of view, the nodes and bulges are unique points of the entire row of points, as they are form a physical point of view. If we see in a standing wave merely a form of motion, we identify the nodes and bulges merely as locations, at which the velocities of the oscillating points vanish and have maxima. However, for example, if the standing wave is a sound wave, at the nodes the density of the air is periodically greatest while at the bulges it does not change at all. If the wave is an electro-magnetic wave, at the bulges the changes of the strength of the electric field are largest, whence a Hertz resonator reacts by strongest formation of sparks; in contrast, at the nodes, no sparks occur. If the standing wave is a light wave (Wiener 1890), the light is strongest at the bulges, weakest at the nodes.

Wiener's principal idea for the demonstration of standing light waves (Fig. 299) was: Mono-colour light meets vertically a plane metal mirror. The entering and reflected waves propagate in mutually opposite directions, whence they generate standing waves, that is, subdivide the space in front of the mirror into certain sections. In a plane, parallel to the mirror, you have everywhere the same state of vibration, whence the bulges and nodes (B and K) form two groups of parallel planes, the distance between two neighbouring planes of the same group being half the wave length (it is bisected by a plane of the other group). Intersect this system of standing waves by a plane E, perpendicular to that of the drawing. The two groups of B- and K-planes then intersect E in two groups of equi-distant lines - B-lines and K-lines - which alternate (they are perpendicular to the plane of the drawing). The respective distance between two of these neighbouring lines depends on the angle between E and the mirror. If it is 90º, the distance is so minute as that of the corresponding intersecting plane (like the wave length of the light); if you reduce it more and more, the B-lines and K-lines move further apart. Wiener has reduced the angle to nearly zero and made the strips of alternating light and dark (about 1/30 wave length wide) visible to the eye. (Also in a thin layer of gelatine, interspersed with fluorescent material, according to Paul Karl Ludwig Drude 1863-1906 and Nernst, the layer fluoresces in equidistant green strips.)

So far we have assumed that each point has only two neighbours, which lie with it on a straight line - in fact, all points were to lie on one straight line. In reality, every point has around it an infinity of neighbours; conceive it as centre of a sphere, that is, to belong simultaneously to an infinity of straight lines. If it becomes the origin of a wave motion, it transfers its vibrations to all neighbours and thus becomes the starting point of infinitely many waves. But, as the motion reaches it, also every other point becomes a starting point of a wave, whence there arise infinitely many waves, which interfere with each other. We cannot a priori (Latin: deductively) predict a law. For an understanding of light waves (you cannot be told too early!), the Principle of Huygens has led to a law. In order to understand the linkup between waves and light, the following will be sufficient: Huygens' Wave Theory of Light perceives the source of light as the starting point of waves which spread in transverse waves - in waves which arise by the particles of an infinitely elastic substance (the light ether) oscillating as in Fig. 283. When these waves reach our eyes, light sensitivity is aroused. The wave length is only a few ten thousandths of a millimetre. At this stage, you need not know more.

Since there are neighbours all around it, the oscillation, which starts at O (Fig. 300) at t=0, propagates in all directions. Hence the wave motion due to the oscillation at O reaches simultaneously all points at equal distance from O and form a spherical surface about O. In other words: There exists at every instant a spherical surface about O, at the points of which the wave motion just arrives, that is, all points of which start simultaneously the wave motion; since all of them move in the same manner and all along have the same phase, they agree in the magnitude and direction of their motion. A surface, the points of which are characterized thereby, is called a wave surface. (In homogeneous, isotropic substances, it is a sphere, in most crystals not a sphere, in some cases an ellipsoid, in others more complicated surfaces.) The radius of this sphere increases as the wave progresses. In this sense, you speak of spherical waves. A very small section of a spherical surface can be conceived to be plane; in this sense, you speak of plane waves.

However, these considerations take account of the fact that O has neighbours in all directions, but do not allow of the fact that each point is also a starting point of a wave. This is just what Huygens' Principle does (Fig. 300). The spherical surface W with radius vt1 corresponds to the instant t1. We ask now: Whereto advances the wave motion during (t2 - t1), if every point of W is a centre of perturbation, that is, causes wave motion? From every point of W arises wave motion, which advances in all directions at the velocity v. Hence there forms around every point an elementary wave, a spherical wave, the radius of which in the interval (t2 - t1) reaches the size v(t2 - t1). Fig. 300 shows some of these elementary waves. All of them are enveloped by a spherical surface, which coincides with the sphere of radius vt2, that is, the points of the spherical wave of radius vt2 coincide with the elementary waves.

In this manner, light waves spread from their source as well as sound waves (their differences are here of no consequence). However, light from a light source reaches us along straight lines (does not bend around corners) in contrast to sound. Why do not also light waves travel around obstacles like sound waves? It is just the analogy between light and sound as wave motions, which leads here t a contradiction: The straight line spreading of light and the generation of shadows could not be explained.

In order to arrive at an explanation, Huygens assumed: Every point P, taken in by a light wave, becomes itself an origin of spherical waves, but these elementary waves cause only a noticeable effect on their envelope. Hence, if Q is a light source and a plane opaque screen S1S2 allows its light to pass through the opening A1A2, the effect of the light from Q at a certain instant t has arrived at a wave surface, which you can find as is shown in Fig. 301. You find as envelope of the elementary waves a spherical surface about Q, which lies only inside the cone, determined by Q and the opening A1A2: Inside this cone, light spreads from Q as if the screen S1S2 did not exist, but nothing at all existed outside it.

Hence Huygens' Principle explains the straight line spreading of light without difficulty. But it does not resolve two objections:
1.
The elementary waves about the points a also have an envelope (C1C2) in the space
between the screen- and light-source, whence light should spread also backwards; but does occur.
2. The construction of Fig. 301 fails, that is light does not spread a straight line; if the opening A1A2 is very small - light
bends around corners and generates diffraction patterns.

Fresnel 1824 has solved this problem. Huygens had assumed that the effect of the light could only be sensed on the envelope of the elementary waves. Fresnel replaced this assumption by the Rule: The elementary waves have influence by crossing each other according to the Principle of interference: Light appears everywhere, where the waves reinforce themselves, in contrast, darkness occurs where they destroy each another. This Fresnel-Huygens Principle explains the straight line spreading of light as well as its diffraction. Fresnel determines the action of light, emanating from Q, at the point P (Fig. 302). (There is now no screen between Q and P!) He starts from a spherical surface about Q (of radius a), which he interprets as a wave surface, subdivides it into annular zones, the centres of which lie on the line QP, and determines their area by the following conditions:

Zone 1. (the central zone) extends to the point M1, which is determined by M1P-M0P = l/2, where l is the wave length;
Zone 2. extends to M2, determined by M2P-M1P = l/2, etc.

Every one of these zones contributes to the action at P. This calculation yields that neighbouring pairs of zones generate opposite signs and Fresnel concludes: Light acts at P as if it was solely due to the action of the elementary waves of half the central zone.

If you place a circular screen with centre at M0 between Q and P, the light at P depends largely on whether the screen covers the central zone and its neighbouring zones or not. It should be imagined that, if it covers half the central zone, there is no light at P. However, that is not so! You can then start the partition into zones at the edge of the screen (that is, from its projection on to the spherical surface) and again, according to the computation, there remains the action of the half first zone (which lies near the screen). There cannot be darkness anywhere on the line M0P. Experiments confirm this conclusion of the theory.

However, for screens, very large compared with the wave length and, moreover, not small compared to the distance M0P, the action of light at P is small, but also, if there is at M0 - a screen with centre at M0, which has not exactly the form of a circle and has the size of many wave lengths. In general, light at P is extremely small, if the screen at M0 has an irregular shape. Hence you can speak of straight line spreading of light, if you induce by means of sufficiently large screens of irregular shape on the line QP darkness at P.

If there stands between P and Q a screen with a circular opening with centre at M0, the action of light at P differs greatly with the size of the hole. If it only leaves free half the central zone, according to the calculation the action at P is the same as if there were no screen (natural intensity). If the opening is twice as large, so that the entire central zone is free, the action is twice as large; if the size of the opening is again doubled, so that the two first central zones remain free, the action is almost zero, etc. Also these conclusions have been confirmed by experiments. Instead of employing screens and openings of variable size, you only have to shift the point of observation P along the line QM0.

Thus, Fresnel's change of Huygens' Principle does not merely explain the straight line spreading of light, but also the deviations from this law - the refraction. But yet it leaves two objections unanswered.
1. It too does not explain, why light only spreads from a wave surface in one
sense, not also backwards (towards its source).
2. Fresnel's computation leads to a not relevant
phase of light excitation at P. These defects of the theory were only removed by Kirchhoff 1882.