H1 Wave motion
A solid elastic body, the form of which has changed as a result of an application of a force, returns to its initial shape, when the force ceases to act and its change of shape has not exceeded the limit of elasticity. But it can return in very different modes. The meaning of this statement is explained by the following experiment:
A fluid at rest in a vessel at rest has the form of the vessel and a horizontal plane as free surface. If the vessel's orientation is changed and then it is held fixed, the fluid assumes a new form; if it is then returned to its initial position, the fluid follows and eventually returns to its initial shape. Even if the conditions are the same, the change and the return to the initial shape is for watery solutions almost instantaneous, while it is very slow for viscous fluids. This difference becomes even clearer as follows: A very viscous fluid returns at hardly noticeable velocity to the initial state of rest and after arriving there stays at rest; in contrast, a watery fluid arrives there at a large velocity, exceeds it, returns to it, etc,. that is, it rocks in gradually decreasing oscillations about the equilibrium position until its velocity (due to friction) is exhausted and it comes to rest at last.
Change of shape and return to its initial shape are characteristic for elastic bodies. While returning after removal of the deforming cause to their initial state, deformed elastic bodies execute motions such as the oscillations of watery solutions and creeping of viscous fluids. If you pull down a heavy mass M, suspended by a spring B (Fig. 281), and then release it, it does not immediately return to the position of rest, but it oscillates vertically about the position of rest with decreasing amplitude and eventually returns to its position of rest: It is a proof that the spring at its initial change of shape transits into a state of motion, during which it alternatively lengthens and shortens and executes oscillations about its state of equilibrium. These vibrations match the behaviour of the readily movable fluid. (The elastic after-effect is the match for the behaviour of viscous fluids.)
Oscillations of a row of elastically linked points
Vibrations, which are maintained by elastic forces, lead to an understanding of wave motion and hence to insight into a large group of physical processes in Acoustics, Optics, Heat and Electricity. (The word wave should not make you think of water waves and such like phenomena from which the term has been taken.) We will begin our study of wave motion from resting, elastically interlinked mass points, which are located closely, at equal distances to each other, on a straight line (Fig. 282). We will assume that the forces acting between neighbours maintain a state of rest and the equilibrium of the row of points. Moreover, we must assume that pairs of neighbours attract as well as repulse each other and that, when the row of points is at rest, the attraction and repulsion are equally large, because then their distance becomes neither larger nor smaller. The magnitude of the mutual interaction between two neighbours - attraction and repulsion - depends on the magnitude of their mutual distance: It grows when it becomes smaller and drops when it becomes larger. But the repulsion grows obviously much faster than the attraction, when the distance becomes smaller, and drops much faster than the attraction, when it becomes larger. - This assumption is due to the observation: In a not deformed body, (in a state of equilibrium) those forces are obviously equal. As a body is compressed (by mutual approach of the mass points), both grow; but as the compressed body, left to itself, returns to its initial shape (and reverts the mutual approach), the repulsion in the compressed body is larger than the attraction, that is, it has grown during compression more than the attraction. Analogous considerations tell us: In the extended body the attraction is larger, that is, the repulsion has decreased by more than the attraction.
These forces maintain the equilibrium of the row of points. However, if only a single point is displaced from its equilibrium position, that is, if its distance from its neighbours is changed, attraction and repulsion start to act and this change of position of a single point perturbs at first the equilibrium position of its neighbours and then in turn those of all other points of the row.
Assume that by any cause whatsoever the point a has been displaced vertically with respect to the row of points to the point a '. (Later on, we will assume, in connection with longitudinal waves, that it has been displaced in the direction of b.) The enlargement of the distance from b reduces the forces acting between the two points, but the attraction by less than the repulsion, whence in the new position repulsion exceeds attraction. Hence b is pulled towards a ', so that it also leaves the straight line. But b cannot displace in the direction a'b, because it is also attracted by g (more correctly speaking, more attracted than repulsed, since then also its distance from g becomes larger.) The point b must therefore move in the direction of the resultant, that is, like a downwards. Eventually, every single point of the row is caused by its neighbours to move like a . Everyone of them starts to move a somewhat later than the preceding one, but this delay is the same all along the row for adjacent neighbours. In order to obtain a representation of this motion of the row, we must first get to know the motion of a single point.
The motions are caused by elastic deformations, indeed by deformations which lie within the bounds of elasticity, that is in the interval for which Hooke's Law applies. The point, which as been moved from its position of rest, will be pulled back to it at each instant by a force, which is proportional to its distance. The force, by which the point wants to return from any position to its position of rest, is obviously equal to that required to hold it at rest in the new position. However, this force is proportional to the deformation, that is, the distance from the position of rest. (Exactly like the force, by which a deformed, elastic body attempts to return to its initial shape, it is equal to the force, which holds it in the deformed state.) Hence the point is attacked by a force directed towards its position of rest, the magnitude of which changes according to the same law, by which the force changes on a pendulum, which has been deflected from its position of rest by a very small angle and is returned to its position of rest. It swings about its position of rest as in Fig. 118 the point P' on the straight line P1P2 through S.
Let the point a in Fig. 283 (1) depart from its position of rest as a result of a push at a certain velocity v in the direction of A. From the instant when it leaves its position of rest, a force attempts to pull it back, whence its velocity decreases and eventually vanishes. Let it have arrived at A as its velocity vanishes. Under the influence of the same force it returns along the same path. It gains now at each point of its track as much velocity as it has lost before at the same point. Hence its velocity increases until it returns to its position of rest to the same value v, at which it departed. As a result of this velocity, it overshoots its position of rest; and now the process repeats itself qualitatively and quantitatively in the direction aA', which previously took place in the direction aA. At each point of this path, it has the same velocity, which it had at the same distance along aA from a, except that the direction of the velocity is now towards A'.
You call the entire motion an oscillation, the distance AA' its amplitude, the time the point takes to travel along the entire path and back its period. The state of the oscillating point, characterized by its instantaneous distance from its position of rest, its velocity and its direction is called its phase. Phases which lie half a period apart are called opposite phases, because the point has in both of them the same distance from the position of rest, but on opposite sides of it, the same velocity, but moves in opposite directions.
You can give a formula for this state of motion, which states the location, the velocity and the direction of the point at each instant. However, we prefer to give you a more illustrative image of the motion of a row of points.
Oscillations of a row of elastically linked points perpendicular to the row: Transverse wave
We ask now: What is the appearance of the row of points once the point a has performed an entire oscillation? We will decompose the oscillation into four parts and examine the form of the row of points after a has moved :
|from 0 to A||from A back to 0||from 0 to A'||from A' back to 0|
Since a moves downwards, one point after the other also moves downwards under the same conditions and according to the same law of motion as a. As a arrives at the return point A, let the perturbation have reached the point d, that is, d has started its downward motion. The row then looks like that in Fig. 283 (2): a is at the return point A and just about to move upwards; all the points between a and d are moving downwards at different distances from their positions of rest and the extreme distance, that is, they form a sine curve.
While a returns to its position of rest, the points between a and d reach one after another their lowest position, from which they return to their position of rest. If a has returned again to its position of rest, d must be at its largest distance from its position of rest, its return point D, because d lags in its motion by 1/4 of the full period behind the motion of a (since its motion only began as a had arrived at A, that is, had covered 1/4 of its amplitude); the motion has already included the point g, which is as far from d as d from a (in its position of rest). The points between a and d are therefore moving upwards, d has arrived at its return point D, the points between d and g are already moving downwards, while g is about to start its motion downwards [Fig. 283 (3)].
As a arrives at its second return point A', d just passes upwards through its position of rest, g has arrived at its lowest point G, since it lags by half a period behind a and a quarter period behind d, and the motion is about to involve the point k, which is as far away from g as was g from d and d from a in the position of rest. The points between a and d are moving upwards towards their second point of return, those between d and g upwards, in order to reach again for the first time their position of rest, those between g and k downwards towards their first point of return, while a is just about to begin its downwards motion [Fig. 283 (4)].
As a returns to its position of rest, that is, has completed once its oscillation, the row of points has the form of Fig. 283 (5). You call this form a wave, its two symmetric halves wave crest and wave trough.
The row of points in Fig. 282 forms a wave, although only in a transferred sense of the every day expression, if (as has already been indicated above) the point a is displaced in the direction of the row of points towards b. The force, the direction of which falls into the row of points, causes the first point to come closer to its neighbour and thereby perturbs the equilibrium of the forces of attraction and repulsion between the points. The perturbation is transferred from point to point and causes the entire row to move.
Each individual point must oscillate about its position of rest (as described above). It is displaced from its position of rest and drawn back by a force, which is proportional to its distance from its position of rest; hence it must move exactly as every single point in the wave motion, which has already been described - except that, corresponding to the direction of the force, it oscillated within the row of points, while the points described earlier oscillated at right angles to it. Obviously, the points in this state of motion never yield a picture like that of Fig. 283; they can only come closer and further away from each other. Fig. 284 demonstrates this form of motion after each 1/4 period after the first point has started its motion: The periodical accumulation and compaction of points change periodically: The row of points does not look like a wave, but nevertheless this motion is called a wave - a longitudinal wave. The reason is: Just imagine the relative positions of the points, for example, at the instant, at which the first point has just completed one cycle (Last row in Fig. 284) and compare it with the initial position of rest (Fig. 284 first row): The first point is at its position of rest, the point, which 1/4 period later (than the first point) has started to oscillate, is at the maximum distance from its position of rest; the point, which started off 1/2 period later, is at rest, etc.
The distance of each individual point sidewards from its state of rest (longitudinal) is exactly as large as the earlier described upwards and downwards distances (transversal). The pendulum type motion is the same, only its direction with respect to the position of rest differs. At the instant when the first point has completed one oscillation, plot above each point vertically to the row the distance, at which it was sidewards from its position of rest at the end of that period. Then you obtain again the wave of Fig. 283. Note: In this graphical mapping of the longitudinal wave the maxima and minima correspond to the compactions and attenuations of the positions of the points, when the transversal wave passes through the position of rest [Fig. 283 (5) g and n].
Velocity of phase propagation
The distance covered by the wave motion during the period (T) of one of its points is called wave length (l); in Fig. 283 (1), it is a - n. If a point oscillates to and fro n times per sec, T = 1/n sec. In 1 sec, the motion therefore propagates by n wave lengths = nl. This distance is called the velocity of propagation (v) of the wave. Hence v = nl and v = l/T. Thus, at the same location of the row of points, the same phase (direction and same position of the row velocity) is repeated always after T sec, and the same phase has after T sec the point of the row, which is l away along the row. You can also say: The phase propagates at the velocity v, whence you call v = l/T the phase velocity (cf. group velocity 521).
Wave of circulating points
The individual points travelled along straight lines (Figs. 283/284). But a wave curve is also formed by circulating points, when the particles execute a motion along a circle somewhat later than the preceding point (Fig. 285). At the beginning, the points lie as in Fig. 283 along a straight line at equal distances, but now they follow circles at uniform velocity in the direction of the arrows. Each point begins its motion somewhat later than its predecessor and this delay is the same all along the row - say, it is 1/12 of the time used by a point to travel around its circle. After 1/4 of this time of circulation the point d begins its motion, after 1/2 of the time the point g, after 3/4 the point k, and when the point a returns to its starting point, n starts to move. Fig. 285 shows the states of the row of points during this process.
There exist yet other kinds of waves, but all of them have in common: The particles forming the wave move within narrow bounds about their position of rest, however, the wave - the geometrical, instantaneous image of all the points - advances in space. What propagates in space is only the perturbation, the force driving the periodic motion of the particles of a definite form. The wave never transfers mass through the space, but only energy, For example, you speak of a wave caused by an earthquake. Also in such a wave, the shaken masses remain close to the location where they were at rest. A report says "that the concussion was registered at that time at A, 20 seconds later on at B". What propagates is the drive of the motion. Using these times and the distance of the two locations, you compute the velocity of propagation of the wave. But this only means: The velocity at which the concussion (earth quake) advances.
A waving field of rice, over which a wind blows, gives the same insight. The grains oscillate in place about their positions of rest, however, the field gives the impression of the presence of an advancing motion. It is just the drive of the motion which propagates. - The particles forming a water wave travel in quite small circles about their initial location. The motion was demonstrated by Ernst Heinrich Weber and Weber in a long, narrow channel (with sides out of glass) by means of pieces of amber, which float in water (amber has the same specific weight as water). You see then the individual particles describing circles in the upper layers, ellipses in the lower layers and only straight lines at the bottom .
Uniform circular motion, decomposed into two pendulum oscillations
We are here not at all interested in water waves. but only in the motion of its particles. Starting from a, let a mass particle describe at uniform velocity a circle about O. Project its location - Fig. 286 after each 1/24 of its motion around the circle - on to the two perpendiculars AB and aD through the point a. When the point arrives at b, g, d, ···, it has moved perpendicularly to aD by ab ', ag', ad ', ··· ) upwards from a. But it has moved simultaneously by ab, ac, ad parallel to aD away from AB (sidewards away from a). The figure demonstrates how the arc length of the circulating mass point, measured from a (along the circle), is linked to its vertical elevation above a (measured AB). The continuously decreasing distances, by which it rises above a to h ', correspond all along to equally long arcs, by which it rises from a towards h. It sinks again by the same growing distances - on the way back from h ' to a - by which it previously rose. The same is repeated as it moves from n to t and returns from t to a. The figure also demonstrates the dependence of the distance along the arc of the mass point from a and its sidewards distances from a (measured along DC).
A mathematical analysis of the motions, projected on to AB and CD yields pendulum motions, the phases of which are displaced relative to each other. In order to demonstrate this, we redraw Fig. 286, using as abscissae the distances travelled by the mass particle along the circle from a , as ordinates the elevations along AB (Curve I) and the sidewards displacements along CD (curve II). We obtain in this manner the sine curves of Fig. 287, which are displaced with respect to each other by 2p /4 = 90º (1/4 period). Hence we can interpret the uniform motion along a circle by the mass particle as the resultant of two pendulum oscillations, which are simultaneously executed - perpendicularly to each other and equal - but with a 1/4 period phase difference.
If you combine two such drives experimentally, you obtain indeed a uniform motion along a circle: Displace a pendulum (Fig. 288) from its position of rest A and hold it at its return point B. If you now release it, it is only subject to a drive back to A, which causes it to swing between B and B'. However, if you push it at B, that is, after 1/4 of its period of oscillation, simultaneously at right angle to its plane of oscillation (in the direction of b), it will describe at uniform velocity a circle about A.
The pendulum of Fig. 288 corresponds to the circulating point of Fig. 286. It is deviated towards a. It receives there a drive perpendicular to CD, which is equal to that which it has already experienced along CD. From then on, it must follow both drives, whence it moves from a along the circle with uniform velocity about O. The oscillation along CD is 1/4 period ahead of that along AB.
For the entire row of points in Fig. 283 (1), you can now envisage the events at a single point as follows: A point of the row has received a push I towards its neighbour
I® · ¼ ¼
and when it has arrived at its largest displacement - that is after 1/4 of its oscillation - it receives the push II perpendicularly to the row of points. If both propagate simultaneously along the row of points, the row of points becomes the wave form of Fig. 285: Interaction of two independent of each other waves can thus generate a new wave, This is readily explained by the composition of the parallelograms of the motions. Every individual point experiences simultaneously two drives, that is, accepts the resulting motion of the two motions; every one like its predecessor, only a little later, where the delay is again equal between two neighbours all along the row.
Magnetic rotary field
The sine pendulum oscillation and the composition of two such by 1/4 period delayed vibrations into a uniform circular motion of a pendulum has an analogue in the magnetic field, generated by a sine like alternating current and the composition of of two such fields into a circulating, rotary field (Ferraris 1888). This analogue is so clear that the figures, which have explained the circular motion of the pendulum and its generation, also demonstrate the rotary field and its generation out of two sine alternating currents. An electric current generates near it a magnetic field, a field, which flows in a circle with a definite direction and magnitude, as we can do for a bar magnet, fixed in space. If the current is an alternating current, the strength of which rises and falls, like in the half wave of Fig. 287, and which reverts its direction periodically by 180º, as is indicated by the position of the half waves below and above the axis, the field strength of the current pulsates correspondingly and changes periodically its direction by 180º. One can say: The field oscillates. If we represent it by a vector - an arrow, the length of which changes in the the same rhythm like the distance of the body of a pendulum from its state of rest; as it passes the origin - the arrow is turned by 180º. Now let there act on this field a second alternating current, which coincides with it completely - except for its phase, it is displaced with respect to the first current by 1/4 of the period - and generates a field at right angle to it (Fig. 287). Pursuing the same reasoning as above, we arrive at the insight, that the field then has constant strength, but changes its direction steadily, that is, it turns like the pendulum oscillating along a circle (Fig. 288). The vector, which represents it, is an arrow of constant length, which rotates at a constant rate, like a magnetic needle on its support. If you apply to an iron ring (Fig. 289) two pairs of coils aa and bb, through which pass two equal alternating currents with a 1/4 period phase difference, there arises inside the ring a rotary field. A freely supported magnetic needle inside the ring will rotate synchronously with the field. Depending on whether two or three currents are applied at phase differences of 90º and 120º, respectively, one speaks of two-phase and three-phase rotatory currents. Since about 1895, the rotary current has occupied a principal role in electrical engineering.
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