H1 Wave motion
Oscillation
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 :
| 1. |
|
2. |
|
3. |
|
4. |
| 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.
Longitudinal
wave
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
II
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.
continue step back