**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*.

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** - r**ight 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.

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 *sam*e 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

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.)}

**Each vibrating point is the
start of a wave. ****Huygens****' Principle****. Wave surface**

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.

If the wave motion advances in
all directions at the velocity *v* cm/sec; it has grown at
time *t*_{1}, that is, after *t*_{1 }seconds,
to the spherical surface about *O *with radius* vt*_{1}
and at a later instant *t*_{2 }to the surface with
radius *vt*_{2}*. *In the interval *t*_{2
}- *t*_{1} seconds, it has advanced along
each radius from the first to the second spherical surface by the
distance *v*(*t*_{2 }- *t*_{1}).
You can envisage the spreading of a wave motion by the
continuously increasing of a sphere about *O*, the radius
of which is equal to the distance, by which the wave has advanced
or by a continuous sequence of concentric spherical surfaces, the
outer one of which yields the bound, to which the wave has
advanced; spherical surfaces, which lie one wave length apart,
correspond to loci of *equal** *phase.

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 *vt*_{1}
corresponds to the instant *t*_{1}. We ask now: Whereto advances the wave motion during
(*t*_{2 }- *t*_{1}), 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 (*t*_{2 }- *t*_{1})
reaches the size *v*(*t*_{2 }- *t*_{1}).
Fig. 300 shows some of these elementary waves. *All* of them are *enveloped**
*by a spherical surface, which coincides with the sphere of
radius *vt*_{2}, that is, the points of the *spherical wave *of radius *vt*_{2 }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 *S*_{1}*S*_{2}*
*allows its light to pass through the opening *A*_{1}*A*_{2},
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 *A*_{1}*A*_{2}: *Inside *this cone, light spreads from *Q*
as if the screen *S*_{1}*S*_{2 }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

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 M*_{1}*,
which is determined by M*_{1}*P-M*_{0}*P
= **l**/2, where **l **is the wave length;
*

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 *M*_{0}*
*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 *M*_{0}*P*.
*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 *M*_{0}*P*,
the action of light at *P *is small, but also, if there is at *M*_{0}
- a screen with centre at *M*_{0}, 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 *M*_{0}* *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 *M*_{0},
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 *QM*_{0}*.*

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.

An envelope of spherical waves
in the sense of Huygens' Principle also occurs in the case of the *head wave**
*of a flying object under certain conditions. The impulse of
the bullet acting on the air particles ahead generates
compactions. A spherical wave spreads from each compaction node
at the *velocity of
sound*. If the missile's velocity is *smaller*,
the spherical waves spread into space and the perturbation fills
an ever growing space around the missile. If the velocity is
larger than that of sound, the common envelope of all spherical
waves at the front tip of the missile form a conical mantle *EDF
*of compacted air, called the* **head wave*
(Fig. 303). We will not give details here. Behind the first head
wave, further compaction waves spread from other protruding parts
of the missile, so that the missile is enveloped by several
conical mantles (as has been shown by photographs). The head wave
is *heard* as a so called *head
wave report*. In the case of missiles
launched at supersonic velocity, you also hear besides the head
wave report the launching noise of the gun and the detonation
noise of the grenade. The head wave *sticks *to the front of
the missile, as long as its velocity is larger than that of
sound, but *detaches* as soon as its velocity drops and passes on at the
normal velocity of sound.

**Velocity of propagation of wave
motion**

At what velocity does wave
motion propagate? An exact
derivation of its formula lies beyond the range of an elementary
presentation. We will therefore only indicate a few details. Wave
motion is only possible due to the *elastic property* of materials,
whence, beyond all, the velocity of propagation must depend on a
substance's elasticity. Moreover, it depends on the size of the
mass, which is made to oscillate. The elasticity is measured by
the coefficient of elasticity* e*, the mass by the density
*d*. The dependence of the velocity of propagation is
given by *v = *(*e*/*d*)^{1/2}.
However, this simple formula only applies when the wave motion
does not provoke temperature changes in the substance and *thereby *a change of *e*; this is a task
of Thermodynamics.

The velocity *v* does *not** *depend on whether the wave length or the
amplitude are large or small. Long waves and short waves, waves
with large and waves with small amplitude propagate equally fast.
However, you must take *one
*matter into account.
In the case of a *longitudinal
wave*, *e* is
the coefficient of the *pressure
**elasticity*, in the case of a transverse wave, that
of the *shear elasticity*. Fluids and gases have *only pressure* elasticity. Hence there can only arise
in them longitudinal waves and transverse waves are confined to *solids*. Of couse, there also arise
longitudinal waves in solids, because they have also pressure
elasticity. Its coefficient is larger than that of shear
elasticity. In a wave motion with transverse and longitudinal
components, the longitudinal wave races ahead of the transverse
wave. (This is what occurs during Earth quakes. Along Earth's *surface*, the most used mean values are for *v*_{long}*
*7.2 km/sec, for* v*_{trans }5.0 km/sec.)
Since the velocity of propagation only depends on the density and
elasticity, waves spread in isotropic bodies equally fast in all directions as
spherical waves. However, in anisotropic bodies, for example, in
certain crystals, the wave surfaces have very complex forms.

We will describe other wave
phenomena, wherever it is specifically required by the
presentation. Reflection of waves is known to everybody through
water waves: If waves encounter an obstacle *AB *(Fig.
306), for example, a shore, which impedes their spreading and *throws them back*, the reflected waves spread apparently
around a centre *a*', which lies as far *behind *the shore as the actual centre *a*
lies in front of it, that is, there appears a new independent
centre of perturbation: The wave *approaching *the wall and the waves *reflected**
*by it *interfere *with each other.