For constant acceleration with no initial displacement

x = displacement t = time v = velocity (change in velocity comes from force) a = constant acceleration (acceleration relates to force) v_{0}= initial velocity

T = Period for one complete rotation ω = angular velocity

f = frequency of motion per unit time

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Simple Harmonic Motion

x = displacement t = time v = velocity (change in velocity comes from force) a = constant acceleration (acceleration relates to force) ω = angular velocity A = Amplitude of motion

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Wave Motion

For transverse waves in a cord

v_{p} = velocity of propagation of the pulse in the cord: m/s S = tension in the cord (intermolecular forces): N μ = mass per unit length: Kg/m

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For longitudinal waves in a solid

Y = Youngs modulus: N/m^{2} ρ = density: M/m^{3}

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For longitudinal waves in a fluid

B = Bulk Modulus: N/m^{2}

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For a traveling wave (Assume θ_{0} = 3π/2)

y_{x}(t) = the vertical position of the cord at a definite horizontal position, x. k = the propagation constant (ω/v_{p})

λ = wavelength

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Energy and Power for a Wave traveling in a Cord

ε = energy per unit length μ = mass per unit length: Kg/m ω = angular velocity: rad/s A = Amplitude of motion: m v_{p} = velocity of propagation of the pulse in the cord: m/s

Published in Annalen der Physik 17 (1905) 132–148 Translator D. TER HAAR Highlights in bold and comments in color by Vinaire

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Here is a brief summary of my comments:

In this paper Einstein shows that light in space is also granular like matter and electricity. These grains may be referred to as “light quanta.” This granularity of light increases with its frequency. In other words, the inherent momentum of light increases with frequency because its velocity is constant. Thus, the energy of light is proportional to the frequency much like energy of a fluid is proportional to its consistency, or the energy of matter is proportional to its inertial mass at constant velocity. This means light is a very light but fast-moving substance.

This paper omits to describe the nature of the granularity of light. This omission has led to the wave-particle discrepancy.Light quanta do not collide with each other like gas molecules. They are not point particles with centers of mass. They interact with electrons by transferring their energy. Their discreteness comes only through such energy interactions, otherwise, they are coalesced with each other like drops of a fluid.

The light of same frequency forms a continuum, though their may be a gradient separation due to different frequencies. This is like the separation between oil and water. Light may be visualized as consisting of “lines,” or “streams” of force, much like the way Faraday visualized it.

The new idea is that light is a substance; and, like any substance, it has a certain consistency. This consistency may be represented mathematically by frequency. The substance of light can be divided indefinitely while it maintains its consistency. That is why we can measure the frequency of light reaching us from the earliest of the galaxies.

Einstein stated, “The energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts…” He then applied this assumption to light quanta also. This is the very assumption that is currently preventing a reconciliation between quantum mechanics and the general theory of relativity.

A quantum is the discrete energy of interaction at the level of radiation. It is not a train of discrete energy particles as Einstein postulated. Light forms a continuum in space of continually increasing consistency, but its energy interactions with electrons are made of discrete quanta. A stream of light can split and coalesce easily as in the double-slit experiment.

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On a Heuristic Point of View about the Creation and Conversion of Light

There exists an essential formal difference between the theoretical pictures physicists have drawn of gases and other ponderable bodies and Maxwell’s theory of electromagnetic processes in so-called empty space. Whereas we assume the state of a body to be completely determined by the positions and velocities of an, albeit very large, still finite number of atoms and electrons, we use for the determination of the electromagnetic state in space continuous spatial functions, so that a finite number of variables cannot be considered to be sufficient to fix completely the electromagnetic state in space. According to Maxwell’s theory, the energy must be considered to be a continuous function in space for all purely electromagnetic phenomena, thus also for light, while according to the present-day ideas of physicists the energy of a ponderable body can be written as a sum over the atoms and electrons. The energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts, while the energy of a light ray, emitted by a point source of light is according to Maxwell’s theory (or in general according to any wave theory) of light distributed continuously over an ever increasing volume.

Einstein is pointing out the inconsistency that, at atomic levels, the Kinetic Theory treats matter and electrons as granular. But Maxwell’s theory treats the electromagnetic state in space, such as, light, to be continuous.

The wave theory of light which operates with continuous functions in space has been excellently justified for the representation of purely optical phenomena and it is unlikely ever to be replaced by another theory. One should, however, bear in mind that optical observations refer to time averages and not to instantaneous values and notwithstanding the complete experimental verification of the theory of diffraction, reflection, refraction, dispersion, and so on, it is quite conceivable that a theory of light involving the use of continuous functions in space will lead to contradictions with experience, if it is applied to the phenomena of the creation and conversion of light.

Einstein raises the possibility that treating light as continuous rather than granular may generate contradictions with experience in some cases.

In fact, it seems to me that the observations on “black-body radiation”, photoluminescence, the production of cathode rays by ultraviolet light and other phenomena involving the emission or conversion of light can be better understood on the assumption that the energy of light is distributed discontinuously in space. According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localised in space, which move without being divided and which can be absorbed or emitted only as a whole.

Einstein provides examples of some situations, such as, “black-body radiation”, photoluminescence, the production of cathode rays by ultraviolet light and other phenomena involving the emission or conversion of light, which are better understood when light is treated as having a granular nature during emission and absorption.

In the following, I shall communicate the train of thought and the facts which led me to this conclusion, in the hope that the point of view to be given may turn out to be useful for some research workers in their investigations.

In this paper, Einstein provides his reasoning why light cannot always be treated as a continuous function.

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1. On a Difficulty in the Theory of “Black-body Radiation’’

To begin with, we take the point of view of Maxwell’s theory and electron theory and consider the following case. Let there be in a volume completely surrounded by reflecting walls, a number of gas molecules and electrons moving freely and exerting upon one another conservative forces when they approach each other that is, colliding with one another as gas molecules according to the kinetic theory of gases.* Let there further be a number of electrons which are bound to points in space, which are far from one another, by forces proportional to the distance from those points and in the direction towards those points. These electrons are also assumed to be interacting conservatively with the free molecules and electrons as soon as the latter come close to them. We call the electrons bound to points in space “resonators”; they emit and absorb electromagnetic waves with definite periods.

*This assumption is equivalent to the preposition that the average kinetic energies of gas molecules and electrons are equal to one another in temperature equilibrium. It is well known that Mr. Drude has theoretically derived in this way the relation between the thermal and electrical conductivities of metals.

In the black-body cavity with reflecting walls, there are gas molecules and electrons moving freely and colliding with each other. Einstein assumes gas molecules to be very small material particles with center of mass. He assumes electrons to be particles that are colliding like gas molecules.

According to present-day ideas on the emission of light, the radiation in the volume considered—which can be found for the case of dynamic equilibrium on the basis of the Maxwell theory—must be identical with the “black-body radiation”—at least provided we assume that resonators are present of all frequencies to be considered.

There are bound “resonator electrons” that emit and absorb electromagnetic waves with definite periods.

For the time being, we neglect the radiation emitted and absorbed by the resonators and look for the condition for dynamic equilibrium corresponding to the interaction (collisions) between molecules and electrons. Kinetic gas theory gives for this the condition that the average kinetic energy of a resonator electron must equal the average kinetic energy corresponding to the translational motion of a gas molecule. If we decompose the motion of a resonator electron into three mutually perpendicular directions of oscillation, we find for the average value Ē of the energy of such a linear oscillatory motion

where R is the gas constant, N the number of “real molecules” in a gramme equivalent and T the absolute temperature. This follows as the energy Ē is equal to ⅔ of the kinetic energy of a free molecules of a monatomic gas since the time averages of the kinetic and the potential energy of a resonator are equal to one another. If, for some reason—in our case because of radiation effects—one manages to make the time average of a resonator larger or smaller than Ē, collisions with the free electrons and molecules will lead to an energy transfer to or from the gas which has a non-vanishing average. Thus, for the case considered by us, dynamic equilibrium will be possible only, if each resonator has the average energy Ē.

The average kinetic energy of a resonator electron must equal the average kinetic energy corresponding to the translational motion of a gas molecule under dynamic equilibrium.

We can now use a similar argument for the interaction between the resonators and the radiation which is present in space. Mr. Planck has derived for this case the condition for dynamic equilibrium under the assumption that one can consider the radiation as the most random process imaginable. He found

Similarly, the average kinetic energy of a resonator electron should also equal the energy of interaction with radiation present in space.

If the radiation energy of frequency v is not to be either decreased or increased steadily, we must have

The two values of average kinetic energy should be equivalent under the conditions of dynamic equilibrium.

This relation, which we found as the condition for dynamic equilibrium does not only lack agreement with experiment, but it also shows that in our picture there can be no question of a definite distribution of energy between aether and matter. The greater we choose the range of frequencies of the resonators, the greater becomes the radiation energy in space and in the limit we get

But they are inconsistent. This was famously known as the ultraviolet catastrophe.

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2. On Planck’s Determination of Elementary Quanta

We shall show in the following that determination of elementary quanta given by Mr. Planck is, to a certain extent, independent of the theory of “black-body radiation” constructed by him.

Planck‘s formula for ρ_{v} which agrees with all experiments up to the present is

For large values ofT/v, that is, for long wavelengths and high radiation densities, this formula has the following limiting form

One sees that this formula agrees with the one derived in section 1 from Maxwell theory and electron theory. By equating the Coefficients in the two formulae, we get

that is, one hydrogen atom weighs 1/N = 1.62 x 10^{-24} g. This is exactly the value found by Mr. Planck, which agrees satisfactorily with values of this quantity found by different means.

We thus reach the conclusion: the higher the energy density and the longer the wavelengths of radiation, the more usable is the theoretical basis used by us; for short wavelengths and low radiation densities, however, the basis fails completely.

In this section Einstein shows that “determination of elementary quanta given by Mr. Planck is, to a certain extent, independent of the theory of “black-body radiation” constructed by him.” The radiation appears continuous per Maxwell’s theory at lower frequencies, but not at higher frequencies.

In the following, we shall consider “black-body radiation,” basing ourselves upon experience without using a picture of the creation and propagation of the radiation.

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3. On the Entropy of the Radiation

The following considerations are contained in a famous paper by Mr. W. Wien and are only mentioned here for the sake of completeness.

Consider radiation which takes up a volume . We assume that the observable properties of this radiation are completely determined if we give the radiation energy ρ(ν) for all frequencies. [This is an arbitrary assumption. Of course, one sticks to this simplest, assumption until experiments force us to give it up.] As we may assume that radiations of different frequencies can be separated without work or heat, we can write the entropy of the radiation in the form

where ϕ is a function of the variables ρand ν. One can reduce ϕ to a function of one variable only by formulating the statement that the entropy of radiation between reflecting walls is not changed by an adiabatic compression. We do not want to go into this, but at once investigate how one can obtain the function ϕ from the radiation law of a black body.

In the case of “black-body radiation”, ρ is such a function of v or that the entropy is a maximum for a given energy, that is,

From this it follows that for any choice of δρas function of v

This is the black-body radiation law. One can thus from the function ϕ obtain the black-body radiation law and conversely from the latter the function ϕ, through integration, bearing in mind that ϕ vanishes for ρ = 0.

In this section Einstein presents Wien’s consideration that entropy of radiation may be determined completely from black body radiation law when the radiation energy is given for all frequencies.

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4. Limiting Law for the Entropy of Monochromatic Radiation for Low Radiation Density

From the observations made so far on “black-body radiation” it is clear that the law

put forward originally for “black-body radiation” by Mr. W. Wien is not exactly valid. However, for large values of v/T, it is in complete agreement with experiment. We shall base our calculations on this formula, though bearing in mind that our results are valid only within certain limits.

First of all, we get from this equation

and then, if we use the relation found in the preceding section

Let there now be radiation of energy E with a frequency between v and v+d v and let the volume of the radiation be . The entropy of this radiation is

If we restrict ourselves to investigating the dependence of the entropy on the volume occupied by the radiation, and if we denote the entropy of the radiation by S_{0}if it occupies a volume V_{0}, we get

This equation shows that the entropy of a monochromatic radiation of sufficiently small density varies with volume according to the same rules as the entropy of a perfect gas or of a dilute solution. The equation just found will in the following be interpreted on the basis of the principle, introduced by Mr. Boltzmann into physics, according to which the entropy of a system is a function of the probability of its state.

In this section Einstein uses Wien’s approximation (valid for higher frequencies of black body radiation) to derive an equation for the entropy of radiation. Thus, Einstein proves that the energy distribution of radiation becomes particle-like at high frequencies. This is an ingenious way of arriving at this conclusion.

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5. Molecular-Theoretical Investigation of the Volume-dependence of the Entropy of Gases and Dilute Solutions

When calculating the entropy in molecular gas theory one often uses the word “probability” in a sense which is not the same as the definition of probability given in probability theory. Especially, often “cases of equal probability” are fixed by hypothesis under circumstances where the theoretical model used is sufficiently definite to deduce probabilities rather than fixing them by hypothesis. I shall show in a separate paper that when considering thermal phenomena it is completely sufficient to use the so-called “statistical probability”, and I hope thus to do away with a logical difficulty which is hampering the consistent application of Boltzmann’s principle. At the moment, however, I shall give its general formulation and the application to very special cases.

If it makes sense to talk about the probability of a state of a system and if, furthermore, any increase of entropy can be considered as a transition to a more probable state, the entropy S_{1} of a system will be a function of the probability W_{1} of its instantaneous state. If, therefore, one has two systems which do not interact with one another, one can write

If one considers these two systems as a single system of entropy S and probability W we have

This last relation states that the states of the two systems are independent. From these equations it follows that

The quantity C is thus a universal constant; it follows from kinetic gas theory that it has the value R/N where the constants R and N have the same meaning as above. If S_{0} is the entropy of a certain initial state of the system considered and W the relative probability of a state with entropy S, we have in general

We now consider the following special case. Let us consider a number, n, moving points (e.g., molecules) in a volume V_{0}. Apart from those, there may be in this space arbitrarily many other moving points of some kind or other. We do not make any assumptions about the laws according to which the points considered move in space, except that as far as their motion is concerned no part of space—and no direction—is preferred above others. The number of the (first-mentioned) points which we are considering be moreover so small that we can neglect their mutual interaction.

There corresponds a certain entropy S_{0} to the system under consideration, which may be, for instance, a perfect gas or a dilute solution. Consider now the case where a part V of the volume V_{0} contains all n moving points while otherwise nothing is changed in the system. This state clearly corresponds to a different value, S_{1} of the entropy, and we shall now use Boltzmann’s principle to determine the entropy difference.

We ask: how large is the probability of this state relative to the original state? Or: how large is the probability that at an arbitrary moment all n points moving independently of one another in a given volume V_{0} are (accidentally) in the volume V?

One gets clearly for this probability, which is a “statistical probability”;

It must be noted that it is unnecessary to make any assumptions about the laws, according to which the molecules move, to derive this equation from which one can easily derive thermodynamically the Boyle-Gay-Lussac law and the same law for the osmotic pressure.

In this section Einstein shows that, when applied to a large number of discrete particles, the use of “statistical probability” is compatible with macroscopic laws of physics.

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6. Interpretation of the Expression for the Volume-dependence of the Entropy of Monochromatic Radiation according to Boltzmann’s Principle

In Section 4, we found for the volume-dependence of the entropy of monochromatic radiation the expression

If we write this equation in the form

and compare it with the general formula which expresses Boltzmann’s principle,

we arrive at the following conclusion:

If monochromatic radiation of frequency v and energy E is enclosed (by reflecting walls) in a volume V_{0}, the probability that at an arbitrary time the total radiation energy is in a part V of the volume V_{0} will be

From this we then conclude:

Monochromatic radiation of low density behaves—as long as Wien’s radiation formula is valid—in a thermodynamic sense, as if it consisted of mutually independent energy quanta of magnitude Rßv/N.

We now wish to compare the average magnitude of the “blackbody” energy quanta with the average kinetic energy of the translational motion of a molecule at the same temperature. The latter is (3/2)RT/N, while we get from Wien’s formula for the average magnitude of the energy quantum :

If monochromatic radiation—of sufficiently low density—behaves, as far as the volume-dependence of its entropy is concerned, as a discontinuous medium consisting of energy quanta of magnitude Rßv/N, it is plausible to investigate whether the laws on creation and transformation of light are also such as if light consisted of such energy quanta. This question will be considered in the following.

In this section, Einstein uses mathematical arguments to conclude that Monochromatic radiation of low density behaves in a thermodynamic sense, as if it consisted of mutually independent energy quanta of magnitude Rßv/N. Each quantum is the energy of one interaction. Einstein mathematically determines the theoretical value of a quantum.

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7. On Stokes’ Rule

Consider monochromatic light which is changed by photoluminescence to light of a different frequency; in accordance with the result we have just obtained, we assume that both the original and the changed light consist of energy quanta of magnitude (R/N)ßv, where v is the corresponding frequency. We must then interpret the transformation process as follows. Each initial energy quantum of frequency v_{1} is absorbed and is—at least when the distribution density of the initial energy quanta is sufficiently low—by itself responsible for the creation of a light quantum of frequency v_{2}; possibly in the absorption of the initial light quantum at the same time also light quanta of frequencies v_{3}, v_{4}, … as well as energy of a different kind (e.g. heat) may be generated. It is immaterial through what intermediate processes the final result is brought about. Unless we can consider the photo-luminescing substance as a continuous source of energy, the energy of a final light quantum can, according to the energy conservation law, not be larger than that of an initial light quantum; we must thus have the condition

This is the well-known Stokes’ rule.

We must emphasize that according to our ideas the intensity of light produced must—other things being equal—be proportional to the incident light intensity for weak illumination, as every initial quantum will cause one elementary process of the kind indicated above, independent of the action of the other incident energy quanta. Especially, there will be no lower limit for the intensity of the incident light below which the light would be unable to produce photoluminescence.

According to the above ideas about the phenomena deviations from Stokes’ rule are imaginable in the following cases:

1. When the number of the energy quanta per unit volume involved in transformations is so large that an energy quantum of the light produced may obtain its energy from several initial energy quanta.

2. When the initial (or final) light energetically does not have the properties characteristic for “black-body radiation” according to Wien’s law; for instance, when the initial light is produced by a body of so high a temperature that Wien’s law no longer holds for the wavelengths considered.

This last possibility needs particular attention. According to the ideas developed here, it is not excluded that a “non-Wienian radiation”, even highly-diluted, behaves energetically differently than a “black-body radiation” in the region where Wien’s law is valid.

In this section Einstein uses the new idea of “energy quanta” to explain the Stokes’ Rule for photoluminescence and indicates new possibilities.

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8. On the Production of Cathode Rays by Illumination of Solids

The usual idea that the energy of light is continuously distributed over the space through which it travels meets with especially great difficulties when one tries to explain photo-electric phenomena, as was shown in the pioneering paper by Mr. Lenard.

According to the idea that the incident light consists of energy quanta with an energy Rßv/N, one can picture the production of cathode rays by light as follows. Energy quanta penetrate into a surface layer of the body, and their energy is at least partly transformed into electron kinetic energy. The simplest picture is that a light quantum transfers all of its energy to a single electron; we shall assume that that happens. We must, however, not exclude the possibility that electrons only receive part of the energy from light quanta. An electron obtaining kinetic energy inside the body will have lost part of its kinetic energy when it has reached the surface. Moreover, we must assume that each electron on leaving the body must produce work P, which is characteristic for the body. Electrons which are excited at the surface and at right angles to it will leave the body with the greatest normal velocity.

The kinetic energy of such electrons is

If the body is charged to a positive potential Π and surrounded by zero potential conductors, and if Π is just able to prevent the loss of electricity by the body, we must have

where ε is the electrical mass of the electron, or

where E is the charge of a gram equivalent of a single-valued ion and P’ is the potential of that amount of negative electricity with respect to the body. [If one assumes that it takes a certain amount of work to free a single electron by light from a neutral molecule, one has no need to change this relation; one only must consider P’ to be the sum of two terms.]

If we put E = 9.6 x 10^{3}, Π x 10^{-8} is the potential in Volts which the body assumes when it is irradiated in a vacuum.

To see now whether the relation derived here agrees, as to order of magnitude, with experiments, we put P’ = 0, v = 1.03 x 10^{15}(corresponding to the ultraviolet limit of the solar spectrum) and ß = 4.866 x 10^{-11}. We obtain Π x 10^{7}= 4.3 Volt, a result which agrees, as to order of magnitude, with Mr. Lenard’s results.

If the formula derived here is correct, Π must be, if drawn in Cartesian coordinates as a function of the frequency of the incident light, a straight line, the slope of which is independent of the nature of the substance studied.

As far as I can see, our ideas are not in contradiction to the properties of the photoelectric action observed by Mr. Lenard. If every energy quantum of the incident light transfers its energy to electrons independently of all other quanta, the velocity distribution of the electrons, that is, the quality of the resulting cathode radiation, will be independent of the intensity of the incident light; on the other hand, ceteris paribus [other things being equal], the number of electrons leaving the body should be proportional to the intensity of the incident light.

As far as the necessary limitations of these rules are concerned, we could make remarks similar to those about the necessary deviations from the Stokes rule.

In the preceding, we assumed that the energy of at least part of the energy quanta of the incident light was always transferred completely to a single electron. If one does not make this obvious assumption, one obtains instead of the earlier equation the following one

For cathode-luminescence, which is the inverse process of the one just considered, we get by a similar argument

For the substances investigated by Mr. Lenard, ΠE is always considerably larger than Rßv, as the voltage which the cathode rays must traverse to produce even visible light is, in some cases a few hundred, in other cases thousands of volts. We must thus assume that the kinetic energy of an electron is used to produce many light energy quanta.

In this section Einstein brilliantly verifies the calculated value of energy quanta from the experimental value obtained from the study of photoelectricity. Here we have the conclusive evidence that energy of light is made up of frequency and not amplitude.

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9. On the Ionization of Gases by Ultraviolet Light

We must assume that when a gas is ionized by ultraviolet light, always one absorbed light energy quantum is used to ionize just one gas molecule. From this follows first of all that the ionization energy (that is, the energy theoretically necessary for the ionization) of a molecule cannot be larger than the energy of an effective, absorbed light energy quantum. If J denotes the (theoretical) ionization energy per gram equivalent, we must have

According to Lenard’s measurements, the largest effective wavelength for air is about 1.9 x 10^{-5} cm, or

An upper limit for the ionization energy can also be obtained from ionization voltages in dilute gases. According to J. Stark the smallest measured ionization voltage (for platinum anodes) in air is about 10 Volt. [In the interior of the gas, the ionization voltage for negative ions is anyhow five times larger.] We have thus an upper limit of 9.6 x 10^{12} for J which is about equal to the observed one. There is still another consequence, the verification of which by experiment seems to me to be very important. If each light energy quantum which is absorbed ionizes a molecule, the following relation should exist between the absorbed light intensity L and the number j of moles ionized by this light:

This relation should, if our ideas correspond to reality, be valid for any gas which—for the corresponding frequency—does not show an appreciable absorption which is not accompanied by ionization.

In this section Einstein tests his ideas to explain the existing experimental observations and further proves the viability of the idea of “energy quantum” or “light quantum”.

Higgs mechanism is the breaking of electroweak symmetry that makes some particles acquire mass. “Electroweak” means the electromagnetic and weak forces coming together. The electromagnetic forces maintain the equilibrium between the electronic and nuclear regions in an atom. The weak forces maintain the equilibrium among the nucleons within the nucleus. “Breaking of symmetry” means some very fundamental shift in interactions, which happens under certain conditions.

It appears that under certain conditions, part of the electronic region very close to the nucleus may condense to become part of the nucleus and vice versa. This is how mass is created or uncreated. This is Higgs mechanism in common English vocabulary. In other words, the frequency (consistency) of a layer of substance near the nuclear boundary may change so it gets converted into the next layer inside or outside the boundary. In other words, substance may condense or un-condense across the nuclear boundary. To discover that this actually happens is a big deal in Physics.

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Summary

The layer of substance just outside the nucleus of an atom may condense 1836 times and become the substance just inside the boundary of the nucleus. The theory underlying Higgs mechanism, now substantiated by the discovery of the Higgs boson, shows that the possibility of this happening is consistent with reality.

In other words, the condensation of substance from electronic to nuclear region (and vice versa) is possible according to the Higgs mathematical theory,

For details on the following concepts, please consult Chapter 2.

SOUND VELOCITY The velocity of sound in air is,

RMS VELOCITY The root-mean-square velocity of the gas molecules themselves is,

WAVE-FRONT Waves in two and three dimensions have a wave-front. It is circular or spherical as shown below.

The wave-front isan imaginary line or surface drawn through the crest (or trough) of one of the ripples at a given instant of time. We are looking at the same phase of the disturbance at all different locations in the fluid. The circular or spherical shape of the wave-front means that the wave propagation of the disturbance is characteristic of the material through which the wave moves. The direction of propagation of the wave at any location is perpendicular to the wave front at that location.

WAVE POWER (TWO DIMENSIONAL) For water ripples, the power transmitted through a unit length parallel to the wave-front is being diluted as the circular wave-front expands to larger circumference. Since the circumference of a ripple increases in proportion to its growing radius R, the power per unit wave-front length must decrease as 1/R.

WAVE POWER (THREE DIMENSIONAL) Similarly, the energy and power of the wave, per unit area perpendicular to the direction of propagation of the wave fall off as 1/R^{2}.

INTENSITY The power per unit area perpendicular to the direction of propagation is called the intensity, I. The intensity for a three-dimensional wave is given by,

I = P/A

PLANE WAVE A wave moving through space in which the wave-front is planar is called a plane wave, and is characterized by the fact that every point on the planar wave-front is in phase at the same time. A small window on the spherical wave front is almost planar if the dimensions are small compared to the distance from the source of the wave. The wave equation for such a wave is exactly the same as for the longitudinal waves in a long tube.

REFLECTION When sound wave-fronts hit a barrier, such as, the side of a mountain, part of the wave reflects and part is transmitted into the barrier. The part of the wave that is reflected has diminished amplitude but the same frequency and velocity as the original wave, and hence the same wavelength.

REFRACTION When a wave travels through a medium of varying densities (for example, layers of air at different temperatures) the velocity of different parts of the wave-front are different, and the direction of propagation of the wave changes as a consequence. This is called refraction.

INTERFERENCE Interference is the effect of having more than one wave passing a given point, and the possibility that the two waves will reinforce or weaken each other as a consequence of the phase difference between the waves.

DECIBEL SCALE To describe the range of sound intensities it is useful to create a logarithmic scale called the decibel scale (db), which gives a quantitative measure to “loudness”, which we label n, and define as:

n = 10 log (I/I_{o})

REVERBERATION The persistence of a sound after its source has stopped, caused by multiple reflection of the sound within a closed space.

REVERBERATION TIME The reverberation time is defined as the time it takes for the intensity of a given steady sound to drop 60 db (or six orders of magnitude in intensity) from the time the sound source is shut off. Reverberation times depend on the total acoustic energy pervading the room, the surface areas of the absorbing materials and their absorption coefficients. A formula that gives good estimates of the reverberation time is given by:

t_{r} = 0.16V / A

where t_{r} is the reverberation time (s), V is the volume of the room (m^{3}) and A is called the absorbing power of the room.

ABSORPTION COEFFICIENT The absorption coefficient of a surface is defined as the fraction of sound energy that is absorbed at each reflection. Thus, an open window has an absorption coefficient of 1 since all the energy passes out of it and none reflects back in. Heavy curtains have a coefficient of about 0.5, and acoustic ceiling tiles have a coefficient of about 0.6.

ABSORBING POWER The absorbing power A is just the sum of the products of the areas of all the absorbing surfaces (m^{2}) and their respective absorption coefficients.

QUALITY When a note on a musical instrument is played, the fundamental is typically accompanied by various overtones (harmonics, i.e., integer multiples of the fundamental) with differing intensity relative to that of the fundamental. The sound of harmonics is pleasing to the ear, and while the note is identified by the listener with the fundamental frequency, the same note from different instruments will sound differently as a consequence of the different harmonic content. These different sound recognitions by the human ear are called the quality of the note.

PITCH The pitch of a note is the human perception of the note as “high” or “low” and is closely related to the frequency but is not identical to it. The pitch involves human subjective sense of the sound. While a higher frequency will be perceived as a higher pitch, the same frequency will be perceived as having slightly different pitches when the intensity is changed. When the human ear hears a fundamental and harmonics it perceives the pitch as that of the fundamental.

BEATS If we have two frequencies that differ only by a few Hz we can indeed detect “interference” effects that oscillate in time slowly enough to be easily detectable. This variable amplitude corresponds to a maximal loudness in the sound, called a beat. The number of beats per second is just the difference of the two frequencies.

DOPPLER SHIFT The Doppler shift is a change in pitch caused by motion of the source of a sound wave through the air (as in the example of the siren of an ambulance) or by the motion of the listener through the air. If the source is moving toward the listener, the sound waves are bunched up, and the listener would detect shorter wavelengths or higher frequencies. If the source is moving away from the listener, the sound waves are more separated, and the listener would detect longer wavelengths or lower frequencies.

For more general case, when the velocity of the listener is included,

SHOCK WAVES When supersonic (faster than the speed of sound) motion occurs a compressional wave, due to the object cutting through the air, is emitted by the traveling body and forms what is called a shock wave. The shock wave moves at a specific angle relative to the direction of motion of the object through the air, and can sometimes be of sufficient intensity to cause a loud booming sound.

R/x is the ratio of the opposite side to the hypotenuse of a right triangle with angle as shown. Then:

The direction of propagation of the shock wave is perpendicular to the wave-front and makes an angle (90° – ) to the direction of motion of the object. Shock waves accompany speeding bullets, and an example in a medium other than air is the bow wave of a speed boat in water.

Wave Motion, Pulse, Amplitude, Transverse Wave, Longitudinal Wave, Velocity Of Propagation, Propagation Constant, Wave Equation, Wavelength, Wave Energy And Power, Wave Reflection, Principle Of Superposition, Interference Patterns, Standing Wave, Nodes And Anti-Nodes, Natural Frequency, Fundamental Frequency, Harmonics, Overtones, Resonant Frequencies, Resonance, Pressure & Displacement Nodes

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GLOSSARY

For details on the following concepts, please consult Chapter 1.

WAVE MOTION The wave motion is a process in which the physical matter itself does not move over significant distances beyond their initial positions, while the energy can be transferred over large distances. The transferred energy can carry information, so that wave motion allows the transfer of information over large distances as well.

PULSE A pulse is a single stroke, vibration, or undulation. The molecules move perpendicular to the direction in which the pulse moves. The shape of the pulse travels as one set of molecules after another go through the vertical motion. The pulse carries the vertical kinetic energy of the moving molecules, and the associated potential energy due to momentary stretching of the cord, in the pulse region.

AMPLITUDE Amplitude is the maximum vertical displacement of the pulse.

TRANSVERSE WAVE The pulse in a cord is an example of a transverse wave, where molecules move to and fro at right angles to the direction of propagation of the wave.

LONGITUDINAL WAVE In a longitudinal wave the molecules actually move to and fro along the direction of the propagation of the wave. This would be a pulse of pressure travelling through the air in a pipe. This air pulse is a primitive example of a sound wave.

VELOCITY OF PROPAGATION The velocity of propagation v_{p} of the pulse in a cord would increase with increase in tension S in the cord. On the other hand, it would decrease with increase in mass per unit length μ.

The propagation of sound in a solid would increase with increase in intrinsic stiffness as measured by the Young’s modulus Y, and with decrease in its density ρ.

The propagation of sound in a fluid would increase with increase in its Bulk modulus B, and with decrease in its density ρ.

PROPAGATION CONSTANT We define the propagation constant for the wave as, k = ω/v_{p}so that

WAVE EQUATION The vertical position of the cord, at a definite horizontal position x along the cord, at any time t is,

This indicates that the vertical displacement of any point x along the cord, the cord exhibits SHM of the same amplitude and frequency with the term in the sine function involving x acting as a phase constant that merely shifts the time at which the vertical motion passes a given point in the cycle.

WAVELENGTH (λ) The wavelength is the spatial periodicity of the wave, i.e., the length along the x-axis that one moves to go through one complete cycle of the wave.

WAVE ENERGY AND POWER For the case of a transverse sinusoidal wave travelling in a cord, or a longitudinal wave travelling in a tube, the energy per unit length is,

And, the power transfer across any point of cross-section is,

WAVE REFLECTION When the far end of the cord is tied down, the reflection is 180° out of phase.

When the far end of the cord is not tied down but free, the reflection is in phase.

A more general case is somewhere in-between these two extremes.

PRINCIPLE OF SUPERPOSITION The actual displacement of molecules from their equilibrium position, at any given location in a medium, at any instant of time, when more than one wave is traveling through that medium, is just the vector sum of the displacements that each would separately have caused at the same location at that same instant of time.

INTERFERENCE PATTERNS When two waves pass the same point in a medium they are said to interfere. If they correspond to long wave trains having the same wavelength, then certain regular patterns can appear, such as points that never move and points that move maximally. Such patterns are called interference patterns.

STANDING WAVE An examination of the actual “superimposed” wave may reveal some points on the cord that seem not to move at all as the waves pass each other, while other points midway between them move up and down with double the amplitude of either wave. The actual wave motion of the cord is therefore not a traveling wave, since in a traveling wave every point in the cord moves up and down in succession. The wave caused by the interference of these two traveling waves is therefore called a standing wave. It has the same frequency.

NODES AND ANTI-NODES The points that don’t move are called nodes, and the points that move maximally are called anti-nodes.

NATURAL FREQUENCY Many physical systems, when stimulated can be made to vibrate or oscillate with definite frequencies. In each of these cases there is a single “natural” frequency associated with the system. In more complex structures, many “natural” frequencies of vibration can occur.

FUNDAMENTAL FREQUENCY The fundamental frequency, f_{1}, is the lowest possible natural frequency.

HARMONICS The integer multiples of the fundamental frequency, f_{n}, are called harmonics.

OVERTONES The overtones are the successive natural frequencies above the fundamental.

RESONANT FREQUENCIES Such “natural” frequencies, which are characteristic of the particular system or structure, are also called the resonant frequencies of the system.

RESONANCE If one stimulates the system at one of the resonant frequencies, one can stimulate huge amplitude oscillations, sometimes to the point of destroying the structure. This is because, when stimulating a system at a resonant frequency it is extremely easy to transfer energy to the system.

PRESSURE & DISPLACEMENT NODES In a longitudinal wave, the pressure variation is zero at the pressure node; and the displacement variation is zero at the displacement node. A pressure node is a displacement anti-node and vice versa.