Particles-to-waves detour: Einstein vs. Planck part 2

Left to right:  James Peebles (Princeton), George Abell (UCLA), Malcolm Longair (Cambridge), and Jaan Einasto (Tartu Observatory, Tõravere, Estonia), at an International Astronomical Union symposium in 1977. From Five Decades of Missing Matter, by Jaco de Swart, Physics Today, August 2024.

 

The first of three things Einstein questioned about Planck’s blackbody radiation formula derivations in 1900 and 1901 was Planck’s use of electromagnetic wave theory instead of his own new quantized energy level theory to derive the equation

 

ρ(υ) = (8πυ²/c³) U_υ

 

relating the cavity blackbody radiation density ρ(υ) to the average resonator energy U_υ at any given arbitrary frequency υ (and arbitrary temperature T, not shown).

Plank’s overall goal was to find ρ(υ), or actually ρ(υ,T), so his next step after deriving this equation would be to find U_υ. As Einstein said in his paper “On the Theory of Light Production and Light Absorption,” published in 1906, “This [equation] reduced the problem of black-body radiation to the problem of determining U_υ as a function of temperature.”  

And here’s what our main physics man in England, Malcolm Longair, pictured above, says about the equation after he shows Planck's derivation of it in Theoretical Concepts in Physics: An Alternative View of Theoretical Reasoning in Physics, 2^nd edition published in 2003:

This is the result which Planck derived in a paper published in June 1899. It is a remarkable formula. All information about the nature of the oscillator has completely disappeared from the problem. There is no mention of its charge or mass. All that remains is its average energy U_υ. The meaning behind the formula is obviously very profound and fundamental in a thermodynamic sense. I find this an intriguing calculation: the whole analysis has proceeded through a study of the electrodynamics of oscillators, and yet the final result contains no trace of the means by which we arrived at the answer. One can imagine how excited Planck must have been when he discovered this basic result.

Well, it’s hard for me to imagine Planck in an excited state, even though he invented the concept, but nevermind that. What did Albert Einstein see as a problem with this equation? It’s what Longair says in the next to last sentence above: “the whole analysis has proceeded through a study of the electrodynamics of oscillators . . .”.  How, Einstein asked in his 1906 paper quoted above, could Planck’s discovery of the discrete nature of the energy of an oscillator make use of the wave-based logic of Maxwell-Lorentz oscillator theory that "does not allow distinguished energy values of a resonator"?

Like a physics bloodhound on the right scent, Einstein was barking up the right tree. But it turned out this time there was just a Cheshire Cat in the tree, and its smile was saying, “Actually, Albert, classical physics is okay for this calculation.”

Almost everybody (and their dog) uses classical physics nowadays in textbook derivations of the factor 8πυ²/c³ in the above equation, and this factor is what Planck found in his “study of the electrodynamics of oscillators.” We now call it a density of states or a mode density, and textbooks obtain it most often by doing a “study” of electromagnetic waves confined to a cubical box or "cavity".  

For instance, referring to electromagnetic waves in a cavity as “field excitations,” Rodney Loudon on page 1 in The Quantum Theory of Light says “The field excitations are then limited to an infinite discrete set of spatial modes determined by the boundary conditions at the cavity walls. The allowed standing wave spatial variations of the electromagnetic field in the cavity are identical in the classical and quantum theories, but the time dependencies of each mode are governed by classical and quantum harmonic-oscillator equations, respectively.”

PAUSE: Here you may well ask “What ARE the ‘classical and quantum harmonic-oscillator equations’ anyway, Rodney?”  Good question! Loudon is kind of skimming over the issue because the wavefunction of a quantum harmonic oscillator in one of its stationary states (yep, an eigenstate, or self-state) has no time dependence. (And neither does an atom in a stationary state, thus the reason it's called a stationary state, although physically it makes no sense.) As pointed out by John Townsend in Quantum Mechanics: A Modern Approach, time dependence only comes from a superposition of quantum harmonic oscillator energy states.  Here’s my chalkboard solution to Townsend’s Problem 7-9, which asks for the position expectation value, <x>, of the superposition of the n and n+1 energy states of a quantum harmonic oscillator:

 

 

This time dependence is what you expect for a classical harmonic oscillator, so I’m not sure what Loudon is saying in regard to “time dependencies” but I’ll come back to that later, maybe. If he’d said temperature dependence, I could understand that. Or energy dependence, i.e., amplitude squared for classical and nhυ for quantum. But anyway! Our subject of the moment is the nature of the calculation of the spatial dependence of cavity radiation in equilibrium with the cavity walls, so I’ll get back to that now. UNPAUSE.

The units of 8πυ²/c³ are inverse volume times inverse frequency, so this factor multiplied by the oscillator average energy gives units of energy per unit volume per unit frequency for ρ(υ).

Since 8π/c³ ≈ 10^-24, the order of magnitude of the mode density at any given frequency is υ² x 10^-24, or (υ x 10^-12)². For microwave frequencies around 10¹² Hz, this average-oscillator-energy to radiation-density conversion factor is unity. Going up or down the frequency scale by a factor of 10 causes the conversion of average oscillator energy to radiation energy to go up or down by a factor of 100.

The higher the frequency of the oscillator (resonator), the more efficiently its energy is converted into radiation. Turn the equation around, however, and you see that the conversion of radiation energy into oscillator energy is more difficult at higher frequencies. Not your great-great-grandfather’s equipartion theorem!

Well, since we haven’t considered temperature dependence here yet, we’re not in the equipartition-or-not-equipartition realm. I'll get to that in my next post when I look at Rayleigh's 1900 calculation versus Planck's 1899 calculation. We are in the equilibrium realm, I remind you, where the resonators are absorbing and emitting radiation at the same rate on average. Which doesn't mean resonators at different frequencies are emitting and absorbing energy at the same rate--that would give a white noise or uniform spectral energy density function instead of the Planck function. (What kind of system would have a white noise spectrum?) It means absorption and emission rates are equal for each resonator, individually. The Planck function ρ(υ) shows what the constant equilibrium absorption and emission rates are at different frequencies.

The conclusion I’m passing along here is one of the answers to my question from my previous post:  “what did Einstein think Planck did wrong, in comparison to what Planck actually did wrong?” Answer: Planck was wrong to use Maxwell-Lorentz oscillator theory to find the conversion factor 8πυ²/c³ (he should have used Rayleigh's "mode count" method), and Einstein was right to be concerned about that. But Einstein’s concern was a Cheshire Cat illusion, as illustrated by Professor Longair’s comments above: “All information about the nature of the oscillator has completely disappeared from the problem.” Planck's result for the radiation density versus oscillator average energy was and is correct.

Next I'll look at Einstein's opinion that the reason it was okay for Planck to use Maxwell-Lorentz theory was because Planck's calculation involves average oscillator energy. The problem with Einstein's opinion in this matter is that average oscillator energy is precisely where Planck made his energy quantization hypothesis. (See the quote from Steven Weinberg in my previous post.) Is there a difference in the average Planck was calculating and the one Einstein had in mind? Perhaps Planck's was a shorter time average? 

(I didn't plan to post this today, and even thought of not posting it today, since this is the 80th anniversary of Hiroshima, the first "application" of physicists' unleashing of uncontrolled nuclear fission energy. "Oh, vey!" said Einstein when he heard about it, which translates from Yiddish as "Woe is me!" or just "Oh, woe!" I haven't found a reference to what Planck, who died in 1947, said about it.) 

(I revised this post on August 9, 2025, Nagasaki Remembrance Day number 80. To remember and not to repeat the application of nuclear weapons, that is the goal.)

Particles-to-waves detour: Einstein vs. Planck part 1

 (Revised 31 July 2025, mainly the footnote.)

Albert Einstein finished writing his paper “On a Heuristic Point of View Concerning the Production and Transformation of Light” on this date in 1905. I’ve recently been writing—trying to write—about Einstein’s criticisms of Planck’s papers of 1900 and 1901 introducing the quantum energy element into physics. This 120^th anniversary of Einstein’s completion of the above paper is a good time to post the first part of what I’ve been working on.

Yes, I’m detouring from my Particles to Waves series before posting Part 3b, the Final Segment. Who knows if it’ll ever get done? Who cares! Who gives a hoot? Well, Woodsy Owl, for one. He wants you to, too! ‘Cause this little detour series will segue nicely back into the Particles to Waves finale, friends, I promise. Einstein’s “Heuristic” paper, after all, introduced the light quantum, a particle-based description of electromagnetic waves. Like several other papers Einstein wrote during the first decade of the 20^th century, this paper was not easy for other physicists to understand and accept. It was different, however, in that it was the only one of these papers Einstein himself considered to be “revolutionary.”

On the other hand, there were the Max Planck papers published at the turn of the century that Einstein, like many other physicists, didn’t understand. Planck’s 1900 and 1901 papers on the blackbody radiation spectrum (the idealized thermal energy radiation spectrum) were themselves revolutionary and provided the basic idea for Einstein and his light quantum.

The questions I want to deal with here are “what did Einstein think Planck did wrong, in comparison to what Planck actually did do wrong, even though he got the right answer?” and “what did Planck actually do, in comparison to what he thought he did?”

 

In deriving what turned out to be the correct formula for the energy spectrum of blackbody radiation, 

                    ρ(υ,T)  =  (8πhυ³/c³) / [exp(hυ/kT) – 1] =  (8πυ²/c³)U_υ ,

Planck in his December 1900 and January 1901 papers postulated the existence of equally spaced discrete, integer-valued, energy levels for electrically charged “resonators” that absorbed and emitted electromagnetic waves—a very strange proposition indeed! In the equation, ρ(υ,T) is the spectral energy density of the radiation and U_υ is the average energy of a resonator. (As noted by Steven Weinberg, it was in the calculation of average oscillator energy that Planck made his "revolutionary suggestion" of discrete energy levels. See page 20 of Weinberg's 1977 Daedalus article, and see last equation below for explicit  U_υ expression.)

The electromagnetic (EM) waves, which Planck called a “stationary radiation field,” and the resonators together have a total energy E_t and are contained in “a diathermic medium with perfectly reflecting walls.” You can think of this system as an insulated box (with air or another gas in it, index of refraction approximately unity) and inside walls made of polished, mirror-like metal, with a few little black dots scattered around on the surface.[i] The black dots are the resonators, needed for thermalizing the enclosed radiation. (Whoa horsie, Rayleigh didn’t need em! But then after doing his reflecting-walls standing-wave analysis, he just used the "Maxwell-Boltzmann doctrine" of equipartition as the thermodynamic basis for his result. How to describe this detail? IN detail! Later!) 

Planck, in his December 1900 paper, says of the total energy, “The question is how in a stationary state this energy is distributed over the vibrations of the resonators and over the various frequencies of radiation present in the medium, and what will be the temperature of the total system.”

He first focuses on a single frequency υ and assigns N resonators with collective energy E to this frequency. Then, boom!, he introduces quantization by saying  this amount of energy E  is considered “to be composed of a very definite number of equal parts.” He introduces the constant h = 6.55 X 10^-27 erg-sec, and says, “This constant multiplied by the common frequency υ of the resonators gives us the energy element ε.” So there it is, the soon-to-be famous ε = hυ. This is for ONE frequency only, an arbitrary frequency, so the equation is not a proportionality in the usual sense. (It’s best to think of “ΔE = hυ,” i.e., the spacing of the energy levels equals hυ.) Every frequency has its own energy element, and Planck used “accents” to label the various frequencies, υ, υ’, υ’’, υ’’’, …, and their corresponding energies ε, ε’, ε’’, ε’’’, ….  

(Since the spectrum of energy vs. frequency of blackbody radiation is continuous, there’s an uncountable infinity of these frequencies. I think that may be why Planck used the accents rather than numerical subscripts 1, 2, 3, etc. for the different frequencies and energies. Subscripts imply discreteness. Later I’ll discuss the normal modes for radiation confined to a box--Dudley Towne's "waves confined to a limited region"-- where there are discrete frequencies, but the blackbody energy-versus-frequency spectrum, or energy versus wavelength spectrum, is continuous.)

Planck associated the energy elements only with the resonators, as if it were the resonators themselves that broke up the EM energy continuum into equal parts during absorption, then after emission the elements somehow melded together into the continuum of EM energies consisting of waves. Planck didn’t believe the EM field itself existed in the form of energy elements, and that’s the reason he didn’t think introducing his ε = hυ energy elements was a revolutionary idea. Here we have our first instance of what Planck thought he did versus what he actually did.

At first Einstein also believed Planck’s theory did not predict the existence of radiation quanta. In his “Heuristic Point of View” paper, Einstein assumed he was going beyond Planck’s work when he explicitly suggested the electromagnetic field was made of energy quanta:  “According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of energy quanta that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole.”

This idea of an electromagnetic field in free space consisting of light quanta is even stranger than Planck’s discrete energy levels for resonators. Planck’s resonators and their integer-valued energy levels nhυ were—and still are—abstractions. The energy of light waves consisting of “a finite number of energy quanta that are localized in points in space” gives a definite mental picture, and not a very believable mental picture, either, given the success of the wave theory of light propagation. Einstein was cautiously suggesting it was possible to think of light as particles or quanta, and he thus used the “heuristic point of view” terminology in his title.

 

Einstein in 1906 realized Planck’s published ideas about energy quanta applied to radiation as well as resonators. He said in his paper “On the theory of Light Production and Light Absorption” that “Planck’s theory makes implicit use of the aforementioned hypothesis of light quanta … the energy of a resonator changes by jumps of integral multiples of (R/N)βυ.” (At this time, instead of directly using Planck’s constant h, Einstein was still using (R/N)β, where R/N = k = Boltzmann’s constant, N = Avogadro's number, and β = h/k.) Thus belatedly realizing Planck has unwittingly gone Full Monty in uncovering the reality of quantization, Einstein then asks: how could Planck justify the use of the Maxwell-Lorentz electromagnetic wave theory  in deriving the very important relation (shown in the equation above)

ρ_{υ  =}  (8πυ²/c³)U_υ

for radiation density ρ_υ as a function of the average energy U_υ of the resonators?  Specifically, Planck derived the frequency dependent coupling constant 8πυ²/c³,  or electromagnetic mode density function, this way, not the average energy itself. He used his quantum energy element idea to calculate average resonator energy, which he found to be

U_υ  =  hυ/[exp(hυ/kT) – 1].

This mixed usage of both Maxwell-Lorentz ("classical" we say now) theory and the quantum-of-energy idea is the first of the three things Einstein believed Planck did wrong. I’ll finish discussing it and move on to the next two things in the next segment of this detour from Particles to Waves.

---

[i] The mirrored walls, being totally reflective, are supposedly incapable of producing thermal radiation in equilibrium with the walls, as discussed by the great Sommerfeld on page 135 (page 142 in online PDF) of his Thermodynamics and Statistical Mechanics:   "when the inner walls of the cavity are made of a perfectly reflecting material and cannot, therefore, influence the rays falling on them, the radiation filling the cavity may become one which is not in equilibrium." However, earlier on the same page he says of the blackbody cavity, "The cavity constitutes a thermodynamic system which is independent of the particular physical and chemical processes of emission and absorption taking place in the walls."   Also, a simple harmonic oscillator, even an electrically charged one, isn’t a thermodynamic entity, but Planck gave his monochromatic electrical resonators randomized amplitudes and phases. The disorder of the random phases and amplitudes, he says, gives the resonators the thermodynamic property of entropy, and thus also the property of having a temperature. This issue of the thermalization of radiation in a box is related to my offhand parenthetic comment above on Rayleigh's use of energy equipartition in doing the classical thermal spectrum calculation in 1900. Blackened walls are used in real blackbody experiments. And Sommerfeld says, on the same page as his above comment on perfectly reflecting walls, "The introduction of a speck of soot into the cavity will turn the radiation into black body radiation. (The speck of dust performs the role of a catalyzer.)" We appreciate your thoughtful and thorough comments, Arnold, but you seem to be waffling a bit on whether it matters what the walls are made of! 

From particles to waves, Part 3a: Eigen Spiel

“. . . when Werner Heisenberg discovered ‘matrix’ mechanics in 1925, he didn’t know what a matrix was (Max Born had to tell him), and neither Heisenberg nor Born knew what to make of the appearance of matrices in the context of the atom. David Hilbert is reported to have told them to go look for a differential equation with the same eigenvalues, if that would make them happier. They did not follow Hilbert’s well-meant advice and thereby may have missed discovering the Schrödinger wave equation.”

                                — Manfred Schroeder, in the Forward to his book Number Theory in Science                                                 and  Communication, 2^nd edition, corrected printing, 1990.

Keeping Up with the Eigens

In Quantum Concepts in Physics (Cambridge University Press, 2013), Malcolm Longair says (page 267), “In seeking a wave equation to describe de Broglie’s matter waves, Schrödinger began by attempting to find an appropriate relativistic wave equation. … These first attempts at the derivation of the relativistic wave equation were never published, but the argument can be traced in Schrödinger’s notebooks and a three-page memorandum he wrote on the eigenvibrations of the hydrogen atom.”

Professor Longair then says on page 268 that “de Broglie’s waves were propagating waves whereas Schrödinger had converted the problem into one of standing waves, like the vibrations of a violin string under tension.”

However, we know from my previous post that the production of standing waves on a string can be done without the string being fixed at both ends. Traveling sine waves of any frequency can be sent in from –∞ on a semi-infinite string and their reflection at the x= 0 end of the string, where the string is tied, will produce a standing wave.

This is where we come to the subject of all things eigen. The standing waves on a semi-infinite string are an example of what “eigenvibrations” are NOT, simply because they can have any frequency. Eigenvibrations occur only at eigenfrequencies, and these are the frequencies that are characteristic of the length of a string tied at both ends and the given boundary conditions at both ends. Indeed, eigenfriends, “characteristic” and “proper” are most often used in math and physics books as the English translation of eigen.

Eigen as “Own”

But let’s see what A Brief Course in German by Peter Hagboldt and F. W. Kauffmann, published in 1946, has to say on the subject. In the back of the book is a section called Vocabulary, which gives the English translation of various German words, including Wien and Wiener, which, just in case you didn’t already know, translate respectively as “Vienna” and “Viennese.” I only recently thought of looking up eigen in the book.

Besides “characteristic” and “proper,” the physics and math books sometimes also translate eigen as “special.” But none of those are what A Brief Course in German says. There, the Vocabulary section says Eigen translates as “own.” That’s it, no foolin’ around with “characteristic” or “proper” or “special” by Hagboldt and Kauffmann.

 

Eigen as “Self”

And “own” itself has a near-synonym in English. Among the Math Stack Exchange answers to a question dated 11 February 2013 and titled “What exactly are eigen-things?” there was one answer I especially liked, written by Alex Chaffee. He (or she) says “eigen means proper only insofar as ‘proper’ means ‘for oneself,’ as in ‘proprietary’ or French propre. Mostly eigen means self-oriented.”

This reference to propre connects with what seems to be a mistranslation used in relativity, where we have something called “proper” time, which some writers on the subject say comes from the French word "propre." It does seem that proper time really should be called “own time,” because “own time” is indeed what you read on your own watch, which never moves relative to you and thus never changes its rate of ticking relative to you. This mistranslation of French may also be why eigen gets mistranslated as “proper” instead of “own.”

Before moving on to discussing eigenwaves (sorry about that) on a finite length of string, I’ll mention one other reference that says eigen refers to self. Last year in the Arkansas Democrat-Gazette, a columnist named Philip Martin wrote about his German grandmother and mentioned that one of the German phrases she sometimes used was eigenlob stinkt. A few readers of this blog, such as Tom Mellett, are no doubt aware of how this phrase translates, but for those who aren’t, here is the English translation:  self-praise stinks.

I encourage you to think in terms of “own” and “self” when you see eigen in the future. To help you with that, here’s the History section from the Wikipedia article on Eigenvalues and Eigenvectors (it helped me) . . . 

 

Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.

In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.

In the early 19th century, Augustin-Louis Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in characteristic equation.

Later, Joseph Fourier used the work of Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur. Charles-François Sturm developed Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.

Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle, and Alfred Clebsch found the corresponding result for skew-symmetric matrices. Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability.

In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory. Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.

At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Hermann von Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.

The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Richard von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G. F. Francis and Vera Kublanovskaya in 1961.

[References aren't cited above, but they are in the Wikipedia article. Also, the applications section of the Wikipedia article on Eigenfunction looks at the 1D wave equation and the Schrö eqn.]

In some cases, there might be confusion over whether "proper" in an English translation of "eigen" means proper in the relativistic sense or in the sense of the eigenvalues, etc. Here's a page from Einstein's 1909 paper "On the present status of the radiation problem," in English (Princeton Einstein Papers Project). In the first paragraph, you'll find the words " at the proper frequency." In the German version , not surprisingly, you'll find "bei der Eigenfrequenz." This usage is the one from relativity. If it hadn't been, the English translation of Eigenfrequenz would be eigenfrequency. Maybe! Translations depend on the translators' knowledge and preferences. 

On page 358 of the English translation of this paper, you'll find our well-known acquaintance, the wave equation (in 3-D), along with some functions in its superposition solution that have t - r/c and t + r/c in their arguments: the "retarded" and "advanced" potentials, respectively. This paper of Einstein's is important because he note's Planck's mistake of assuming equal (a priori) probabilities for Boltzmann's "complexions" and 

Pause.

15 January 2025: Incorrect statement there, sorry! The problem Einstein is pointing out is that in order to use The Big W as the number of complexions in the entropy formula S = k*log(W), the complexions must be equally weighted, i.e., each must "be equally probable on the basis of statistical considerations." This is something Planck didn't address, but he did accomplish it without being aware of it.  In following Boltzmann's statistical method as presented in Boltz's 1877 paper relating the entropy and the probability of all the "state  distributions" of imaginary discrete kinetic energy elements distributed among gas molecules, Planck, like Boltzmann, actually used Bose-Einstein statistics, long before its significance in physics was discovered. Einstein didn't realize that Plank's complexions are equally probable under the "statistical considerations" used by Boltz and Planck, which were (are) Bose-Einstein considerations.  A bit of irony in that, and also in Boltzmann being the physicist who first used B-E statistics but being known eponymously for Maxwell-Boltzmann statistics.

Unpause.

In the 1909 paper, Einstein finds (equation 36) a wave and a particle term in the mean square fluctuations of thermal (blackbody) radiation--early evidence for the wave-particle duality.

To be continued in Part 3b.

From particles to waves, part 2b

 So far, I’ve discussed the meaning of periodic reflective boundary conditions for the 1-D wave equation and looked at the general solution, or what might be called the necessary generic form of the solution, to this equation as it applies to a small disturbance (y-direction small-amplitude motion) on a taut string having linear density σ and tension τ. This solution is written as

y(x,t) = f(x-ct) + g(x+ct)

or (see end of previous post)

y(x,t) = f(t-x/c) + g(t+x/c),

where c = (τ/σ)^1/2 is the speed of the disturbance, which is actually two disturbances when both terms in either of these equations are nonzero—the f-disturbance travelling to the right and the g-disturbance travelling to the left.

I’ll first look at how we get from the generic traveling wave solution to a traveling sine wave solution for the infinitely long string, then look at the form the generic solution takes for the semi-infinite string (where there’s a boundary at one end), and finally look at what happens when the traveling wave (the “incident wave”) on a semi-infinite string is a sine wave.  Part 3 (the next, and last, post on this subject) will be a return to the string fixed at both ends and a look at what happens to the generic form of the solution and the sine wave solution in that case. The beginning of Schrödinger’s wave mechanics in 1926 will also be discussed in Part 3. 

(The titles of Erwin's first four wave mechanics papers were "Quantization as an Eigenvalue Problem" Parts 1, 2, 3, 4. Since many people, myself included, have eigenproblems understanding the proper eigencontext of the prefix "eigen," I will include some eigencomments in my Part 3.)

The generic form of the 1-D wave equation solution above is derived without consideration of initial conditions or boundary conditions and has a unique speed associated with it, but no single frequency or wavelength like we might expect from a wave. But, as you probably realize, if it did have a frequency and wavelength in place of speed—if we just tried to substitute λν = c or ω/k = c into the generic solution—it wouldn’t be a generic solution anymore because f and g would have to be pure sine waves having their own frequencies ν or ω, and wave numbers k,  with f and g necessarily expressible, for instance, as Acos(kx ± ωt + φ), where A, k, ω, and φ are constants, although we can have a superposition of sine waves of different amplitudes and frequencies. 

Dudley Towne says in Section 1.7 of Wave Phenomena, where he’s still considering waves on a string of infinite length:

 

Contrary to the impression which may be created by the fact that waves of sinusoidal form are the most frequently cited examples of waves, it is to be noted that neither wavelength nor periodicity is an essential characteristic of a wave. An initial waveform of any desired shape determines an allowable solution to the wave equation.

 

After making this cautionary statement, Towne gives several reasons why sinusoidal waves are fundamental in the study of wave phenomena: 1) a sine wave is “one of the simplest analytic functions which is bounded on an infinite interval,” 2) it represents a pure tone in sound and a “spectral color” of light, and 3) it “achieves an overwhelming importance through Fourier’s theorem.” Dudley also mentions dispersion: “In some contexts involving the phenomenon of dispersion, the wave equation is not satisfied except for waves of sinusoidal form, and then only when a wave propagation velocity appropriate to the given frequency is substituted.”

 

Sinusoidal Progressive Wave on an Infinite String

For the 1-D wave equation, how do we get to a traveling sine wave solution, y(x,t) = Acos(kx±ωt + φ), from the generic traveling wave solution, y(x,t) = f(x-ct) + g(x+ct)? The same way we always get from the general to the particular when dealing with differential equations: initial conditions and boundary conditions. For the infinite string, there are no boundaries, so we just have initial conditions. And for partial differential equations like the 1-D wave equation, it’s really initial functions we have to deal with, a subject I’ll mention again further down in this post (see the paragraph below that starts with “Now”).

Right now, I just want to particularize the generic solution to the case of a traveling sine wave. That means we start with an initial sinusoidal disturbance on our infinitely long string, courtesy of Towne’s Section 1.7, “Description of a Progressive Sinusoidal Wave,” where he says

 

An initial waveform which is sinusoidal is described by the function

y(x, 0) = Re {ye^ikx}

where k and y are given constants. If the complex amplitude y is written in the form

y = y_me^iφ,

the real amplitude y_m determines the maximum displacement of the string from the x-axis and the phase φ determines the position of the curve with respect to translation parallel to the x-axis. … When the quantity (kx) increases by 2π, the function y(x,0) goes through one cycle of its values. The corresponding increment in x is, of course, the wavelength, λ. Thus k(x  + λ) = kx + 2π, or kλ = 2π, or k = 2π/λ. The parameter k is referred to as the wave number and may be thought of as the number of waves contained in a distance of 2π meters.

 

Towne chooses φ = 0, which makes the initial waveform a cosine function—but let’s keep the solution in exponential form for the moment. Next, we declare the initial waveform to be in motion in the positive x direction, making g(x+ct) in the generic solution of the wave equation identically equal to zero. The travelling wave solution is now

 

y(x,t) = f(x-ct) =  Re {y_me^ik(x-ct)}.

 

Towne says, “Note that if we focus on a particular particle of the string, say the particle [located at] x = x₁, the motion of this particle as a function of time is sinusoidal.” In a footnote, he says “in any progressive wave the curve which describes the history of a single particle is of the same geometric form as the wave profile of the string.” Then he continues on about the motion being sinusoidal, but I won’t quote him from this paragraph, except to say this is where he brings angular frequency into the argument: as the wave travels, the particles of the string undergo simple harmonic motion, with angular frequency ω = kc. Making this substitution in the above equation, we get

 

y(x,t) = Re {y_me^i(kx-ωt)} = y_mcos(kx-ωt).

 

Thus, starting from the generic form y(x,t) = f(x-ct) + g(x+ct), we’ve arrived at our desired expression for a traveling sinusoidal wave.

Now I’ll mention the use of “initial functions,” as discussed in Towne’s last section in Chapter One, “Initial Conditions Applied to the Case of a String of Infinite Length,” which, by the way, he says “may be omitted without loss of continuity.” (Dr. Rolleigh didn't omit this subject in his notes.)  I’ll omit the derivations but state the main idea. Since the wave equation is a 2^nd order partial differential equation, knowing the initial positions and initial velocities of all the particles on the string—the initial wave profile and the initial velocity profile—allows the functions f and g to be uniquely determined. As Towne says at the beginning of Chapter Three (Boundary Value Problems, where we’re about to go now), “the solution is uniquely determined for an infinite string by the requirements that y(x,t) and dy/dt reduce at t=0 to given functions.”

 

Sinusoidal Progressive Wave on Semi-Infinite String

The semi-infinite string extends from minus infinity on the x-axis to x=0, where it’s tied or fixed or anchored to a solid wall—the boundary. The 1-D wave equation must be satisfied within the interval – ∞ < x < 0. Towne chooses the form of the generic solution to be

y(x,t) = f(t-x/c) + g(t+x/c)                 (3-1)

“to simplify algebraic manipulations,” he says. I’ll just use two pages from the book to show the general waveform case and the sinusoidal waveform case. The boundary condition is y(0,t) = 0.

y(x,t) to vanish, regardless of the value of t. The fixed end is a node, and the spacing between successive nodes is a half-wavelength. Unlike the progressive wave, the sinusoidal waveform does not undergo translation parallel to the x-axis. This type of motion is referred to as a standing wave.

d) The amplitude of the simple harmonic motion of an individual particle depends on its location. The particles halfway between the nodes have the largest amplitude for their motion. These halfway positions are referred to as antinodes.

 

Before moving on to the string having boundaries at both ends, I’d like to note a couple of things about the semi-infinite string.

First, as you can surmise from the picture of the sine wave(s) above, there is no restriction on the frequency (or wavelength)  the standing wave can have. This is shown by the standing wave solution itself,

 

y(x,t) = 2y_msin(kx) sin(ωt),

 

where k and ω can take on a continuous range of values. (And now that we do have a sinusoidal wave, we necessarily have the relation ω/k = c = (τ/σ)^1/2.) This range of continuous values is in contrast to waves propagating in a “confined region,” such as on a string fixed at both ends, where the allowed frequencies are determined by the dimensions of the region, such as the length of the string.  This latter case has a discretely-infinite set of frequencies that are characteristic of the dimensions of the limited region--a periodicity in the frequency space and k-space. In the former case, there is a continuously-infinite number of possible sinusoidal standing wave frequencies that can be formed.

Secondly, the standing wave solution is a product of a function only of x and a function  only of t, something I’ll be returning to in Part 3—and something probably familiar to you from the usual “separation of variables” technique. I don’t like to invoke this rather abstruse technique unnecessarily, and neither does Dudley Towne, so I’ll follow his Wave Phenomena Chapter 15 discussion in Part 3.