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

 

Albert Einstein finished writing his revolutionary 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) =  (8πυ²/c³)U_υ  =  (8πhυ³/c³) / [exp(hυ/kT) – 1],

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.

The electromagnetic 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 in it 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, he got the classical equipartition 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 frequencies 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, 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, 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 a single resonator? Planck used his quantum energy element idea to calculate this average energy, which he found to be

U_υ  =  hυ/[exp(hυ/kT) – 1],

but used Maxwell-Lorentz wave-based oscillator theory to calculate the relation of average oscillator energy to spectral radiation energy density. This 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.

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[i] The mirrored walls, being totally reflective, are supposedly incapable of producing thermal radiation, but I can’t agree with that supposition, since any material is going to absorb and emit thermal radiation (Kirchhoff’s radiation law). Also, a simple harmonic oscillator, even an electrically charged one, isn’t a thermodynamic entity, but Planck’s electrical resonators are considered to have randomized amplitudes and phases, while maintaining a constant frequency. The disorder of the random phases and amplitudes, says Planck, gives the resonators the thermodynamic property of entropy. This thermalization of radiation question is related to my offhand parenthetic comment above on Rayleigh's use of energy equipartition in doing the thermal spectrum calculation in 1900.

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. \But 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 all 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 is known 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.

From Particles to Waves, Part 2a

In part one, the emphasis was on the meaning of periodic reflective boundary conditions in the case of the one-dimensional wave equation. This is also called a rigid or fixed boundary condition. Other possible b.c.'s for the string are the free and the circular conditions. (The circular b.c. for a string is apparently the same as the more general periodic b.c., which I mistakenly said in earlier versions of these posts is the name for what is really a reflective b.c.) In Part 3, the final part, the emphasis will be on solutions to the wave equation for reflective boundaries, but first it may be helpful to be reminded of what the “most general solution” is to the 1-D wave equation with no boundary or initial conditions prescribed. 

This generic solution is most easily found by using a "difference of two squares" method to factor the equation. I first saw this method used in some notes written by Prof. Richard Rolleigh for a second-semester classical mechanics class I took at Hendrix College. (Yeah, two semesters of classical mechanics were required, and ditto for classical E&M.) The overall subject of Dr. Rolleigh's 14-page handout is "Classical Mechanics of Continuous Media." The key idea is that the 1-D wave equation, 

∂ ²y/∂x² — (1/c²) ∂²y/∂t² = 0,                                          (1-4)    

(see the middle part of Wikipedia's Wave Equation entry) can be factored into

(∂ /∂x + (1/c) ∂ /∂t)( ∂ /∂x – (1/c) ∂ /∂t)) y(x,t) = 0. 

The factored form of the equation makes it simpler to look for the necessary relation between x and t that makes the left-hand side identically equal to zero: in the first factor we have the requirement that the first partial of y with respect to x equals (1/c) times the first partial of y with respect to t , and in the second factor that the first partial of y with respect to x is (-1/c) times the first partial with respect to t. Since the factors commute with each other, an expression satisfying either requirement is a solution.

The simplest relations between x and t that satisfy these requirements are x-ct and x+ct. (Below we’ll see, courtesy of Dudley Towne, that these are the only relations between x and t that fit the 1-D wave equation as a function of position. We’ll also see [Exercise 2] how these relations can be re-written as a more useful time dependence.) We are thus looking for any functions f(x-ct) and g(x+ct). To see for yourself how this works, try each of these in the factored form of the equation. Hint: you should write these as f(u) and g(v), where u=x-ct and v=x+ct, and then do the chain rule partial derivatives.

One more standard thing needs to be pointed out before I reproduce three pages from Chapter 1 of Towne’s book: The functional form f(x-ct) represents a waveform travelling to the right on the string, and the form g(x+ct) represents a (possibly different) waveform traveling to the left. To show this for f(x-ct) we can move time ahead by Δt, so that distance is changed by Δx = cΔt, giving

f(x+cΔt, t+Δt) = f(x+cΔt – c(t+Δt)) =  f(x+cΔt – ct - cΔt)) =f(x-ct)).

The waveform’s vertical displacement is unchanged, meaning the waveform has moved to the right at speed c. (How would you do the same calculation for g(x+ct)?)

Now for the Towne pages, wherein we see how the sum (superposition!) of f(x,t) and g(x,t) given above provides the necessary form of the one dimensional wave equation solution. Prior to this page, Towne has shown f(x-ct) to be a solution, by doing what I mentioned above as a chain rule hint for using f(u).

…pressed by the general solution. We are not as fortunate in the case of the two-and three-dimensional wave equations, for which such a convenient form of general solution does not exist."

Towne’s general description of the motion of the string, and of the variables, constants, and assumptions involved in the 1-D wave equation, are also worth reproducing here, and the same goes for his first two textbook problems in Chapter 1, which I’ll give as exercises for the interested reader:

 

The motion of the string can be specified by a function y(x,t) which is a function of the two independent variables x and t. Thus, for example, the graph of y versus x determined by the equation y = y(x, t₁) depicts the shape of the string at fixed time t₁, and the graph of y versus t  determined by the equation y = y(x₁,t) specifies the transverse displacement of the single particle located at x₁ as a function of the time. The latter graph is sometimes referred to as the history of the particle at x₁. … Assume that the string in the undisturbed configuration has a uniform linear mass density, σ, and is under uniform tension T. Also assume that any changes in either of these quantities which occur during the motion of the string are sufficiently small so that they may be neglected. … The equation [1-4] is a valid representation of the physical conditions in the system only so long as the inclination of the string remains everywhere small. [Meaning the slope is small, so ∂y /∂x << 1. Not really what's shown in the drawings above.] … The particles of the string are moving in a transverse direction, whereas the waveform propagates along the string. The “propagation” is one of form, but not of substance.

 

 

Exercises

1. Which of the following are solutions to the one-dimensional wave equation for transverse waves on a string? [For use in (c), the wave speed, or propagation speed of the waveform in the x direction, is given by c = (T/σ)^1/2.]

 

a) x²-2xct+c²t²

b) 10(x²- c²t²)

c) σx² + Tt²

d) {sin[(x-ct)³]}^1/3

e) 2x-3ct 

f) 10(sin x)(cos ct)

 

2. If h(u) is an arbitrary twice-differentiable function of u, show by direct calculation that y(x,t) = h(t + x/c) satisfies the one-dimensional wave equation. What relation does this solution have to the general solution written as y(x,t) = f(x-ct) + g(x+ct)?