The Role of Harmonic Frequencies in Natural Phenomena

Question

I am deeply fascinated by the apparent intrinsic relationship between harmonic frequencies and the natural world. This relationship is evident not only in the mathematics that simplify our description of the universe but also in the very phenomena that resonate with these frequencies. My curiosity compels me to explore why harmonic frequencies are so prevalent in physics and nature, where other waveforms seem to take a backseat.

Mathematical Foundation of Harmonic Frequencies:

It begins with the mathematical underpinnings. Can other periodic functions besides $\sin(kx)$, $\cos(kx)$, or $e^{ikx}$ fully describe the spectrum of functions? If harmonics are indeed the exclusive basis, my puzzle may be resolved. However, if a broader set of functions exists, it challenges my understanding of why nature specifically aligns with harmonic frequencies.

Physical Phenomena and Harmonic Dominance:

Several physical examples underscore my point:

1. Photon Energy Equation ($E=hf$): Why does the energy of a photon rely on the frequency of the associated harmonic wave rather than any other waveform basis, like triangular or square waves?

2. Planck's Black Body Radiation: Planck's oscillators are based on energy $hf$, which again points to harmonic frequency. What about this formulation aligns it so closely with harmonic oscillations?

3. Color Perception: Our eyes discern colors based on the harmonic frequency of light. Does this imply that our perception is tuned to harmonics, or is there an underlying reason for this specificity?

4. Quantization of Harmonic Oscillator Energy: Again, we see the quantification in terms of harmonic frequency quanta $\hbar \omega$. Is there a fundamental principle that necessitates this?

5. De Broglie Wavelength: This concept refers to harmonic (sinusoidal) waves. In applications like electron microscopy, the required wavelength for resolving small structures depends on harmonic frequency. Similarly, edge diffraction and wave refraction through slits hinge on harmonic frequency (Again, why harmonic? We could decompose the wave on a triangular, square or any other waveform basis (i.e. a basis composed of infinite versions of the same periodic function, but each one with a different frequency, in the same way the Fourier Transform uses a basis of infinite periodic functions, each with a different frequency, those are, sines and cosines.)).

The prevalence of harmonics in these instances seems more than coincidental, suggesting an underlying principle that I am eager to understand. Could there be a foundational reason for the dominance of harmonic waves in the description of physical reality?

I appreciate any insights into this intriguing aspect of physics and welcome discussion that can illuminate the unique role of harmonic frequencies in our universe.

Answer 1

It's partly mathematical (harmonic functions are a handy tool which we know lots about) and partly physical (many real waves are close to being harmonic and those that aren't can be treated as sum of many harmonic ones).

Waves in the real world are rarely exactly sinusoidal (harmonic), but when the wave is periodic, often it seems close enough to harmonic to us that we think of it as being harmonic even if that is not exactly so, because that is the simplest and most intuitive model of periodic motion we have.

For example, light from a laser is a wave, and has one strongly dominant frequency, but it is not a perfect periodic wave, and not a perfect harmonic wave, as there are many close but different frequency components and also chaotic behaviour in the wave.

Or for sounds of musical instruments when they play a note, this is not a perfect harmonic oscillation either, there are details in the sound called timbre which we recognize and use to distinguish various instrument. This difference of timbre shows the waves of same frequency made by different instruments deviate from each other, so not all of them can be a pure sine wave.

However, harmonic wave is the simplest mathematical model of a periodic wave which real periodic waves are often similar to. Mathematically, we could analyze real non-harmonic waves using different families of basis functions. The important condition is that they form a complete basis so any function can be expressed as a sum of components proportional to members of the basis. This is is possible in many different ways; e.g. a set of piece-wise constant functions or piece-wise linear functions can form a complete set (a basis in our space of functions). But using such ugly functions with discontinuities or sharp points would be mathematically cumbersome and such patterns are not easily seen in the wild in propagating waves. Harmonic functions are the easiest to work with in analysis and are close to what we see in nature.

It takes some special circumstances to observe harmonic wave in some region of space. Either the production of the wave is close to harmonic and the medium just preserves this pattern in the propagating wave (e.g. vibrating voice chords or a saxophone body producing close to harmonic wave in the air), or the medium where the wave propagates interacts with the incoming non-harmonic wave and creates harmonic waves propagating in different directions, out of the incoming non-harmonic wave (dispersion prism or diffraction grating turning white light beam into coloured light beams).

In both cases, harmonic motion of matter has to do with the fact that a matter element interacting with the wave is bound to some equilibrium position, and often (not always) force pushing it back (the restoring force) obeys the Hooke law - the force is proportional to displacement from the equilibrium position. If the linear restoring force is dominant force, the motion is close to harmonic.

The easiest model to see the production of harmonic waves out of non-harmonic wave in a medium, is the classical Drude-Lorentz model of dielectric polarization: a material medium made of molecules where a linear restoring force is acting on charged particles, so polarization is proportional to effective electric field. This model implies that a non-harmonic EM wave (white light) in such a medium is gradually split into a spectrum of harmonic waves, each propagating with a different speed, and under certain conditions, also in a different direction.

This linearity is such a useful model (and harmonic oscillations are so common) partly because often oscillations are quite small. For small enough oscillations, the restoring force is always a linear function of the displacement. It takes big enough oscillations to make the force behave non-linearly and then we get generation of non-harmonic waves. E.g. second-harmonic generation using strong laser light in crystals. These non-harmonic waves can be analyzed in terms of multiple harmonic waves of different frequencies, starting with the natural frequency and its multiples (second, third harmonic or higher).

Answer 2

Why does the energy of a photon rely on the frequency of the associated harmonic wave rather than any other waveform basis, like triangular or square waves?

The prevalence of harmonics in these instances seems more than coincidental, suggesting an underlying principle that I am eager to understand. Could there be a foundational reason for the dominance of harmonic waves in the description of physical reality?

If harmonics are indeed the exclusive basis, my puzzle may be resolved.

The last bit is true. The answer, sadly, is nothing deep. Physics deals with a lot of first order approximations, truncations of Taylor expansions, etc, leaving us with a lot of linear ODE to solve. The exponential function is the unique eigenfunction of the differential operator, i.e. $\frac{\mathrm d\ }{\mathrm dx}e^{kx}=ke^{kx}$, and that cements their foundational appearance in so many aspects of physics. What you are seeing, thus, is that physics deals with a lot of 2$^\text{nd}$ order linear ODE and PDE.

Triangular and square waves do not solve these differential equations.

It also is particularly helpful that Fourier analysis shows us that all physically meaningful functions can be Fourier decomposed, and so on. Dualities like Parseval–Plancherel identity is even central to quantum theory. There are many nice things to learn here, but it is not some deep connection inside Nature.