---
slug: convergence-c18
title: WAVES / OSCILLATORY TRANSMISSION
canonical: https://miscsubjects.com/a/convergence-c18
---

## The Claim

The universe sends signals. Light crosses vacuum. Sound compresses air. Neurons fire. Hearts beat. Populations oscillate. Gravitational waves ripple spacetime. Chemical mixtures pulse with color. These look like one thing. They are not.

The word "wave" hides a split. Two mathematics live under one name. One is linear. The other is nonlinear. One scales. The other resets. Both transmit. Both oscillate. Both carry information through rhythmic disturbance. But their equations differ. Their physics differ. Their convergence is functional, not mathematical.

## Definitions

- **Linear wave**: Energy propagates. Amplitude scales with input. Superposition holds. Waves pass through each other unchanged. The mathematics is d'Alembert.
- **Excitable pulse**: Fixed amplitude. Threshold response. Pulses annihilate on collision. No superposition. The mathematics is Hodgkin-Huxley.
- **Wave equation**: ∂²u/∂t² = c²∇²u. Second-order linear PDE. d'Alembert derived it in 1746 from a vibrating string.
- **Limit cycle**: Closed orbit in phase space. Heartbeats live here. Neurons live here. Predator-prey cycles live here. Van der Pol found them in 1927.
- **Threshold**: Minimum stimulus that triggers full response. All-or-none. No partial firing.
- **Refractory period**: After firing, the medium needs recovery time. It cannot fire again immediately. This creates directionality.
- **Convergence**: Same word, two mathematics. Same function, different physics. Same name, different truth.

## The Logic

You see waves everywhere. Ocean rises and falls. Sound travels through air. Light crosses void. Gravity ripples spacetime. You think one mathematics explains all. It does not.

Linear waves obey d'Alembert's equation. [SOURCE:noether-1918|type:mathematical] Conservation laws underwrite every wave. Noether proved this in 1918. Every continuous symmetry yields a conserved quantity. Momentum conservation underwrites wave propagation. Energy conservation underwrites oscillation. Add two waves. They pass through unchanged. Small push makes small ripple. Big push makes big ripple. The mathematics is clean. It holds across thirty-three orders of magnitude. Gamma rays at 10⁻¹² meters. Radio waves at 10⁴ meters. Gravitational waves at 10²¹ meters. Same equation. [SOURCE:schrodinger-1944|type:theoretical] Matter waves obey it too. Every particle is a wave. de Broglie proved this in 1924. λ = h/p. Schrödinger wrote the equation in 1926. It swallowed atomic physics.

Then you look at a nerve. A neuron fires. Signal travels down axon. But this is not a wave. It is a pulse. Fixed-size pulse. Weak stimulus does nothing. Strong stimulus triggers exact same pulse. Two pulses meet. They destroy each other. They do not pass through. [SOURCE:wiener-1948|type:theoretical] Wiener saw this in 1948. Cybernetics is the study of messages. Messages need discrete states. The neuron chose discrete pulses. Not continuous waves. This is not accident. It is optimization. Discrete signals resist noise better.

The heart beats the same way. Cardiac cell depolarizes. It hits threshold. It fires. Then enters refractory time. It cannot fire again. Pulse moves through tissue. Two pulses collide. They cancel. Spiral waves form. Rotors kill. [SOURCE:prigogine-1977|type:theoretical] These are dissipative structures. Far-from-equilibrium processes. Not linear waves. Prigogine won the Nobel Prize for this in 1977. Living systems maintain themselves away from equilibrium. They export entropy. They pulse.

Population cycles work the same. Wolves eat rabbits. Rabbits crash. Wolves starve. Wolves crash. Rabbits rebound. Numbers oscillate. But dynamics are nonlinear. They are limit cycles. Not sine waves. [SOURCE:darwin-1859|type:empirical] Natural selection shaped these oscillations. They carry ecological information. Lotka and Volterra modeled them in 1925. The equations look simple. Two coupled differential equations. But the solutions are limit cycles. Closed orbits in phase space. Not propagating waves. Temporal oscillation, not spatial transmission.

Chemical oscillations too. Belousov and Zhabotinsky discovered them in the 1950s. Mixtures pulsed with color. Waves of oxidation spread through liquid. But these were excitable pulses. Not linear waves. The BZ reaction became the canonical example. Turing predicted such patterns in 1952. He wrote about morphogenesis. Reaction-diffusion equations. Chemicals react and diffuse. Patterns emerge. Stripes. Spots. Spirals. But these are not solutions to the wave equation. They are solutions to nonlinear parabolic PDEs. [SOURCE:turing-1936|type:mathematical]

The word "wave" betrayed you. It made you think one mathematics rules everything. It does not. Two separate phenomena wear same name. Both transmit oscillations. Both move through media. But linear waves scale continuously. Excitable pulses reset discretely. [SOURCE:ashby-1956|type:theoretical] Ashby understood this in 1956. Requisite variety demands distinct channels. The brain needs both. Continuous potentials for graded signals. Discrete spikes for long-distance transmission. It uses both. EEG shows linear superposition. Action potentials show all-or-none. The brain is a hybrid architecture.

## The Evidence

d'Alembert solved the wave equation in 1746. He studied a vibrating string. He proved the mathematics. [SOURCE:shannon-1948|type:mathematical] Fourier followed in 1822. He broke waves into harmonic components. Any waveform equals superposition of sinusoids. This is the foundation of signal analysis. Shannon built information theory on this in 1948. The Nyquist-Shannon sampling theorem. Any bandlimited signal can be reconstructed. From discrete samples. The continuous and the discrete meet here. Fourier bridges them.

Maxwell unified light and electromagnetism in 1865. He wrote the field equations. Light became a wave. Speed fixed at c. No medium required. [SOURCE:schrodinger-1944|type:theoretical] Schrödinger put quantum mechanics in wave form in 1926. The wave equation swallowed atomic physics. Dirac relativized it in 1928. Quantum field theory grew from this. Every particle is an excitation of a field. Fields obey wave equations. The Standard Model is a wave machine.

Meanwhile, Hodgkin and Huxley measured the squid giant axon in 1952. They recorded action potentials. They built a nonlinear model. Ion channels open and close. Voltage-gated sodium rushes in. Potassium rushes out. The pulse propagates. But it is not the wave equation. It is a reaction-diffusion system with threshold. FitzHugh and Nagumo simplified it in 1961-1962. They found the limit cycle. The mathematics is Van der Pol, not d'Alembert. Hodgkin and Huxley won the Nobel Prize in 1963. They did not derive the wave equation. They derived excitability.

Lotka and Volterra modeled predator-prey oscillations in 1925-1926. They found limit cycles in phase space. Not physical waves. Population waves in time, not space. The logistic equation with predation. dx/dt = αx − βxy. dy/dt = δxy − γy. Simple rules. Complex oscillation. The Lotka-Volterra equations are not wave equations. They are coupled nonlinear ODEs. Their solutions orbit. They do not propagate.

Belousov and Zhabotinsky discovered chemical oscillations in the 1950s. Mixtures pulsed with color. Waves of oxidation spread through liquid. But these were excitable pulses, not linear waves. Winfree documented spiral waves in excitable media in 1987. He wrote When Time Breaks Down. Rotors in cardiac tissue can kill. Ventricular fibrillation is spiral chaos. Keener and Sneyd wrote the textbook in 1998. Mathematical Physiology. They showed that heart, nerve, and chemical reaction share a mathematical skeleton. It is not the wave equation. It is excitable dynamics. [SOURCE:von-neumann-1966|type:theoretical] Von Neumann saw this earlier. His cellular automata showed the same thing. Discrete excitation. Threshold. Recovery. Self-reproduction emerges from local rules. Universal construction. The automaton is an excitable medium. It computes. It builds. It lives.

The scale is staggering. Electromagnetic waves span gamma rays to radio. Gravitational waves hit LIGO in 2015. Ripples in spacetime curvature. Two black holes merged. The wave carried energy. It stretched arms four kilometers long. By less than a proton's width. [SOURCE:wilson-1971|type:mathematical] Renormalization group explains why same mathematics appears at different scales. Wilson won the Nobel Prize for this in 1982. Critical phenomena look the same regardless of microscopic details. Universality classes. The wave equation belongs to every class. Neural membranes operate at nanometers. Population cycles span continents. [SOURCE:mandelbrot-1967|type:mathematical] Fractal scaling connects them. Power laws govern event sizes. Mandelbrot showed this in 1967. Coastlines. Rivers. Markets. Neurons. The same scaling appears everywhere.

Brains use both architectures. EEG shows oscillations. Delta, theta, alpha, beta, gamma. These are linear superpositions of neural activity. Masses of neurons firing together. Creating continuous potentials. But individual neurons fire pulses. Action potentials. All-or-none. The brain is a hybrid. Linear waves on top. Nonlinear pulses underneath. [SOURCE:kauffman-1993|type:theoretical] Kauffman showed life sits at the edge. Between order and chaos. Where networks self-organize. Where computation emerges. [SOURCE:bak-1987|type:empirical] Bak showed the deeper pattern. Neural avalanches show self-organized criticality. Power-law size distributions. Information transmission maximized at the critical seam. The brain operates at the edge. Between order and chaos. Where waves and pulses coexist. Where computation happens.

## The Honest Limits

Whitham stated the rival clearly in 1974. Linear and Nonlinear Waves. All real waves become nonlinear at high amplitude. The linear wave equation is small-amplitude approximation. It is convenience, not deep truth. Shock waves form. Solitons emerge. The KdV equation governs them. The nonlinear Schrödinger equation governs fiber optics. Burgers' equation governs turbulence. Every linear wave is a lie at some scale.

Winfree and Keener argue the linguistic convergence is the only convergence. We call them both waves because they move and oscillate. But a nerve pulse is a chemical reaction. A light wave is an electromagnetic field. They share no underlying equation. The action potential is a phase transition. Membrane depolarization. Ion channel gating. It is thermodynamics, not field theory. Light is field theory. Not thermodynamics. The names rhyme. The physics does not.

The mathematical-objection camp says this. The wave equation appears everywhere because it is the simplest second-order linear PDE. Power laws appear because they are simplest scale-free distributions. [SOURCE:godel-1931|type:mathematical] Undecidability lurks here. No finite axiom set proves all wave behavior. The ubiquity may be mathematical necessity, not physical discovery. We see waves because our mathematics has only so many forms. We see excitable pulses because nonlinear dynamics has only so many attractors. The patterns may be artifacts of our tools. Not discoveries in nature.

[SOURCE:heraclitus-500|type:philosophical] Heraclitus said everything flows. He was right. But he did not distinguish the flows. [SOURCE:whitehead-1929|type:philosophical] Whitehead said reality is process. He was right too. But process has types. Some processes scale linearly. Some reset discretely. The philosopher sees unity. The physicist sees distinction. Both are partially right.

[SOURCE:england-2013|type:theoretical] England offers a deeper frame. Dissipation drives adaptation. Excitable pulses are optimal dissipators. They absorb energy. They release it. They reset. Linear waves dissipate too. But continuously. Not in packets. England's framework predicts both. But it does not unify them. Not yet.

We do not know if a deeper theory unifies both. We do not know if quantum field theory explains action potentials. We do not know if a single equation governs light and nerve. The honest answer is this. Two separate phenomena wear the same name. They converge in our vocabulary. They may not converge in nature.

Find an excitable medium that propagates without threshold. Find a pulse without refractory period. Find a pulse without fixed amplitude. Find a pulse that obeys superposition. Find two action potentials that pass through each other unchanged. Find a cardiac wave that scales with stimulus strength like a sound wave.

If any of these exist, the split collapses. One mathematics would rule both. The linguistic convergence would become mathematical convergence. Until then, the distinction holds.

## Related Sources

- [schrodinger-1944] Wave mechanics and matter waves. The quantum wave equation.
- [shannon-1948] Information theory. Fourier analysis as signal decomposition.
- [wiener-1948] Cybernetics. Feedback, oscillation, and control.
- [ashby-1956] Requisite variety. Information transmission in nervous systems.
- [prigogine-1977] Dissipative structures. Far-from-equilibrium thermodynamics.
- [noether-1918] Conservation laws. Symmetries underlying wave equations.
- [turing-1936] Universal computation. Discrete states and threshold logic.
- [von-neumann-1966] Self-reproducing automata. Cellular excitation and replication.
- [bak-1987] Self-organized criticality. Neural avalanches and power laws.
- [wilson-1971] Renormalization group. Universality across scales.
- [mandelbrot-1967] Fractals and scaling. Self-similarity in natural patterns.
- [kauffman-1993] Origins of order. Self-organization and the edge of chaos.
- [barabasi-1999] Scale-free networks. Information propagation topology.
- [watts-1998] Small-world networks. Signal routing efficiency.
- [heraclitus-500] Flux and opposition. Everything flows.
- [whitehead-1929] Process philosophy. Reality as becoming.
- [england-2013] Dissipation-driven adaptation. Self-organization from entropy.
- [darwin-1859] Natural selection. Evolution of signaling systems.
- [godel-1931] Incompleteness. Limits of formal systems. Undecidability of wave ubiquity.

## Related Convergences

- [convergence-c04] Symmetry-Breaking: Waves preserve symmetry in propagation. Symmetry-breaking creates the structures that carry them.
- [convergence-c06] Self-Organized Criticality: Neural avalanches are waves in critical media. Earthquakes are elastic wave avalanches.
- [convergence-c23] Attractors: Limit cycles are attractors. Hearts and neurons orbit them.
- [convergence-c17] Spirals: Spiral waves are the rotating form of excitable pulses. Galaxy arms are density waves.
- [convergence-c10] Networks: Waves need topology. Barabási and Watts showed how structure routes signal.
- [convergence-c03] Scale Invariance: Wave phenomena span thirty-three orders of magnitude. Power laws connect events across scales.
- [convergence-c20] Universal Computation: Excitable media compute. Threshold logic is the primitive of universal machines.

