# 11a. Sound From Energy Flow Chapter 11 recovered Newton's second law as the integrated continuity equation for momentum flux across the boundary of a stable bounded configuration. That result applies to a single localized mode responding to an external organized field. This chapter asks the next question: what happens when a large collection of such modes — the bulk matter medium — is disturbed? The answer is a wave. The wave equation that governs it is not a new postulate. It is the coarse-grained, longitudinal mode of the electromagnetic wave equation already in hand. Propagation speed, dispersion, and dissipation all follow from the electromagnetic structure of the medium. ## The medium Matter in a source-free Maxwell universe is aggregate toroidal closures of electromagnetic energy flow (Ch 8, Ch 10). Each toroidal closure is characterized by a winding pair $(m, n)$ — charge and spin as topological invariants — and a scalar amplitude of trapped energy $E$. Its mass is $$ M = E/c^2. $$ Its exterior carries an axial charge field that falls as $1/r^2$, sustained by the non-contractible through-hole flux of the $m$-winding. A bulk material is a regular arrangement of $n_0$ such toroids per unit volume, at equilibrium separation $d_0 \sim n_0^{-1/3}$. This equilibrium is set by the balance of charge coupling between neighbors: at $d_0$ the net force on each toroid from its neighbors vanishes. ## The perturbation Displace the toroid centered at position $\mathbf{x}$ by a small vector $\boldsymbol{\xi}(\mathbf{x}, t)$ from its equilibrium position. The restoring force comes from the axial charge coupling. The interaction energy between two toroidal axial charge fields at separation $d$ is of the form $$ U(d) \sim -\frac{Q^2}{d}, $$ where $Q$ is the through-hole flux strength fixed by the winding number $m$. For a small displacement $\xi$ from equilibrium: $$ U(d_0 + \xi) \approx U(d_0) + \tfrac{1}{2}\,U''(d_0)\,\xi^2, $$ with restoring stiffness $$ K_\text{local} = U''(d_0) \sim \frac{Q^2}{d_0^3} > 0. $$ This is a purely electromagnetic quantity: $Q$ comes from the charge winding, $d_0$ from the equilibrium of the same charge coupling. ## Continuum limit: the wave equation Pass to the continuum. With $n_0$ toroids per unit volume: $$ \rho = n_0\,M = \frac{n_0 E}{c^2} \qquad\text{(mass density, from aggregate trapped EM energy)} $$ $$ K = n_0\,K_\text{local} \qquad\text{(bulk stiffness, from charge coupling between neighbors)} $$ The equation of motion for the displacement field in the continuum limit is $$ \rho\,\partial_t^2\,\boldsymbol{\xi} = K\,\nabla^2\boldsymbol{\xi}. $$ This is a wave equation. The propagation speed is $$ v_s = \sqrt{\frac{K}{\rho}}. $$ Both $K$ and $\rho$ are derived from the electromagnetic structure of the toroids. No mechanical postulate enters. For the longitudinal (compressional) mode, set $p \propto \nabla \cdot \boldsymbol{\xi}$ and take the divergence. The result is $$ \partial_t^2 p - v_s^2\,\nabla^2 p = 0. $$ This is the sound wave equation, recovered from Maxwell transport. ## The operator identity The electromagnetic wave equation is $$ \partial_t^2 \mathbf{F} - c^2\,\nabla^2 \mathbf{F} = 0. $$ The acoustic wave equation is $$ \partial_t^2 p - v_s^2\,\nabla^2 p = 0. $$ The operator $\partial_t^2 - v^2\,\nabla^2$ is the same at both scales. The speed changes; the structure does not. The sound wave equation is not a new law — it is the same operator acting on the density perturbation of a medium whose constituents obey the fine-scale EM equation. ## Why $v_s \ll c$ is forced $$ \frac{v_s^2}{c^2} = \frac{K}{\rho\,c^2} = \frac{K_\text{local}}{Mc^2} \sim \frac{Q^2/d_0^3}{E}. $$ The numerator $Q^2/d_0^3$ is the interaction energy gradient between neighboring toroids — a near-field quantity. The denominator $E$ is the full self-energy of a single toroid. For the medium to be stable — for toroids to remain distinct, non-merging configurations — the inter-toroid coupling must be much weaker than the toroid self-energy: $$ \frac{Q^2}{d_0^3} \ll E \quad \Rightarrow \quad v_s \ll c. $$ The inequality $v_s \ll c$ is not a parameter choice. It is the stability condition of the medium. ## Dissipation and its disappearance Dissipation of sound occurs when the driving perturbation excites partial internal circulations within the toroidal closures that do not re-close into stable modes. Those open circulations radiate outward as electromagnetic transport — what macroscopic acoustics calls heat. The energy returns to the electromagnetic field, not to a separate thermal substance. Whether this happens depends on the frequency of the driving perturbation relative to the internal mode structure of the toroids. Each toroidal closure has a discrete set of natural internal frequencies $\omega_0 < \omega_1 < \omega_2 < \cdots$, set by its winding numbers and geometry (Ch 8). Three regimes follow: **Sub-resonant regime, $\omega \ll \omega_0$.** The perturbation is too slow to excite any internal circulation. The toroids move as rigid bodies under the charge coupling, adiabatically tracking the displacement field. No internal circulation is excited — partial or otherwise — so no radiation escapes. Propagation is elastic and lossless. **Resonant regime, $\omega = \omega_n$.** The driving frequency matches a natural mode of the toroidal closure. The perturbation couples into a complete standing wave inside the closure — a circulation that does re-close. No partial open circulations arise, so no radiation escapes from this coupling channel. The energy is absorbed into the stable internal mode, not dissipated. This is absorption without radiation. **Inter-resonance regime, $\omega_n < \omega < \omega_{n+1}$.** The driving frequency lies between natural modes. The perturbation excites partial circulations that cannot settle into a complete standing wave. These open circulations radiate. This is the dissipative regime. The structure is: $$ \omega \ll \omega_0 : \quad \text{elastic, lossless propagation} $$ $$ \omega = \omega_n : \quad \text{resonant absorption into stable modes} $$ $$ \omega_n < \omega < \omega_{n+1} : \quad \text{dissipation by EM radiation} $$ Sound has an acoustic band structure — transparent windows and absorbing peaks — derived entirely from the internal electromagnetic mode structure of the toroidal closures. No phonon postulate is needed. ## The coherent limit: superfluidity The most extreme case: if all toroids in the medium occupy the same mode — every winding aligned, every phase synchronized — then a perturbation either couples to the collective mode coherently or does not couple at all. The effective $\omega_0 \to 0$ and the entire acoustic spectrum falls in the sub-resonant regime. Propagation is lossless at all frequencies. This is the superfluid. Sound propagates without dissipation because the medium has no partial-mode degrees of freedom to radiate into. Superfluidity is not a separate phenomenon added to the framework. It is the limit of full toroidal coherence, in which the dissipation mechanism is structurally absent. ## What this closes Acoustics requires no postulates beyond source-free Maxwell transport: - The acoustic wave equation has the same operator as the EM wave equation; only the speed changes, and that change is derived. - Propagation speed $v_s = \sqrt{K/\rho}$ comes entirely from electromagnetic quantities ($K$ from charge coupling, $\rho$ from trapped-energy mass). - $v_s \ll c$ is forced by medium stability. - Dissipation is frequency-gated by the internal EM mode structure of the toroids: absent below $\omega_0$, absent at resonances, present between them. - The coherent limit of the same structure recovers dissipation-free propagation — superfluidity — without additional postulates. Acoustics is a slow, longitudinal, coarse-grained regime of Maxwell transport.