Wave Propagation and the Equal-Speed Condition

01Background and perturbation action

A common vacuum limiting speed for open longitudinal and transverse modes is an ASD ground-state constitutive relation. This page derives the propagation spectrum from the action and shows how order-parameter feedback changes the effective coefficients that must be matched.

Propagation depends on the background, restoring response and time response. Start with an approximately homogeneous matter-free and external-field-free state. The length-dimensional perturbation $\mathbf u(\mathbf x,t)$ satisfies $|\nabla\mathbf u|\ll1$, with wavelengths much larger than the microscopic cutoff. Reference position is $\mathbf x$ and evolution parameter is $t$. Constant coefficients give Chapter 4's action density:

The first term supplies positive dynamic weight; the other two supply volume and rotation-type restoration. With $[\rho_s]=\mathrm{kg\,m^{-3}}$ and $[K_L]=[K_T]=\mathrm{J\,m^{-3}}$, every term is energy density and $S_2=\int\mathcal L_2dt\,d^3x$ has action units. This quadratic form and its parameters are constitutive inputs, without being derived solely from the word “space” or dimensions.

$K_L$ is the longitudinal coefficient of this state model, distinct from the bulk modulus of ordinary elasticity. Comparisons must retain the selected model before introducing another constitutive structure.

02Variation and boundary terms

Write $s=\nabla\cdot\mathbf u$ and $\mathbf w=\nabla\times\mathbf u$. Both are dimensionless; $s$ is dilation rather than source strength, and $\mathbf w$ is curl. An allowed variation gives

Integration by parts yields $-\rho_s\partial_t^2\mathbf u$ from time derivatives. Using $\nabla\cdot(s\delta\mathbf u)=\nabla s\cdot\delta\mathbf u+s\nabla\cdot\delta\mathbf u$ gives $+K_L\nabla s$. The curl term gives $-K_T\nabla\times\mathbf w$. Temporal endpoints and spatial surfaces remain:

$\mathbf n$ is the outward normal. Fixed temporal variations together with decaying fields, periodic boundaries or appropriate displacement/natural conditions remove $\mathcal B$. Arbitrary interior variations then require

Both sides have force-density units. Longitudinal restoration follows $\nabla s$, and transverse restoration follows minus curl of curl. Positive restoring coefficients therefore enter stable free-wave operators. A boundary that actually supplies work cannot be discarded unconditionally.

03Projection, polarization and spectrum

The identity $\nabla^2\mathbf u=\nabla(\nabla\cdot\mathbf u)-\nabla\times(\nabla\times\mathbf u)$ rewrites restoration as $K_T\nabla^2\mathbf u+(K_L-K_T)\nabla(\nabla\cdot\mathbf u)$. Constant coefficients and compatible boundaries let projections commute with this operator:

Insert $\mathbf u_A=\operatorname{Re}[\mathbf U_Ae^{i(\mathbf k\cdot\mathbf x-\omega_A t)}]$, with length-dimensional amplitude, angular frequency and inverse-length wave vector. Time differentiation gives $-\omega_A^2$ and the Laplacian $-k^2$:

Equivalently, the Fourier restoring matrix is $K_Tk^2\delta_{ij}+(K_L-K_T)k_i k_j$. Its parallel eigenvalue is $K_Lk^2$, with two perpendicular eigenvalues $K_Tk^2$. This explains one longitudinal and two transverse vector polarizations. The gapless constant-coefficient branches have phase and group speeds $c_A$; a strict front still belongs to the underlying initial-value equation and its effective range.

Vector polarization does not supply tensor gravitational waves. Chapter 5 shows that a single plane wave's compatible strain from single-valued displacement has zero transverse-traceless projection. An independent tensor branch needs additional variables and a spectrum.

04What the common-speed matching means

ASD chooses a common limiting speed $c$ in the long-wave matter-free and external-field-free ground state. This quadratic model therefore requires

The left side matches restoring coefficients to kinetic weight; the derived ratios then produce the result on the right. The SI numerical definition of light speed does not determine an unknown longitudinal coefficient. Twelve neighbours and an FCC second moment alone also do not prove equal vector-sector speeds.

Ordinary isotropic elasticity has $c_L^2=(K+4G/3)/\rho$ and $c_T^2=G/\rho$ from a different strain energy. Its $K,G$ cannot simply replace $K_L,K_T$, and $K=G$ does not prove equality. The ASD task is to obtain the required coefficient relation from actual microscopic response.

The reference axis is the propagation direction. Longitudinal displacement is parallel and transverse displacement perpendicular. Dots sample a continuum and displacement is exaggerated. Coefficient controls illustrate a homogeneous uncoupled model, rather than particle motion, an observed lattice or instrument readings.

05How waves carry energy

Kinetic and restoring terms form

Chapter 5's sector normalization $Z_A=\rho_s$, $Z_Ac_A^2=K_A$ gives the same integrated energy under compatible boundaries. With the matched speed $c$, flux and source power are

Multiplying the sector equation by $\partial_t\mathbf u_A$ and applying the product rule splits gradients into energy derivatives and flux divergence. Energy density has $\mathrm{J\,m^{-3}}$ units, flux $\mathrm{W\,m^{-2}}$, and the source dot velocity is power density. A source-free system without net boundary flow conserves energy; outward radiation carries energy out of a region.

Absence of explicit dissipation does not remove geometric spreading. A localized source can have $1/R$ far-zone amplitude and $1/R^2$ flux density while retaining finite spherical power. Receiver coupling controls detection efficiency rather than redefining field energy.

06Why an ordered background changes the spectrum

The local dual-field candidate includes energy $g_LsF(\Theta)$. For a nonzero radial vacuum amplitude, take $\Theta=v_0+\eta$, where $\eta$ is an amplitude fluctuation rather than Chapter 4's density symbol. Then $F=2v_0\eta+O(\eta^2)$ and quadratic mixing is $\lambda s\eta$, with $\lambda=2g_Lv_0$. Set $W''(v_0)=I_\Theta\Omega_\Theta^2$, $c_\Theta^2=A_\Theta/I_\Theta$, and require $\Omega_\Theta^2>0$.

The linear longitudinal and amplitude equations contain $\lambda\nabla\eta$ and $\lambda s$. Eliminating their plane-wave amplitudes gives

Positive $Z_u,I_\Theta$ are kinetic coefficients and $c_u$ is the bare displacement speed; the transverse branch has no such amplitude mixing in this truncation. For a gapless root $\omega^2=\alpha k^2+O(k^4)$, the leading determinant gives $(c_u^2-\alpha)\Omega_\Theta^2=\lambda^2/(Z_uI_\Theta)$:

This calculates how feedback shifts longitudinal low-wave-number speed. Positive gradient response requires at least $c_u^2>\lambda^2/(Z_uI_\Theta\Omega_\Theta^2)$. Equality needs higher orders; a negative value is not ordinary stable propagation. The full stability domain also depends on remaining fields, gradient energy and allowed wave numbers.

Equal bare coefficients therefore do not ensure equal physical speeds. The manuscript considers a microscopic bare longitudinal/transverse difference or a source invariant with zero first derivative in vacuum. Those possibilities need full-spectrum checks rather than independent fitting presented as a mechanism. OP-SD-002 continues this task.

07Beyond homogeneous coefficients

These steps apply to the stated homogeneous small-amplitude long-wave models. Derivatives act on varying coefficients, projections may fail to commute, and coupling requires the full multicomponent spectrum. Near a cutoff, return to discrete dispersion. Equal long-wave speeds do not identify particles, polarizations or instrument readings.

Free longitudinal waves connects sources, initial data and reception. L–T coupling examines mutual driving at finite amplitude. Detailed equation relationships remain in the auxiliary index and equation pages.