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---
## From excitation to reception
A free longitudinal spatial wave is a time-dependent propagating compression state with energy and flux fixed by its action. It connects an excitation source to a receiver; a measurable signal additionally requires the receiver and its reference standard to respond.
A free longitudinal spatial-wave process needs an excitation site, a propagation background and a material receiver. Time-dependent source-region volume or connection states excite longitudinal displacement and density. Outside the source, a homogeneous equation describes free evolution. Receiver coupling then converts the arriving field into length, phase, frequency or energy-exchange readings.
This page takes homogeneous constant coefficients and long-wave small amplitudes. Length-dimensional $\mathbf u_L$ is curl-free, and $\sigma=-\nabla\cdot\mathbf u_L$ is compression-positive linear relative density. ASD's “phonon” language borrows a name for propagating excitations or possible mode quantization, without establishing an observed spatial lattice or replacing space by an ordinary acoustic material.
## Sources and initial data determine propagation
With longitudinal displacement source $\mathbf J_L$ and positive normalization $Z_L$, minus the divergence of its equation gives
$s_\sigma=-Z_L^{-1}\nabla\cdot\mathbf J_L$ has $\mathrm{s^{-2}}$ units. It vanishes outside the source, but free propagation still requires initial $\sigma$ and $\partial_t\sigma$. Both must arise from longitudinal displacement with the same boundary and global number/volume constraints.
For an isotropic infinite-space retarded response, set $R=|\mathbf r-\mathbf r'|$:
$$
\sigma(\mathbf r,t)=\sigma_{\rm in}(\mathbf r,t)+\frac{1}{4\pi c^2}\int\frac{s_\sigma(\mathbf r',t-R/c)}{R}\,d^3r'.
$$
$\sigma_{\rm in}$ is the homogeneous initial-data solution, and $\mathbf r'$ the source position. A finite starting time restricts the source history to responses already established and causally able to arrive. The $1/(4\pi c^2)$ normalization belongs to $\partial_t^2-c^2\nabla^2$. Changing to $c^{-2}\partial_t^2-\nabla^2$ requires changing source normalization too. Reflecting or finite-domain boundaries require another Green kernel.
## Energy and geometric spreading
Source-free energy follows the original sector action:
Here $A=L$, with energy density $\mathcal E_L$ and flux $\mathbf S_L$. At $\mathbf J_L=0$, only boundary flow changes energy. The density representation must also retain
and its $1/k^2$ kinetic kernel, rather than choosing an arbitrary local density energy. Wave vector $\mathbf k$ excludes the separately constrained zero mode; the [longitudinal field](/en/theory/space-dynamics/longitudinal-field) explains the transformation.
Localized sources can give $1/R$ spherical amplitudes, $1/R^2$ flux densities and finite total power. A nondissipative free equation does not imply no spreading, infinite material penetration or no absorption and scattering. Material coupling and boundaries determine those responses. Transverse divergence-free fields can spread geometrically as well.
## Different backgrounds, dressing and particle candidates
A static longitudinal background is fixed by steady sources and boundaries. A free wave is a time-dependent propagating solution after excitation or initial preparation. A matter-associated structure evolves with a moving core without necessarily carrying independent far-zone radiation. Related degrees of freedom still require separate sources, spectra and boundaries.
Chapter 4 explores a neutrino candidate dominated by longitudinal response with weak orientation coupling. This is a particle-identification proposal: longitudinal displacement does not supply half-integer spin, Fermi statistics, chirality or flavour response. The internal-phase model has further propagation and detection-projection conditions, so a homogeneous density wave is not an established neutrino solution.
Detected gravitational-wave polarization likewise does not follow from the word “longitudinal.” Chapter 5 separates source energy, scalar/independent-tensor propagation and interferometer differential response. A common speed replaces none of these calculations. Static gravity, neutrino candidates and tensor observations are not used here as proof that a free longitudinal spatial wave has been identified.
## How a receiver produces a reading
Reception first needs a coupling between material states and the longitudinal field, then displacement, transition or other receiver responses. Rulers, frequency standards and transfer chains belong to the instrument. Common response can cancel a signal. Arriving density, speed or energy alone is not an observable prediction.
Chapter 5 gives a concrete long-wavelength differential-interferometer example. For arm unit vectors $\mathbf a,\mathbf b$, define $D_{ij}=(a_i a_j-b_i b_j)/2$. If the scalar effective strain includes a breathing term $\alpha_B\sigma_g(\delta_{ij}-n_i n_j)$ and a longitudinal term $\alpha_L\sigma_g n_i n_j$, the trace condition $D_{ii}=0$ leaves only $(\alpha_L-\alpha_B)\sigma_gD_{ij}n_i n_j$ in the scalar reading. Here $\sigma_g$ denotes the scalar variable of this candidate gravitational channel, $\mathbf n$ the propagation direction, and $\alpha_B,\alpha_L$ responses still to be calculated from matter and light propagation. Adding more detectors of the same type does not alone remove this ideal long-wavelength degeneracy. Finite-arm responses, other receivers or independent theoretical relations require separate calculations.
Existing directions seek differential length, phase or frequency responses and identifiable channel conversion. Each needs source strength, propagation transfer functions, receiver response, reference and noise models. Mechanical acoustics, temperature and ordinary electromagnetic cross-coupling require the same apparatus calculation. There is no independently justified sensitivity number here, and nondissipative waves do not establish communication performance.
[OP-SD-005](/en/open-problems/space-dynamics/longitudinal-wave-detection) records detection as a specific task. The [test matrix](/en/tests/space-dynamics) retains candidate directions and constraint classifications, without recording a confirmed free longitudinal spatial-wave detection. Quantitative discrimination requires a completed source–propagation–receiver model.