Wave geometry, modes and singularities bridge v1

Status: typed vocabulary and local-to-global bridge; no new physical identification Trigger: an illustrative chain-of-circles/water-wave image shared by Matheorems Boundary: the image motivates the question but is not evidence that every wave is circular or generated by tangent circles.

Why this bridge exists

The corpus already types causal response, resonance, reciprocal noise, geometry, memory and detection. It did not yet provide a compact join from local curve construction and moving wavefronts to envelopes, caustics, cusps, pressure waves, phonons, dispersion, resonance and shocks.

These objects can coexist in one experiment, but they are not interchangeable. Their resemblance in a photograph is not an identification theorem.

Local transformations and circles

For a regular planar curve parameterized by arc length,

The curvature function \(\kappa(s)\), together with one initial frame, reconstructs the curve up to a rigid motion. Discretely this is an ordered sequence of small rotations and translations in \(SE(2)\). The osculating circle at a point is a local reader with radius \(1/|\kappa|\); it is not automatically a physical wavefront.

A chain of tangent circles is even less restrictive. Unless its radius update, turning rule and parameterization are supplied, many different paths fit the description. Tangency is a constraint, not a dynamics.

Moving fronts and envelopes

For an isotropic source at \(X(s)\), a simple constant-speed family can be written

The family envelope obeys

This says that neighbouring fronts touch. If the fronts arise as a projection of a ray or phase manifold, a rank loss in that projection produces a caustic. A generic cusp requires a further degeneracy,

plus the appropriate nonzero higher derivative and transversality conditions. Finite wavelength, diffraction, damping and detector resolution regularize the geometric-optics divergence.

The construction is dimension- and medium-dependent. Isotropic point propagation gives circles in two spatial dimensions and spheres in three. An anisotropic constitutive law gives different wavefront indicatrices. A dispersive medium does not transport every frequency or packet at one common speed.

Material and spectral modes

A pressure wave is a classical compression/rarefaction disturbance with a material constitutive law and evolution equation. In a bounded linearized problem it may decompose into normal modes.

A normal mode is an eigenmotion of a declared operator and its boundary conditions. A phonon is a quantum of a normal mode of a lattice or effective elastic medium. A pressure disturbance is therefore not automatically a phonon; the carrier and quantization matter.

For a dispersion branch

the packet velocity is

When the map from wavevector to group-velocity direction loses rank, many modes can propagate toward the same spatial direction, producing wave or phonon focusing and caustics.

Resonance is a different reader

A resonance belongs to the complete driven response problem. In the minimal damped example,

Poles select dynamical modes, residues record probe coupling and linewidths record damping. A normal mode can remain dark to a particular probe. A caustic can geometrically concentrate broadband propagation without being a resonance. Conversely, a spatially uniform resonator can ring strongly without forming a caustic.

The useful compression is

Shocks are not cusps

A shock is a discontinuity in a nonlinear weak solution, produced when characteristics intersect and selected by jump plus entropy/admissibility conditions. A cusp is a singularity of an envelope or projection. The two can occur together in a nonlinear propagation problem, but neither implies the other.

Non-equivalence firewall

Operational readers

1. What propagates? Record the field, mode branch, support/front and polarization.
2. How does it propagate? Record dispersion, group velocity, attenuation and the metric or constitutive tensor.
3. Where does it concentrate? Record projection Jacobians, rank loss, caustic class and finite-wavelength smoothing.
4. What rings? Record retarded poles, residues, linewidths, probe coupling and boundary/loading conditions.
5. Is it nonlinear? Record amplitude dependence, characteristic crossing, harmonic generation and admissible jump conditions.
6. What reaches the observer? Record detector transfer, bandwidth, phase reference, noise and covariance.

These readers prevent a photograph, spectrum or time trace from silently answering a differently typed question.

Existing corpus joins

• Dynamic response, geometry and detection already supplies the drive-to-record chain and the correct complete-response definition of resonance.
• Conjugate-frequency reality keeps complex dispersive response distinct from static covariance.
• The reciprocal Keldysh split keeps causal propagation and reciprocal noise in their appropriate analytic types.
• Detector frontier register already names frequency, phase and causal-lag spectroscopy as an operational detector family.

Ambient-context boundary

This bridge is not organized around MOND, gravitational retardation, M44 or extra dimensions. Those topics were ambient when the bridge was noticed. Any later use must independently supply a carrier, operator, constitutive law, dimensionality, boundary conditions and measurable outputs. No application inherits support merely because its diagram contains circles, waves, cusps or resonances.

Sources

• Forward wave fundamental solutions and strong Huygens support: <https://math.mit.edu/~dyatlov/res/res_final.pdf>
• Water-wave dispersion, phase and group velocity: <https://ocw.mit.edu/courses/18-357-interfacial-phenomena-fall-2010/229785489c81648500f89044a97c0e51MIT18357F10_Lecture19.pdf>
• An example of modern osculating-circle geometry: <https://arxiv.org/abs/2112.03614>

Verdict

Admit this as a neutral local-to-global geometry and mode-typing bridge. Require a separate physical law before instantiating it in any application.