# 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](https://x.com/matheorems/status/2114714945204199814) **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, \[ \gamma'(s)=T(s), \qquad T'(s)=\kappa(s)N(s). \] 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 \[ F(x,t;s)=|x-X(s)|^2-c^2(t-s)^2=0. \] The family envelope obeys \[ F=0, \qquad \partial_sF=0. \] 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, \[ F=\partial_sF=\partial_s^2F=0, \] 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 \[ \omega=\omega_n(k), \] the packet velocity is \[ v_g=\nabla_k\omega_n(k). \] 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, \[ \chi_R(\omega)={1\over k-m\omega^2-i\gamma\omega}. \] 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 \[ \boxed{ \text{poles select what rings};\quad \text{dispersion selects how it moves};\quad \text{projection geometry selects where it concentrates}.} \] ## 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 | tempting identification | required correction | |---|---| | caustic = resonance | geometric concentration and spectral amplification are different mechanisms | | cusp = shock | projection singularity is not automatically a field discontinuity | | pressure wave = phonon | quantized carrier and mode branch must be supplied | | tangent circle = wavefront | a local geometric constraint is not a propagation law | | envelope = superposition | superposition retains amplitude and phase, not only tangency | | normal mode = observed resonance | the probe residue can vanish | | group velocity = phase velocity | they differ in a dispersive medium | | shared visible silhouette = shared mechanism | forward operators and controls must agree | ## 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](../admissibility/admissibility_06_dynamic_response_geometry_detection_metastability.md) already supplies the drive-to-record chain and the correct complete-response definition of resonance. - [Conjugate-frequency reality](../../claims/1000-1099/1024_conjugate_frequency_reality_pairs_but_does_not_remove_the_imaginary_residual_mode_so_spin23_is_static_or_reciprocal.md) keeps complex dispersive response distinct from static covariance. - [The reciprocal Keldysh split](../../claims/1000-1099/1025_a_reciprocal_keldysh_kernel_places_noise_in_h2r_while_retarded_response_remains_complex_and_fdt_preserves_the_split.md) keeps causal propagation and reciprocal noise in their appropriate analytic types. - [Detector frontier register](detector_frontier_register_v1.json) 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: - Water-wave dispersion, phase and group velocity: - An example of modern osculating-circle geometry: ## 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.