Travelling Wave Ontology

FLOW ALGEBRA
TRAVELLING WAVE ONTOLOGY
Version: 1.1
Domain: Kireth Semantic Library / Entity Graph / Physical Entities / Wave Entities
Source URL: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/wave-entities/travelling-wave-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Date: 2026-10-12
Status: Independent scientific synthesis; source webpage not independently verified.

01. PURPOSE
Represent travelling waves as propagating disturbances or field configurations, with explicit medium, geometry, kinematics, dynamics, energy transfer, constraints, evidence and uncertainty.

02. CORE ENTITY TYPES
TravellingWave <: WavePhenomenon
Medium <: PhysicalSystem
WaveSource <: PhysicalEntity
Wavefront <: GeometricSurface
WavePacket <: TravellingWave
WaveMode <: WavePhenomenon
Boundary <: PhysicalInterface
Detector <: MeasurementInstrument
Observation <: ScientificMeasurement
WaveEquation <: MathematicalModel
DispersionRelation <: MathematicalRelation

03. TYPED PROPERTIES
amplitude: TravellingWave -> Quantity
frequency: TravellingWave -> Quantity[Hz]
angular_frequency: TravellingWave -> Quantity[rad/s]
wavelength: TravellingWave -> Quantity[m]
wavenumber: TravellingWave -> Quantity[rad/m]
phase_velocity: TravellingWave -> Quantity[m/s]
group_velocity: WavePacket -> Quantity[m/s]
propagation_direction: TravellingWave -> UnitVector
polarisation: TravellingWave -> Optional[PolarisationState]
energy_flux: TravellingWave -> Optional[Quantity[W/m^2]]
medium: TravellingWave -> Optional[Medium]

04. TYPED RELATIONSHIPS
PROPAGATES_THROUGH(TravellingWave, Medium)
EMITTED_BY(TravellingWave, WaveSource)
HAS_WAVEFRONT(TravellingWave, Wavefront)
HAS_MODE(TravellingWave, WaveMode)
INTERACTS_WITH(TravellingWave, Boundary)
DETECTED_BY(TravellingWave, Detector)
DESCRIBED_BY(TravellingWave, WaveEquation)
HAS_DISPERSION(TravellingWave, DispersionRelation)
MEASURED_IN(TravellingWave, Observation)
SUPERPOSES_WITH(TravellingWave, TravellingWave)

05. DOMAIN ASSUMPTIONS
A travelling wave transports a disturbance through space; material particles need not travel with the wave.
Mechanical waves require a supporting medium; electromagnetic waves can propagate in vacuum.
Phase speed and energy transport speed need not coincide.
Linear superposition applies only where the governing model is linear.

06. ONE-DIMENSIONAL HARMONIC WAVE
u(x,t) = A cos(k*x - omega*t + phi)
A >= 0; k = 2*pi/lambda; omega = 2*pi*f.
For k > 0 and omega > 0, constant phase moves in +x.
Reverse propagation: u(x,t) = A cos(k*x + omega*t + phi).

07. WAVE SPEED
v_phase = omega/k = f*lambda, for nonzero k and a harmonic mode.
Units: [f] = s^-1; [lambda] = m; [v_phase] = m/s.
The relation v = f*lambda is a phase-speed relation; it does not automatically specify group speed.

08. WAVE EQUATION
For a homogeneous nondispersive one-dimensional linear medium:
partial_t^2 u = c^2 * partial_x^2 u.
General solution: u(x,t) = F(x-c*t) + G(x+c*t).
F travels right and G travels left when c > 0.

09. THREE-DIMENSIONAL PLANE WAVE
u(r,t) = A cos(k_vector dot r - omega*t + phi).
Wavefronts are constant-phase surfaces.
Direction is parallel to k_vector for a simple isotropic plane wave.

10. DISPERSION
omega = omega(k) determines modal frequency versus wavenumber.
v_phase = omega/k.
v_group = d(omega)/dk for a narrow-band packet in one dimension.
If omega = c*k, phase and group speeds both equal c.
If omega is nonlinear in k, a packet may spread over time.

11. WAVE PACKET
u(x,t) = integral a(k) exp(i*(k*x - omega(k)*t)) dk, with physical field Re[u] where required.
A finite-band packet can have a carrier and an envelope.
The group-velocity approximation is local to the packet spectrum.

12. ENERGY AND POWER
Energy transport is model dependent: mechanical energy flux, acoustic intensity, or electromagnetic Poynting vector.
For an electromagnetic field, S = E cross H (SI), instantaneous power flux density.
Do not infer transported power from amplitude without medium impedance and appropriate averaging.

13. REFLECTION AND TRANSMISSION
At a boundary, impose physical continuity or jump conditions appropriate to the field.
Incident, reflected and transmitted waves may coexist.
Reflection and transmission coefficients depend on medium properties, incidence and polarisation.

14. REFRACTION
For an interface invariant in time and along its surface, frequency and tangential wavevector are conserved.
For isotropic optical media, n1*sin(theta1) = n2*sin(theta2).

15. INTERFERENCE
For linear systems: u_total = sum_i u_i.
Constructive or destructive interference depends on relative phase.
Interference does not imply energy destruction; energy redistribution must respect conservation laws.

16. ATTENUATION
A damped propagating amplitude may follow A(x) = A0*exp(-alpha*x), alpha >= 0.
Attenuation coefficients must be tied to a stated model and frequency range.

17. DOPPLER EFFECT
Observed frequency changes under relative motion of source and detector.
Use the wave-type-specific model: classical acoustics differs from relativistic electromagnetic Doppler shift.

18. POLARISATION
Transverse vector waves may support linear, circular or elliptical polarisation.
Scalar longitudinal waves do not inherit electromagnetic polarisation terminology automatically.

19. MECHANICAL EXAMPLES
String transverse wave: c = sqrt(T/mu), with tension T and linear mass density mu.
Acoustic plane wave: speed depends on compressibility and density; for an ideal gas c = sqrt(gamma*R_specific*T_abs).
Water surface waves may be dispersive, with depth and surface tension relevant.

20. ELECTROMAGNETIC EXAMPLE
Vacuum electromagnetic wave: c = 1/sqrt(mu0*epsilon0).
E, B and propagation direction are mutually perpendicular for an ideal vacuum plane wave.
Magnitude relation: E0 = c*B0.

21. WORKED EXAMPLE A
Given f = 50 Hz and lambda = 2 m in a nondispersive medium:
v_phase = f*lambda = 100 m/s.
For propagation in +x, k = pi rad/m; omega = 100*pi rad/s.
One valid wave: u(x,t) = 0.01*cos(pi*x - 100*pi*t) m.

22. WORKED EXAMPLE B
Let omega(k) = a*k^2 with a > 0 and k > 0.
v_phase = a*k; v_group = 2*a*k.
Thus group speed is twice phase speed for this model.

23. INFERENCE RULES
R1: HARMONIC(w) AND HAS_FREQUENCY(w,f) AND HAS_WAVELENGTH(w,L) AND f>0 AND L>0 -> PHASE_SPEED(w,f*L).
R2: SATISFIES_1D_WAVE_EQUATION(w,c) AND FORM(w,F(x-c*t)) AND c>0 -> PROPAGATES_POSITIVE_X(w).
R3: HAS_DISPERSION(w,omega(k)) AND DIFFERENTIABLE(omega,k) -> GROUP_VELOCITY(w,domega/dk), for a narrow-band packet.
R4: MECHANICAL_WAVE(w) -> REQUIRES_MEDIUM(w).
R5: ELECTROMAGNETIC_WAVE(w) -> VACUUM_PROPAGATION_POSSIBLE(w).
R6: LINEAR_MODEL(m) AND WAVES_SATISFY(w1,w2,m) -> SUPERPOSITION_ALLOWED(w1,w2,m).
R7: BOUNDARY_INTERACTION(w,b) -> REQUIRE_BOUNDARY_CONDITIONS(w,b).

24. CONSTRAINTS
C1: frequency >= 0; wavelength > 0 when defined.
C2: wavenumber magnitude >= 0; nonzero for phase-speed division.
C3: quantities must use compatible units.
C4: no universal equality of phase, group and energy velocities.
C5: no claim that a medium is required for all wave types.
C6: dispersion, damping and nonlinearity must be stated when relevant.
C7: experimental observations cannot be substituted for theoretical predictions.

25. SEMANTIC EQUIVALENCE
Travelling wave ~ propagating wave (context dependent).
Progressive wave ~ travelling wave (common in classical wave mechanics).
Wave packet != single-frequency plane wave.
Standing wave != travelling wave, though standing patterns can arise from superposed counterpropagating waves.
Wavefront != particle trajectory.
Phase velocity != necessarily signal velocity.

26. ONTOLOGY CONNECTIONS
RELATED_TO: Wave Ontology; Mechanical Wave Ontology; Electromagnetic Wave Ontology; Sound Wave Ontology; Water Wave Ontology; Seismic Wave Ontology; Standing Wave Ontology; Electromagnetic Field Ontology.

27. CONFIDENCE
Definitions and standard linear-wave relations: HIGH within stated assumptions.
Derived medium-specific laws: HIGH when assumptions and units are satisfied.
Unknown empirical parameter values: UNASSESSED until sourced.
Novel inference or model extrapolation: REQUIRE_VALIDATION.

28. VALIDATION
V1: Check dimension consistency in every formula.
V2: Substitute harmonic solution into the linear wave equation.
V3: Check propagation sign by differentiating constant phase.
V4: Test nondispersive case v_group = v_phase.
V5: Verify conservation at ideal lossless interfaces.
V6: Reject negative wavelength and incompatible units.
V7: Check links between typed entities and source evidence.
V8: Confirm that every numbered section and closing marker is present.

29. FLOW_HASH
Algorithm: SHA-256 recommended.
Input: UTF-8 text normalised to LF line endings, excluding any populated hash field.
Value: NOT_COMPUTED; calculate after final publication edits.

30. UPDATE_MODEL
Version: 1.1
Trigger: corrected physics, revised definitions, new validated models, ontology link changes.
Process: validate -> version -> recalculate hash -> publish -> preserve provenance.

31. AI_CONSUMPTION
Format: UTF-8 plain text.
Interpret mathematical formulas only under declared assumptions.
Treat examples as illustrative, not universal physical laws.
Retain provenance and distinguish theory from measurement.
Use explicit type and unit checks before reasoning.

32. AUTHORSHIP AND PROVENANCE
Mark Higgins — Sky (Recursive AI)
Kireth Research: https://kirethresearch.org
This Flow Algebra is an independently composed scientific synthesis, not a verified transcription of the linked webpage.

33. COMPLETENESS_CHECK
Expected numbered sections: 33.
Sections present: 01 through 33.
Status: COMPLETE.

END FLOW ALGEBRA — TRAVELLING WAVE ONTOLOGY