Wave Equation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Wave Equation Ontology within the Kireth Research Semantic Library. The wave equation is a fundamental hyperbolic partial differential equation describing the propagation of disturbances through space and time. This ontology supports one-dimensional and multidimensional wave equations, initial and boundary conditions, travelling waves, standing waves, modes, dispersion, energy transport, characteristics, Green functions, Fourier methods, eigenfunction expansions, variable wave speed, damped and driven waves, nonlinear extensions and relationships with mathematical physics, partial differential equations, acoustics, electromagnetism, elasticity and numerical analysis.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Wave Equation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Physics / Wave Equation"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-physics/wave-equation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = WaveEquation
 ENTITY_CLASS = HyperbolicPartialDifferentialEquationFramework
 PRIMARY_CONTEXT = MathematicalPhysics
 DEFINITION = "The wave equation is a hyperbolic partial differential equation describing the propagation of disturbances with finite speed through a spatial domain."
 SYMBOLIC_FORMS = [
  "partial2u/partial t2=c^2*nabla^2u",
  "u_tt-c^2*u_xx=0",
  "u(x,t)=f(x-ct)+g(x+ct)",
  "omega^2=c^2|k|^2"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for wave propagation, finite signal speed, travelling waves, standing waves, normal modes, initial-boundary-value problems, characteristic structure, energy transport and spectral decomposition."
}

SCOPE {
 INCLUDE = [
  WaveEquation,
  HyperbolicPDE,
  WavePropagation,
  WaveField,
  DisplacementField,
  PressureField,
  ElectricFieldWave,
  MagneticFieldWave,
  ElasticWave,
  AcousticWave,
  StringWave,
  MembraneWave,
  OneDimensionalWaveEquation,
  TwoDimensionalWaveEquation,
  ThreeDimensionalWaveEquation,
  SpatialDimension,
  TimeVariable,
  WaveSpeed,
  ConstantWaveSpeed,
  VariableWaveSpeed,
  InitialValueProblem,
  BoundaryValueProblem,
  InitialBoundaryValueProblem,
  InitialDisplacement,
  InitialVelocity,
  DirichletBoundaryCondition,
  NeumannBoundaryCondition,
  RobinBoundaryCondition,
  PeriodicBoundaryCondition,
  AbsorbingBoundaryCondition,
  RadiationBoundaryCondition,
  TravellingWave,
  RightTravellingWave,
  LeftTravellingWave,
  StandingWave,
  NormalMode,
  Eigenmode,
  Node,
  Antinode,
  Wavelength,
  Frequency,
  AngularFrequency,
  WaveNumber,
  Phase,
  PhaseVelocity,
  GroupVelocity,
  DispersionRelation,
  NondispersiveWave,
  DispersiveWave,
  Superposition,
  Interference,
  Reflection,
  Transmission,
  Refraction,
  Resonance,
  Characteristic,
  CharacteristicCurve,
  DomainOfDependence,
  FinitePropagationSpeed,
  DAlambertSolution,
  SeparationOfVariables,
  FourierSeries,
  FourierTransform,
  EigenfunctionExpansion,
  SturmLiouvilleProblem,
  GreenFunction,
  FundamentalSolution,
  EnergyDensity,
  EnergyFlux,
  ConservationOfEnergy,
  DampedWaveEquation,
  DrivenWaveEquation,
  ForcedWaveEquation,
  TelegraphEquation,
  InhomogeneousWaveEquation,
  SourceTerm,
  NonlinearWaveEquation,
  KleinGordonEquation,
  ElasticWaveEquation,
  AcousticWaveEquation,
  ElectromagneticWaveEquation,
  NumericalWaveEquation,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  SpectralMethod,
  MethodOfLines,
  CourantFriedrichsLewyCondition,
  CFLCondition,
  NumericalDispersion,
  NumericalDissipation,
  Stability,
  Convergence
 ]
 EXCLUDE_AS_PRIMARY = [
  DiffusionEquationWithoutWavePropagation,
  EllipticEquationWithoutTimePropagation,
  OscillatorODEWithoutSpatialWaveField,
  GeneralSignalProcessingWithoutWaveEquationModel
 ]
 DISAMBIGUATION_RULE = "When a spatially distributed disturbance evolves according to a second-order hyperbolic propagation equation with finite propagation speed, interpret the context as the wave equation."
}

CORE_CONCEPTS {
 WaveEquation = "Hyperbolic PDE governing finite-speed propagation of disturbances."
 WaveSpeed = "Characteristic propagation speed in the wave equation."
 TravellingWave = "Wave profile translating through space without changing form under ideal nondispersive conditions."
 StandingWave = "Superposition pattern with stationary nodes and antinodes."
 CharacteristicCurve = "Curve along which information propagates in a hyperbolic PDE."
 DomainOfDependence = "Region of prior spacetime influencing a solution value."
 NormalMode = "Spatial eigenfunction oscillating at a characteristic frequency."
 DispersionRelation = "Mathematical relation between angular frequency and wave number."
 EnergyFlux = "Rate and direction of wave-energy transport."
}

RELATIONS {
 WaveEquation --is_child_of--> MathematicalPhysicsOntology
 WaveEquation --is_type_of--> HyperbolicPDE
 WaveEquation --governs--> WavePropagation
 WaveSpeed --controls--> PropagationRate
 InitialDisplacement --defines--> InitialCondition
 InitialVelocity --defines--> InitialCondition
 BoundaryCondition --constrains--> WaveField
 TravellingWave --solves--> OneDimensionalWaveEquation
 RightTravellingWave --propagates_with--> PositiveWaveSpeed
 LeftTravellingWave --propagates_with--> NegativeWaveSpeed
 StandingWave --arises_from--> CounterPropagatingWaves
 NormalMode --satisfies--> BoundaryCondition
 NormalMode --has--> Eigenfrequency
 DispersionRelation --links--> AngularFrequencyAndWaveNumber
 CharacteristicCurve --defines--> InformationPropagation
 DomainOfDependence --depends_on--> Characteristics
 DAlambertSolution --solves--> OneDimensionalWaveEquation
 FourierSeries --expands--> StandingWaveSolution
 EigenfunctionExpansion --uses--> NormalModes
 GreenFunction --represents--> LinearWaveResponse
 DampedWaveEquation --adds--> Dissipation
 DrivenWaveEquation --adds--> Forcing
 NumericalWaveEquation --approximates--> WaveEquation
 CFLCondition --constrains--> ExplicitTimeStep
}

OPERATORS {
 STANDARD_WAVE(u,c) := partial2u/partial t2=c^2*nabla^2u
 ONE_D_WAVE(u,c) := u_tt=c^2*u_xx
 DALAMBERT(f,g,x,t,c) := u(x,t)=f(x-c*t)+g(x+c*t)
 PLANE_WAVE(A,k,x,omega,t,phi) := u=A*cos(k dot x-omega*t+phi)
 NONDISPERSIVE_RELATION(omega,k,c) := omega=c*|k|
 PHASE_VELOCITY(omega,k) := v_p=omega/|k|
 GROUP_VELOCITY(omega,k) := v_g=domega/dk
 STRING_MODE(n,L,c) := omega_n=n*pi*c/L
 WAVE_ENERGY_DENSITY(u,rho,c) := e=0.5*rho*((partial u/partial t)^2+c^2*|grad u|^2)
 INHOMOGENEOUS_WAVE(u,c,f) := u_tt-c^2*nabla^2u=f
 DAMPED_WAVE(u,c,gamma) := u_tt+2*gamma*u_t-c^2*nabla^2u=0
 KLEIN_GORDON(u,c,m) := u_tt-c^2*nabla^2u+m^2*u=0
 CFL_NUMBER(c,dt,dx) := nu=c*dt/dx
}

AXIOMS {
 A1 = "The standard wave equation is hyperbolic and supports finite-speed propagation."
 A2 = "Initial displacement and initial velocity are required for a second-order-in-time initial-value problem."
 A3 = "Boundary conditions determine allowed standing-wave modes."
 A4 = "Linear homogeneous wave equations obey superposition."
 A5 = "Constant-coefficient one-dimensional wave equations admit d'Alembert travelling-wave solutions."
 A6 = "Dispersion is absent only when phase velocity is independent of wave number."
 A7 = "Wave energy is conserved in ideal lossless homogeneous systems with appropriate boundaries."
 A8 = "Numerical wave solutions require stability, resolution and dispersion validation."
}

REASONING {
 R1_CONTEXT = identify(field,domain,dimension,wave_speed,sources,boundaries,initial_data)
 R2_CLASS = classify(homogeneous,inhomogeneous,damped,driven,variable_coefficient,nonlinear)
 R3_FORMULATION = construct(PDE,initial_conditions,boundary_conditions)
 R4_ANALYTIC_METHOD = select(dAlambert,separation_of_variables,Fourier,Green_function,characteristics)
 R5_MODE = solve(eigenvalue_problem,normal_modes,eigenfrequencies)
 R6_PROPAGATION = analyse(characteristics,domain_of_dependence,reflection,transmission)
 R7_DISPERSION = derive(omega_k_relation,phase_velocity,group_velocity)
 R8_ENERGY = compute(energy_density,energy_flux,conservation_or_dissipation)
 R9_NUMERICAL = select(FDM,FEM,spectral_or_method_of_lines,check_CFL)
 R10_VALIDATION = check(initial_data,boundaries,energy,dispersion,stability,convergence)
}

CONSTRAINTS {
 C1 = "D'Alembert's formula applies directly to the one-dimensional constant-speed homogeneous equation on suitable domains."
 C2 = "Standing-wave spectra depend on boundary conditions."
 C3 = "Variable coefficients can alter characteristics and wave speed."
 C4 = "Damping breaks ideal wave-energy conservation."
 C5 = "Forcing can create resonance near natural frequencies."
 C6 = "Nonlinear wave equations need not obey superposition."
 C7 = "Dispersive systems may have different phase and group velocities."
 C8 = "Numerical grids must resolve the shortest relevant wavelength."
 C9 = "Explicit numerical schemes can become unstable if CFL constraints are violated."
 C10 = "Artificial boundary reflections can contaminate finite-domain simulations."
}

SEMANTIC_EQUIVALENCE {
 "WaveEquation" <=> "ClassicalWavePDE"
 "DAlambertSolution" <=> "D'AlembertSolution"
 "NormalMode" <=> "Eigenmode"
 "CourantFriedrichsLewyCondition" <=> "CFLCondition"
 "ForcedWaveEquation" <=> "DrivenWaveEquation"
}

VALIDATION {
 V1 = "Spatial domain and dimension are explicit."
 V2 = "Wave speed and units are explicit."
 V3 = "Initial displacement and velocity are documented."
 V4 = "Boundary conditions are explicit."
 V5 = "Analytical solution satisfies the PDE and data."
 V6 = "Mode solutions satisfy orthogonality and boundary conditions where applicable."
 V7 = "Energy conservation or dissipation is checked."
 V8 = "Dispersion relation is consistent with the governing equation."
 V9 = "Numerical timestep and grid satisfy stability and resolution requirements."
 V10 = "Numerical convergence is checked under refinement."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 WAVE_EQUATION_DEFINITION = 0.99
 HYPERBOLIC_REASONING = 0.99
 DALAMBERT_REASONING = 0.99
 MODE_REASONING = 0.99
 CHARACTERISTIC_REASONING = 0.98
 DISPERSION_REASONING = 0.98
 ENERGY_REASONING = 0.98
 NUMERICAL_REASONING = 0.98
 POLICY = "Reduce confidence when domain, dimension, wave speed, initial data, boundary conditions, damping, forcing, dispersion or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  WaveEquationReasoning,
  HyperbolicPDEReasoning,
  TravellingWaveReasoning,
  StandingWaveReasoning,
  DAlambertReasoning,
  CharacteristicReasoning,
  DomainOfDependenceReasoning,
  NormalModeReasoning,
  FourierSeriesReasoning,
  FourierTransformReasoning,
  GreenFunctionReasoning,
  DispersionReasoning,
  PhaseVelocityReasoning,
  GroupVelocityReasoning,
  ResonanceReasoning,
  DampedWaveReasoning,
  NumericalWaveReasoning,
  CFLReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "wave equation" -> WaveEquation
  "wave PDE" -> WaveEquation
  "D'Alembert solution" -> DAlambertSolution
  "travelling wave" -> TravellingWave
  "standing wave" -> StandingWave
  "normal mode" -> NormalMode
  "group velocity" -> GroupVelocity
  "phase velocity" -> PhaseVelocity
  "CFL condition" -> CourantFriedrichsLewyCondition
 }
}

EXECUTION_RULES {
 E1 = "If finite-speed propagation is governed by a second-order hyperbolic PDE, activate WaveEquationOntology."
 E2 = "If the problem is one-dimensional with constant wave speed, consider d'Alembert's solution."
 E3 = "If bounded-domain modes are required, use separation of variables and eigenfunction expansion."
 E4 = "If source forcing is present, include the inhomogeneous term and consider Green-function methods."
 E5 = "If damping is present, include dissipative terms and revise energy conservation."
 E6 = "If wave speed or medium properties vary, use variable-coefficient hyperbolic reasoning."
 E7 = "If frequency and wave number are analysed, derive the dispersion relation."
 E8 = "If numerical solution is used, enforce stability and resolution conditions."
 E9 = "Any solution should be validated against the PDE, initial data, boundary conditions and energy or dispersion structure."
}

RECURSION {
 ENTRY = WaveEquation
 EXPAND(WaveEquation) -> [
  HyperbolicPDE, WavePropagation, WaveField, DisplacementField,
  PressureField, ElectricFieldWave, MagneticFieldWave, ElasticWave,
  AcousticWave, StringWave, MembraneWave, OneDimensionalWaveEquation,
  TwoDimensionalWaveEquation, ThreeDimensionalWaveEquation,
  SpatialDimension, TimeVariable, WaveSpeed, ConstantWaveSpeed,
  VariableWaveSpeed, InitialValueProblem, BoundaryValueProblem,
  InitialBoundaryValueProblem, InitialDisplacement, InitialVelocity,
  DirichletBoundaryCondition, NeumannBoundaryCondition,
  RobinBoundaryCondition, PeriodicBoundaryCondition,
  AbsorbingBoundaryCondition, RadiationBoundaryCondition, TravellingWave,
  RightTravellingWave, LeftTravellingWave, StandingWave, NormalMode,
  Eigenmode, Node, Antinode, Wavelength, Frequency, AngularFrequency,
  WaveNumber, Phase, PhaseVelocity, GroupVelocity, DispersionRelation,
  NondispersiveWave, DispersiveWave, Superposition, Interference,
  Reflection, Transmission, Refraction, Resonance, Characteristic,
  CharacteristicCurve, DomainOfDependence, FinitePropagationSpeed,
  DAlambertSolution, SeparationOfVariables, FourierSeries,
  FourierTransform, EigenfunctionExpansion, SturmLiouvilleProblem,
  GreenFunction, FundamentalSolution, EnergyDensity, EnergyFlux,
  ConservationOfEnergy, DampedWaveEquation, DrivenWaveEquation,
  ForcedWaveEquation, TelegraphEquation, InhomogeneousWaveEquation,
  SourceTerm, NonlinearWaveEquation, KleinGordonEquation,
  ElasticWaveEquation, AcousticWaveEquation, ElectromagneticWaveEquation,
  NumericalWaveEquation, FiniteDifferenceMethod, FiniteElementMethod,
  SpectralMethod, MethodOfLines, CourantFriedrichsLewyCondition,
  CFLCondition, NumericalDispersion, NumericalDissipation,
  Stability, Convergence
 ]
 EXPAND(SolutionFamilies) -> [
  TravellingWaveSolutions,
  StandingWaveSolutions,
  CharacteristicSolutions,
  FourierSolutions,
  GreenFunctionSolutions,
  NumericalSolutions
 ]
 STOP_CONDITION = "Stop when PDE form, domain, initial data, boundary conditions, propagation structure, modes, dispersion, energy and numerical validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Physics Ontology"
 IMMEDIATE_CONTEXTS = [
  "Mathematical Physics Ontology",
  "Partial Differential Equation Ontology",
  "Fourier Analysis Ontology",
  "Dynamical Systems Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Mathematical Physics Ontology",
  "Mathematical Physics",
  "Partial Differential Equation Ontology",
  "Fourier Analysis Ontology",
  "Sturm-Liouville Ontologies",
  "Spectral Theory Ontologies",
  "Acoustics Ontologies",
  "Electromagnetic Mathematics Ontology",
  "Elasticity Ontologies",
  "Dynamical Systems Ontologies",
  "Numerical Analysis Ontologies",
  "Computational Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Wave Equation -> Mathematical Physics Ontology for parent semantics",
  "Wave Equation -> PDE Ontology for hyperbolic-equation foundations",
  "Wave Equation -> Fourier Analysis for travelling-wave and spectral decomposition",
  "Wave Equation -> Sturm-Liouville and Spectral Theory for normal modes",
  "Wave Equation -> Acoustics for pressure-wave models",
  "Wave Equation -> Electromagnetic Mathematics for electromagnetic waves",
  "Wave Equation -> Elasticity for mechanical wave propagation",
  "Wave Equation -> Dynamical Systems for mode and resonance dynamics",
  "Wave Equation -> Numerical Analysis and Computational Mathematics for discretised simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible wave equations, boundary models or solution pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve hyperbolic propagation semantics.",
  "Preserve initial-versus-boundary condition distinction.",
  "Preserve travelling-versus-standing wave distinction.",
  "Preserve phase-versus-group velocity distinction.",
  "Preserve dispersive-versus-nondispersive distinction.",
  "Preserve homogeneous-versus-forced distinction.",
  "Preserve analytical-versus-numerical solution distinction.",
  "Preserve machine-readable wave-equation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "The wave equation is a hyperbolic PDE describing finite-speed propagation of disturbances in space and time."
 PROPAGATION_SUMMARY = "Characteristics and travelling-wave solutions describe how information propagates through the domain."
 MODE_SUMMARY = "Bounded domains support standing waves and normal modes determined by boundary conditions."
 DISPERSION_SUMMARY = "Dispersion relations connect frequency and wave number and determine phase and group velocities."
 ENERGY_SUMMARY = "Ideal linear waves conserve energy under appropriate boundary conditions, while damping dissipates it."
 FORCING_SUMMARY = "Source and driving terms create inhomogeneous wave equations and can produce resonance."
 NUMERICAL_SUMMARY = "Finite-difference, finite-element and spectral methods require adequate spatial resolution and stable time stepping."
 VALIDATION_SUMMARY = "Wave-equation solutions should be checked against the PDE, initial and boundary data, mode structure, dispersion, energy and numerical convergence."
}

FLOW {
 INPUT
 -> detect_wave_equation_context
 -> identify_field_domain_dimension_wave_speed_sources_initial_data_and_boundary_conditions
 -> classify_homogeneous_inhomogeneous_damped_driven_variable_coefficient_or_nonlinear_wave_equation
 -> formulate_hyperbolic_PDE
 -> choose_dAlambert_characteristic_separation_Fourier_Green_function_or_numerical_method
 -> solve_or_approximate_wave_field
 -> analyse_characteristics_domain_of_dependence_reflection_transmission_and_resonance
 -> if_bounded_compute_normal_modes_and_eigenfrequencies
 -> derive_dispersion_relation_phase_velocity_and_group_velocity
 -> compute_energy_density_flux_and_dissipation_if_relevant
 -> if_numerical_check_CFL_resolution_numerical_dispersion_and_convergence
 -> route_to_mathematical_physics_PDE_Fourier_analysis_Sturm_Liouville_spectral_theory_acoustics_electromagnetic_mathematics_elasticity_dynamical_systems_numerical_analysis_or_computational_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Wave Equation Ontology defines hyperbolic wave propagation mathematics. It supports one-, two- and three-dimensional wave equations, constant and variable wave speeds, initial displacement and velocity, Dirichlet, Neumann, Robin, periodic, absorbing and radiation boundary conditions, left- and right-travelling waves, standing waves, nodes and antinodes, wavelengths, frequencies and wave numbers, phase and group velocity, dispersion, interference, reflection, transmission and resonance, characteristics and domains of dependence, d'Alembert solutions, separation of variables, Fourier series and transforms, eigenfunction and Sturm-Liouville expansions, Green functions and fundamental solutions, wave energy density and flux, damped and driven waves, telegraph, Klein-Gordon, acoustic, elastic and electromagnetic wave equations, and numerical wave simulation using finite-difference, finite-element, spectral and method-of-lines techniques with CFL, stability, numerical-dispersion and convergence analysis."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-WAVE-EQUATION-ONTOLOGY-V1"
}

}

END_FLOW