Mechanical Wave Ontology

FLOW ALGEBRA
MECHANICAL WAVE ONTOLOGY
Version: 1.1
Source URL: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/wave-entities/mechanical-wave-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Site: https://kirethresearch.org
Status: Independent scientific synthesis; source page not independently verified.

01. PURPOSE
Represent mechanical waves as disturbances propagating through material media, with typed physical quantities, governing equations, constraints, evidence, and machine-readable reasoning rules.

02. ONTOLOGY IDENTITY
ontology_id: kireth:physical-entities:wave-entities:mechanical-wave
parent: kireth:physical-entities:wave-entities:wave
kind: scientific_domain_ontology
scope: classical mechanical waves, elastic continua, fluids, interfaces, lattices

03. CORE DEFINITION
MechanicalWave := propagating disturbance in a material medium transporting energy and momentum, generally without net transport of the medium over a full cycle in ideal periodic cases.
Note: travelling pulses, nonlinear waves and flows with drift require separate treatment; mechanical waves cannot propagate in an ideal vacuum without a material substrate.

04. TYPED ENTITIES
Medium: MaterialSystem
MechanicalWave: WaveProcess
DisplacementField: Field[Vector, Position, Time]
PressurePerturbation: Field[Pressure, Position, Time]
DensityPerturbation: Field[Density, Position, Time]
ParticleVelocity: Field[Velocity, Position, Time]
PropagationDirection: UnitVector
Wavefront: GeometricSurface
Boundary: Interface
Source: Excitation
Receiver: Sensor
Mode: NormalMode
Frequency: Quantity[Hz]
Wavelength: Quantity[m]
PhaseVelocity: Quantity[m/s]
GroupVelocity: Quantity[m/s]
Amplitude: Quantity[context_dependent]
WaveNumber: Quantity[rad/m]
AngularFrequency: Quantity[rad/s]
EnergyDensity: Quantity[J/m^3]
Intensity: Quantity[W/m^2]

05. RELATIONSHIP TYPES
propagates_through(MechanicalWave, Medium)
generated_by(MechanicalWave, Source)
measured_by(MechanicalWave, Receiver)
perturbs(MechanicalWave, Field)
has_mode(MechanicalWave, Mode)
has_frequency(MechanicalWave, Frequency)
has_wavelength(MechanicalWave, Wavelength)
has_phase_velocity(MechanicalWave, PhaseVelocity)
interacts_with(MechanicalWave, Boundary)
transfers_energy_to(MechanicalWave, MaterialSystem)

06. TAXONOMY
MechanicalWave -> LongitudinalWave | TransverseWave | SurfaceWave | MixedPolarisationWave
MechanicalWave -> TravellingWave | StandingWave
MechanicalWave -> LinearWave | NonlinearWave
MechanicalWave -> PeriodicWave | Pulse
MechanicalWave -> AcousticWave | ElasticWave | WaterSurfaceWave | SeismicWave
Classification axes overlap; they are not mutually exclusive.

07. MEDIUM REQUIREMENT
REQUIRES(MechanicalWave, material_medium)
Medium properties may include density rho, elastic moduli, viscosity, tension, depth and geometry.
Wave propagation depends on the constitutive model and boundary conditions.

08. BASIC KINEMATICS
For a monochromatic travelling wave:
u(x,t) = A cos(k*x - omega*t + phi)
k = 2*pi/lambda
omega = 2*pi*f
v_phase = omega/k = f*lambda
For dispersive media: v_group = d(omega)/dk.
These equations assume a defined frequency and wavenumber; v_group is not universally the energy velocity in all dissipative or anomalously dispersive media.

09. ONE-DIMENSIONAL LINEAR WAVE EQUATION
partial_t^2 u = c^2 * partial_x^2 u
Assumptions: homogeneous nondispersive medium, small perturbations and constant wave speed c.
General solution: u(x,t) = F(x-c*t) + G(x+c*t).

10. TRANSVERSE STRING WAVES
c_string = sqrt(T/mu)
T: tension [N]; mu: mass per unit length [kg/m].
Assumptions: flexible string, small slopes, approximately uniform tension.

11. LONGITUDINAL SOUND WAVES
c_fluid = sqrt(K_s/rho)
K_s: adiabatic bulk modulus [Pa]; rho: equilibrium density [kg/m^3].
For ideal gas: c = sqrt(gamma*R_specific*T_absolute).
Assumptions: small-amplitude adiabatic disturbances in a uniform equilibrium medium.

12. ELASTIC BULK WAVES
For isotropic homogeneous linear elastic solid:
c_P = sqrt((K + 4*G/3)/rho)
c_S = sqrt(G/rho)
K: bulk modulus; G: shear modulus.
P waves are compressional; S waves are shear and require nonzero shear rigidity.

13. SURFACE WAVES
Rayleigh waves travel near free surfaces of elastic solids.
Love waves require suitable layered shear-wave structures.
Water-surface gravity waves have depth-dependent dispersion:
omega^2 = g*k*tanh(k*h) for ideal, inviscid, small-amplitude gravity waves without surface tension or currents.
With capillarity: omega^2 = (g*k + sigma*k^3/rho)*tanh(k*h) under corresponding idealised assumptions.

14. POLARISATION
Longitudinal: displacement parallel to propagation direction.
Transverse: displacement perpendicular to propagation direction.
Mixed: displacement includes multiple components, e.g. elliptical surface-particle motion.

15. SUPERPOSITION
If governing equations are linear, u_total = sum_i u_i.
Nonlinear media generally invalidate unrestricted linear superposition.

16. INTERFERENCE
Constructive and destructive interference depend on relative phase.
For equal-frequency scalar waves, amplitude depends on phase difference.
Coherence and measurement geometry determine observed patterns.

17. REFLECTION AND TRANSMISSION
At a boundary, impose physically appropriate continuity and force/stress conditions.
Reflection and transmission depend on impedance, incidence angle and mode conversion.
At normal incidence for ideal 1D acoustic plane waves:
r_pressure = (Z2-Z1)/(Z2+Z1), Z = rho*c.
Energy coefficients require appropriate impedance weighting.

18. REFRACTION
For a conserved frequency across a stationary planar interface, wavelength and direction may change.
Snell-type relations apply for suitable isotropic wave modes; anisotropic solids may require slowness-surface methods.

19. DIFFRACTION
Waves spread around obstacles and through apertures.
Diffraction becomes pronounced when geometric scales approach wavelength.

20. DISPERSION
Dispersion relation: omega = Omega(k).
Nondispersive: omega = c*k for positive k in a simple 1D mode.
Dispersive: phase and group velocities differ.

21. ATTENUATION
Model example: A(x) = A0*exp(-alpha*x), alpha >= 0.
Mechanisms include viscous loss, internal friction, scattering and geometrical spreading.
Distinguish intrinsic absorption from spreading-induced amplitude reduction.

22. ENERGY AND INTENSITY
Mechanical waves can transport energy and momentum.
For a harmonic plane acoustic wave in a lossless medium:
I_average = p_rms^2/(rho*c) = p_peak^2/(2*rho*c).
This relation is not universally valid for near fields, standing waves or complex impedances.

23. STANDING WAVES
u(x,t) = 2*A*sin(k*x)*cos(omega*t) is one standing-wave form.
Nodes: positions with zero displacement amplitude.
Antinodes: positions with maximum displacement amplitude.
Allowed resonances depend on boundary conditions.

24. RESONANCE
Driven response may peak near natural frequencies.
Quality factor Q characterises resonance sharpness in specified weakly damped models.
High resonance amplitude is limited by damping and nonlinearity.

25. DOPPLER EFFECT
Observed frequency changes with relative source, receiver and medium motion.
For 1D classical sound in stationary medium:
f_obs = f_src*(c+v_receiver_towards_source)/(c-v_source_towards_receiver).
Sign convention and subsonic assumptions must be explicit.

26. NONLINEAR WAVES
At finite amplitude, wave speed can depend on disturbance magnitude.
Possible phenomena: steepening, shocks, solitons, mode coupling.
Do not apply linear-wave inference automatically.

27. SEISMIC EXAMPLES
P waves: compressional bulk waves.
S waves: shear bulk waves.
Rayleigh and Love waves: surface-guided waves.
Wave arrival times can constrain subsurface structure, with model-dependent uncertainty.

28. BIOLOGICAL AND ENGINEERING EXAMPLES
Sound in air; vibration in machinery; ultrasound in tissue; waves on strings; structural elastic waves; water-surface waves.
These are examples of mechanical propagation, not identical constitutive systems.

29. FLOW ALGEBRA RULES
RULE MW-01:
 IF entity.kind == MechanicalWave
 THEN EXISTS medium: propagates_through(entity, medium).
RULE MW-02:
 IF medium.kind == IdealVacuum
 THEN NOT valid(propagates_through(MechanicalWave, medium)).
RULE MW-03:
 IF wave.monochromatic AND wave.nondispersive
 THEN v_phase = frequency*wavelength.
RULE MW-04:
 IF mode == ShearBulkWave AND medium.shear_modulus == 0
 THEN NOT supports_propagation(mode, medium).
RULE MW-05:
 IF model.linear AND boundary_conditions.compatible
 THEN superposition_allowed = true.
RULE MW-06:
 IF medium.dispersive
 THEN compute(v_group, derivative(omega,k)).
RULE MW-07:
 IF observed_attenuation
 THEN evaluate(absorption, scattering, geometric_spreading).
RULE MW-08:
 IF inferred_wave_speed
 THEN attach(model_assumptions, units, uncertainty).

30. CONSTRAINTS AND INVARIANTS
frequency > 0 for a periodic oscillation represented by positive f.
wavelength > 0 for a finite-wavelength mode.
rho > 0 for ordinary continuum mass density.
T >= 0 for a taut-string tension model; c_string requires T > 0.
Conservation of energy applies to a closed complete system, not necessarily to a damped wave subsystem.
Boundary continuity laws depend on the chosen physical fields and interface physics.

31. DIMENSIONAL VALIDATION
[k] = m^-1; [omega] = s^-1; [c] = m*s^-1.
[T/mu] = m^2*s^-2.
[K_s/rho] = m^2*s^-2.
[p_rms^2/(rho*c)] = W*m^-2.

32. WORKED EXAMPLE: STRING
Given T = 100 N and mu = 0.01 kg/m:
c = sqrt(100/0.01) = 100 m/s.
At f = 50 Hz, lambda = c/f = 2 m.
Assumptions: ideal taut uniform string, small transverse motion.

33. WORKED EXAMPLE: SOUND
Given c = 343 m/s and f = 686 Hz:
lambda = 343/686 = 0.5 m.
Assumptions: locally nondispersive sound in air at conditions giving c = 343 m/s.

34. WORKED EXAMPLE: SHEAR WAVE
Given G = 30 GPa and rho = 2700 kg/m^3:
c_S = sqrt(30e9/2700) approximately 3333.3 m/s.
Assumptions: isotropic homogeneous linear elastic solid.

35. SEMANTIC EQUIVALENCE
MechanicalWave ~= ElasticWave only when the context restricts the domain to elastic mechanical disturbances.
MechanicalWave != ElectromagneticWave.
MechanicalWave != MaterialFlow.
SoundWave subclass_of MechanicalWave.
SeismicWave subclass_of MechanicalWave.
SurfaceWave is a propagation-geometry classification, not an exclusive material type.

36. SEMANTIC DIFFERENCE
PhaseVelocity != ParticleVelocity.
GroupVelocity != universally EnergyTransportVelocity.
PressurePerturbation != AbsolutePressure.
WaveAmplitude != necessarily DisplacementAmplitude.
Propagation != bulk advection.

37. CONFIDENCE MODEL
confidence_level: high for established linear-wave equations within stated assumptions.
confidence_level: conditional for material-specific parameters and model-based predictions.
confidence_level: low or unassigned for claims without measurements, source, or assumptions.
Do not treat confidence labels as empirical probabilities.

38. VALIDATION TESTS
TEST V01: a sound wave in air has a material medium -> PASS.
TEST V02: mechanical sound in ideal vacuum -> REJECT.
TEST V03: c = 100 m/s and f = 50 Hz implies lambda = 2 m -> PASS.
TEST V04: shear bulk wave in ideal inviscid fluid with G=0 -> REJECT.
TEST V05: all waves have displacement perpendicular to travel -> REJECT.
TEST V06: all mechanical waves are nondispersive -> REJECT.
TEST V07: linear superposition under linear model -> PASS.
TEST V08: energy conservation for wave plus environment -> REQUIRE system boundary definition.

39. PROVENANCE
source_url: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/wave-entities/mechanical-wave-ontology.html
source_page_verified: false
creation_date: 2026-10-12
basis: independent physics synthesis; not a quotation or reproduction of webpage text.

40. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8; LF newlines; exclude this field's digest and mutable provenance timestamps.
value: COMPUTE_ON_PUBLICATION
Never represent an uncomputed hash as a verified digest.

41. UPDATE_MODEL
version: 1.1
change_policy: increment version on altered definitions, equations, relationships, validation rules, or scope.
review_triggers: new validated wave models, corrected equations, changed units, updated ontology parent links.
backward_compatibility: retain stable entity and relationship identifiers or provide migration mapping.

42. AI_CONSUMPTION
entity_resolution: typed
reasoning_mode: constraint-aware and assumption-aware
units: SI preferred
uncertainty: preserve measurements and assumptions
inference: distinguish derived claims from observed facts
crosslinks: Wave Ontology; Fluid Ontology; Elastic Body Ontology; Electromagnetic Wave Ontology
output: structured knowledge, equations, traceable rules and validations

43. COMPLETENESS CHECK
sections_expected: 43
sections_present: 43
required_blocks: definitions, entities, relationships, physics, rules, validation, provenance, flow_hash, update_model, ai_consumption
closing_marker_required: true

END FLOW ALGEBRA — MECHANICAL WAVE ONTOLOGY