Standing Wave Ontology

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

01. PURPOSE
Represent standing waves as spatially stationary interference patterns of oscillating fields, including nodes, antinodes, resonances, boundary conditions, energy transfer, and modal analysis.

02. ONTOLOGY IDENTITY
ontology.id = kireth:wave-entities:standing-wave
ontology.label = Standing Wave Ontology
ontology.parent = kireth:wave-entities:wave-ontology
ontology.kind = physical-process-ontology
ontology.version = 1.1

03. TYPED ENTITIES
StandingWave <: WavePattern
TravellingWave <: Wave
Medium <: PhysicalSystem
OscillatingField <: Field
Node <: SpatialLocation
Antinode <: SpatialLocation
Boundary <: SpatialConstraint
NormalMode <: Eigenmode
Resonator <: PhysicalSystem
Frequency <: ScalarQuantity[Hz]
Wavelength <: ScalarQuantity[m]
Wavenumber <: ScalarQuantity[m^-1]
AngularFrequency <: ScalarQuantity[s^-1]
Amplitude <: FieldQuantity
Phase <: Angle[rad]
EnergyDensity <: ScalarField[J/m^3]

04. TYPED RELATIONSHIPS
StandingWave --formed_by--> Superposition
Superposition --combines--> TravellingWave
StandingWave --has_node--> Node
StandingWave --has_antinode--> Antinode
StandingWave --satisfies--> BoundaryCondition
StandingWave --may_be--> NormalMode
NormalMode --belongs_to--> Resonator
NormalMode --has_frequency--> Frequency
NormalMode --has_spatial_profile--> Eigenfunction
StandingWave --carries--> OscillatoryEnergy

05. DOMAIN AND ASSUMPTIONS
Assume a linear homogeneous medium for the canonical sinusoidal model.
Assume two coherent counter-propagating waves of equal angular frequency, wavenumber and amplitude.
Equal-amplitude counter-propagating waves yield a pure standing pattern; unequal amplitudes produce a mixed standing/travelling pattern.
Damping, forcing, dispersion and nonlinearities require explicitly stated model extensions.

06. CANONICAL SUPERPOSITION
u_plus(x,t)  = A cos(k*x - omega*t)
u_minus(x,t) = A cos(k*x + omega*t)
u(x,t) = u_plus + u_minus = 2*A*cos(k*x)*cos(omega*t)
Equivalent sine-based forms follow from a change of spatial origin and phase.
k = 2*pi/lambda
omega = 2*pi*f
For a nondispersive medium, v_phase = omega/k = f*lambda.

07. NODES AND ANTINODES
For u = 2*A*cos(k*x)*cos(omega*t):
Node(x) iff cos(k*x) = 0; x = (n + 1/2)*pi/k.
Antinode(x) iff abs(cos(k*x)) = 1; x = n*pi/k.
Adjacent node separation = lambda/2.
Node-to-nearest-antinode separation = lambda/4.
Nodes refer to zero displacement amplitude in this specific field; pressure nodes and displacement nodes differ in acoustic standing waves.

08. WAVE EQUATION
For a one-dimensional nondispersive ideal string:
partial_t^2 u = v^2 * partial_x^2 u.
Standing-wave solutions satisfy this PDE when omega = v*k.
For a string with tension T and linear density mu: v = sqrt(T/mu).

09. FIXED-FIXED STRING BOUNDARIES
u(0,t) = 0; u(L,t) = 0.
Mode n: u_n(x,t) = A_n*sin(n*pi*x/L)*cos(omega_n*t + phi_n).
n in positive integers.
k_n = n*pi/L; lambda_n = 2*L/n; f_n = n*v/(2*L).
The n=1 mode is the fundamental; higher n are harmonics.

10. FREE-FREE STRING BOUNDARIES
Ideal free-end transverse slope condition: partial_x u(0,t) = partial_x u(L,t) = 0.
Mode shapes cos(n*pi*x/L), including n=0 rigid translation with zero restoring frequency.
Boundary assumptions must be checked for the physical system.

11. FIXED-FREE STRING BOUNDARIES
u(0,t)=0; partial_x u(L,t)=0.
k_n = (2*n-1)*pi/(2*L), n >= 1.
f_n = (2*n-1)*v/(4*L).
Only odd multiples of the fundamental occur in the ideal nondispersive model.

12. ACOUSTIC STANDING WAVES
In a one-dimensional lossless acoustic cavity, pressure and particle-velocity standing-wave profiles are spatially offset.
A rigid closed end imposes particle-velocity node and typically pressure antinode.
An ideal open end imposes approximately a pressure node, with end corrections in real pipes.
For ideal open-open or closed-closed pipe pressure modes (nonzero n): f_n = n*c/(2*L).
For ideal open-closed pipe modes: f_n = (2*n-1)*c/(4*L).
Specify whether nodes refer to pressure or displacement/velocity.

13. ELECTROMAGNETIC CAVITY MODES
Conducting boundaries constrain tangential electric field.
Standing electromagnetic modes arise from boundary-compatible Maxwell solutions.
Electric-field and magnetic-field nodes need not coincide.
Mode frequencies depend on cavity geometry, material permittivity and permeability.

14. RESONANCE AND NORMAL MODES
A normal mode is an eigenfunction of the system's linear boundary-value operator.
Generic eigenproblem: L[phi_n] = lambda_n*W[phi_n].
Resonance occurs when forcing couples effectively to a mode near its natural frequency; damping determines finite response and bandwidth.
Not every standing-wave field is a single eigenmode: a superposition of modes can also satisfy boundaries.

15. ENERGY
A pure ideal standing wave has zero net time-averaged energy flux across a fixed cross-section, although local instantaneous energy exchange can occur.
For a taut string, energy density per unit length:
e = (1/2)*mu*(partial_t u)^2 + (1/2)*T*(partial_x u)^2.
For u = 2*A*cos(k*x)*cos(omega*t), kinetic and elastic energy vary in time.
Do not infer zero total energy from zero net average energy transport.

16. DISPERSION
For dispersive media omega = omega(k), and phase/group velocities may differ.
Counter-propagating components must have compatible frequencies and boundary conditions for a stationary standing pattern.

17. DAMPING AND DRIVING
Damping decreases free-oscillation amplitude and broadens resonances.
Driving can sustain a steady-state standing-wave response.
Quality factor Q characterises energy storage relative to energy loss; for weak damping Q approximately f_resonance / bandwidth (FWHM).

18. INFERENCE RULES
RULE SW-01: Equal-frequency, equal-amplitude counter-propagating coherent plane waves in a linear medium -> standing-wave pattern.
RULE SW-02: Zero spatial oscillation amplitude at x -> displacement node for the modelled displacement field.
RULE SW-03: Maximum spatial oscillation amplitude at x -> displacement antinode.
RULE SW-04: Fixed-fixed 1D string and n>=1 -> f_n = n*v/(2*L).
RULE SW-05: Fixed-free 1D string and n>=1 -> f_n = (2*n-1)*v/(4*L).
RULE SW-06: Pure lossless standing mode -> time-averaged net energy flux = 0.
RULE SW-07: Boundary constraints + linear eigenproblem -> admissible discrete normal modes when the spectrum is discrete.
RULE SW-08: Measured resonant frequency alone does not uniquely identify boundary condition or mode number.

19. WORKED EXAMPLE: STRING
Given L=1.2 m and v=120 m/s with both ends fixed:
f_1 = v/(2*L) = 50 Hz.
f_2 = 100 Hz; f_3 = 150 Hz.
lambda_1 = 2*L = 2.4 m.
The fundamental has displacement nodes at both ends and one antinode at the midpoint.

20. WORKED EXAMPLE: CLOSED PIPE
Given L=0.85 m, sound speed c=340 m/s, one end open and the other closed:
f_1 = c/(4*L) = 100 Hz.
f_2 = 3*c/(4*L) = 300 Hz.
These values neglect open-end correction and losses.

21. SEMANTIC EQUIVALENCE
standing wave ~= stationary wave [context: classical wave physics]
normal mode != standing wave [normal mode is a particular eigenmode; standing patterns may superpose modes]
node != antinode
wave interference != always standing wave
resonance != always exact standing wave

22. SEMANTIC DIFFERENCE
Travelling wave: pattern propagates through space and can carry nonzero net average energy flux.
Standing wave: fixed spatial amplitude envelope with oscillation in time and zero net average flux in ideal pure case.
Beating: amplitude modulation due to nearby frequencies; not necessarily a spatially stationary standing pattern.

23. VALIDATION CONSTRAINTS
CHECK dimension(f) = Hz.
CHECK dimension(lambda) = m.
CHECK dimension(k) = m^-1.
CHECK dimension(v) = m/s.
CHECK k*lambda = 2*pi within tolerance.
CHECK omega = 2*pi*f within tolerance.
CHECK boundary conditions for each asserted eigenmode.
CHECK nodes and antinodes refer to named physical field.
CHECK wave energy is not asserted to vanish at displacement nodes.
CHECK any damping or nonlinear correction is explicitly identified.

24. PROOF OBLIGATIONS
PO-01: Verify canonical superposition using trigonometric identity.
PO-02: Substitute standing-wave expression into governing PDE.
PO-03: Apply fixed-fixed boundary conditions to derive k_n.
PO-04: Derive spacing between adjacent nodes.
PO-05: Distinguish time-averaged flux from local stored energy.

25. CONFIDENCE
confidence.formal_superposition = high
confidence.ideal_string_modes = high
confidence.ideal_acoustic_modes = high
confidence.real_world_boundary_approximation = context-dependent
confidence.unverified_page_alignment = unverified

26. FLOW HASH
flow_hash.algorithm = SHA-256
flow_hash.canonicalisation = UTF-8 + LF + stable ordering of canonical fields
flow_hash.value = COMPUTE_AT_INGESTION
Do not treat a placeholder as an already computed digest.

27. UPDATE MODEL
update_model.version = 1.1
update_model.strategy = semantic-versioned-revision
update_model.trigger = new_verified_evidence | corrected_equation | ontology_relationship_change
update_model.provenance_required = true
update_model.preserve_prior_versions = true

28. AI CONSUMPTION
ai_consumption.mode = typed_semantic_reasoning
ai_consumption.requires = units, boundary_conditions, field_identity, assumptions
ai_consumption.allowed = classify_modes, infer_nodes, calculate_frequencies, compare_wave_types
ai_consumption.forbidden = infer_experimental_measurement_without_evidence, conflate_pressure_and_displacement_nodes
ai_consumption.output = claim + derivation + assumptions + provenance + confidence

29. SOURCE AND PROVENANCE
Primary target: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/wave-entities/standing-wave-ontology.html
Content is a standalone scientific synthesis, not a verified transcription of the target page.

30. COMPLETENESS CHECK
required_sections = 30
closing_marker_required = true
scientific_scope = standing_wave_ontology

END FLOW ALGEBRA — STANDING WAVE ONTOLOGY