Wave Equation Ontology

flow "WaveEquationOntology" v1.0

# WAVE EQUATION ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/mathematical-physics-analysis/wave-equation-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Mathematical Physics Analysis"
ENTITY := "Wave Equation"

WAVE_EQUATION := {
 type: HyperbolicPartialDifferentialEquation,
 domain: MathematicalPhysicsAnalysis,
 purpose: "Represent propagation of disturbances through space and time with finite characteristic speed, including vibration, acoustics, electromagnetism and related wave phenomena."
}

CONCEPTS := {
 WAVE_EQUATION,
 HYPERBOLIC_PDE,
 WAVE_SPEED,
 DISPLACEMENT_FIELD,
 INITIAL_CONDITION,
 BOUNDARY_CONDITION,
 CHARACTERISTIC,
 D_ALEMBERT_SOLUTION,
 STANDING_WAVE,
 TRAVELLING_WAVE,
 EIGENMODE,
 ENERGY,
 FOURIER_METHOD,
 GREEN_FUNCTION,
 DISPERSION,
 DAMPING
}

# ---------- STANDARD FORM ----------
in n spatial dimensions:

partial_squared_u/partial_t_squared
 =
c^2
*
LAPLACIAN(u)

u
 -> WAVE_FIELD

c
 -> WAVE_SPEED

WAVE_EQUATION
 -> SECOND_ORDER_HYPERBOLIC_PDE

# ---------- ONE DIMENSION ----------
u_tt
 =
c^2*u_xx

where

u
 =
u(x,t)

ONE_DIMENSIONAL_WAVE_EQUATION
 -> canonical_propagation_model

# ---------- INITIAL CONDITIONS ----------
u(x,0)
 =
f(x)

u_t(x,0)
 =
g(x)

SECOND_ORDER_TIME_EQUATION
 -> requires(INITIAL_DISPLACEMENT_AND_INITIAL_VELOCITY)

INITIAL_VALUE_PROBLEM
 := {
 WAVE_EQUATION,
 f,
 g
}

# ---------- D'ALEMBERT SOLUTION ----------
on whole real line:

u(x,t)
 =
1/2
*
[
 f(x-c*t)
 +
 f(x+c*t)
]
+
1/(2*c)
*
integral_from_(x-c*t)_to_(x+c*t)(
 g(s)
 ds
)

D_ALEMBERT_SOLUTION
 -> exact_solution_for(1D_CONSTANT_SPEED_WHOLE_LINE_IVP)

# ---------- TRAVELLING WAVES ----------
right-moving wave:

u(x,t)
 =
F(x-c*t)

left-moving wave:

u(x,t)
 =
G(x+c*t)

GENERAL_HOMOGENEOUS_1D_SOLUTION
 =
F(x-c*t)
+
G(x+c*t)

TRAVELLING_WAVE
 -> profile_translated_without_distortion
    in(NONDISPERSIVE_CONSTANT_SPEED_MODEL)

# ---------- CHARACTERISTICS ----------
characteristic curves:

x-c*t
 =
constant

and

x+c*t
 =
constant

CHARACTERISTICS
 -> paths_of(INFORMATION_PROPAGATION)

WAVE_SPEED
 -> slope_structure_of(CHARACTERISTICS)

# ---------- FINITE PROPAGATION SPEED ----------
disturbance at point
 (x_0,t_0)

-> influences only points inside
 characteristic_cone

FINITE_PROPAGATION_SPEED
 -> fundamental_hyperbolic_property

DOMAIN_OF_DEPENDENCE
 -> bounded_by(CHARACTERISTICS)

# ---------- CAUSAL CONE ----------
in multidimensional constant-speed wave equation:

|x-x_0|
 <=
c*(t-t_0)

-> causal_influence_region

LIGHT_CONE_ANALOGUE
 -> geometric_representation_of(FINITE_SIGNAL_SPEED)

# ---------- BOUNDARY VALUE PROBLEM ----------
on finite interval:

0<x<L

possible boundaries:

DIRICHLET:
 u(0,t)=0
 u(L,t)=0

NEUMANN:
 u_x(0,t)=0
 u_x(L,t)=0

ROBIN:
 alpha*u + beta*u_x = 0

BOUNDARY_CONDITION
 -> determines(REFLECTION_AND_MODE_STRUCTURE)

# ---------- SEPARATION OF VARIABLES ----------
assume

u(x,t)
 =
X(x)*T(t)

then

X*T_double_dot
 =
c^2*X_double_prime*T

divide:

T_double_dot/(c^2*T)
 =
X_double_prime/X
 =
-lambda

SEPARATION
 -> produces(SPATIAL_EIGENVALUE_PROBLEM_AND_TEMPORAL_ODE)

# ---------- FIXED STRING EIGENMODES ----------
for

u(0,t)=u(L,t)=0

spatial equation:

X_double_prime
+
lambda*X
 =
0

eigenvalues:

lambda_n
 =
(n*pi/L)^2

eigenfunctions:

X_n(x)
 =
sin(n*pi*x/L)

NORMAL_MODES
 -> discrete_frequency_spectrum

# ---------- NATURAL FREQUENCIES ----------
omega_n
 =
c*n*pi/L

frequency:

f_n
 =
omega_n/(2*pi)

MODE_NUMBER
 n
 -> determines(NATURAL_FREQUENCY)

# ---------- FOURIER SERIES SOLUTION ----------
u(x,t)
 =
sum_from_n_1_to_infinity(
 [
  A_n*cos(omega_n*t)
  +
  B_n*sin(omega_n*t)
 ]
 *
 sin(n*pi*x/L)
)

FOURIER_COEFFICIENTS
 -> determined_by(INITIAL_DATA)

FOURIER_METHOD
 -> decomposes(WAVE_INTO_NORMAL_MODES)

# ---------- STANDING WAVE ----------
u(x,t)
 =
X(x)*T(t)

STANDING_WAVE
 -> fixed_spatial_nodes_and_antinodes

STANDING_WAVE
 -> superposition_of(OPPOSITE_TRAVELLING_WAVES)

# ---------- ENERGY ----------
for 1D string:

E(t)
 :=
1/2
*
integral(
 u_t^2
 +
c^2*u_x^2
 dx
)

under suitable boundary conditions:

dE/dt
 =
0

UNDAMPED_WAVE_EQUATION
 -> CONSERVES(ENERGY)

# ---------- ENERGY METHOD ----------
multiply PDE by
 u_t

integrate over domain

-> derive(ENERGY_IDENTITY)

ENERGY_ESTIMATE
 -> supports(
 UNIQUENESS,
 STABILITY,
 CONTINUOUS_DEPENDENCE
)

# ---------- FORCED WAVE EQUATION ----------
u_tt
-
c^2*LAPLACIAN(u)
 =
F(x,t)

F
 -> SOURCE_OR_FORCING

FORCED_WAVE
 -> response_to(EXTERNAL_INPUT)

solution
 -> homogeneous_part
    +
    forced_part

# ---------- DAMPED WAVE EQUATION ----------
u_tt
+
gamma*u_t
-
c^2*LAPLACIAN(u)
 =
0

gamma>0
 -> DAMPING

energy
 -> decreases_over_time

DAMPED_WAVE
 -> combines(HYPERBOLIC_PROPAGATION_AND_DISSIPATION)

# ---------- VARIABLE SPEED ----------
u_tt
 =
div(
 c(x)^2
 *
 grad(u)
)

or other physically appropriate variable-coefficient forms

VARIABLE_COEFFICIENT_WAVE_EQUATION
 -> spatially_dependent_propagation

REFRACTION_AND_SCATTERING
 -> may_result_from(MEDIUM_VARIATION)

# ---------- HIGHER DIMENSIONS ----------
u_tt
 =
c^2*LAPLACIAN(u)

in
 R^n

RADIAL_SYMMETRY
 -> may_reduce(PDE_DIMENSION)

GEOMETRY
 -> influences(WAVEFRONT_EVOLUTION)

# ---------- THREE DIMENSIONS ----------
for 3D free-space wave equation:

disturbance
 -> propagates_on_or_within(SPHERICAL_WAVEFRONTS)

Huygens_principle
 -> dimension_dependent_property

THREE_DIMENSIONAL_WAVE_PROPAGATION
 -> canonical_acoustic_and_electromagnetic_model

# ---------- GREEN FUNCTION ----------
wave operator:

Box_c
 :=
partial_t^2
-
c^2*LAPLACIAN

Green function:

Box_c G
 =
delta_space
*
delta_time

RETARDED_GREEN_FUNCTION
 -> causal_response_kernel

solution
 -> convolution_of(GREEN_FUNCTION_WITH_SOURCE)

# ---------- FOURIER TRANSFORM ----------
spatial Fourier transform:

u_hat(k,t)

PDE becomes:

u_hat_tt
+
c^2*|k|^2*u_hat
 =
0

each Fourier mode
 -> HARMONIC_OSCILLATOR

FOURIER_TRANSFORM
 -> diagonalises(CONSTANT_COEFFICIENT_SPATIAL_WAVE_OPERATOR)

# ---------- DISPERSION RELATION ----------
plane wave:

u
 =
A
*
exp(
 i*(k dot x - omega*t)
)

substitution gives:

omega^2
 =
c^2*|k|^2

therefore

omega
 =
c*|k|

for positive-frequency branch

PHASE_VELOCITY
 =
omega/|k|
 =
c

GROUP_VELOCITY
 =
domega/d|k|
 =
c

STANDARD_WAVE_EQUATION
 -> NONDISPERSIVE

# ---------- DISPERSIVE GENERALISATION ----------
other wave equations may have

omega
 =
omega(k)

with

phase_velocity
 !=
group_velocity

DISPERSION
 -> frequency_dependent_propagation

STANDARD_LINEAR_WAVE_EQUATION
 -/> represents_all_wave_dispersion

# ---------- REFLECTION ----------
wave reaches boundary

-> boundary condition determines
 REFLECTED_WAVE

fixed boundary
 -> phase inversion in simple 1D string context

free boundary
 -> different reflection condition

BOUNDARY
 -> modifies(WAVE_SUPERPOSITION)

# ---------- RESONANCE ----------
periodic forcing frequency
 -> approaches(NATURAL_FREQUENCY)

response amplitude
 -> may_grow_large
    depending_on(DAMPING_AND_FORCING)

RESONANCE
 -> mode_selective_amplification

# ---------- ACOUSTICS ----------
small pressure perturbation
 p

under suitable homogeneous-medium assumptions:

p_tt
 =
c_sound^2
*
LAPLACIAN(p)

ACOUSTIC_WAVE
 -> application_of(WAVE_EQUATION)

# ---------- ELASTIC STRING ----------
transverse displacement
 u(x,t)

string tension
 T

linear density
 rho

wave speed:

c
 =
sqrt(T/rho)

STRING_WAVE_EQUATION
 -> derived_from(NEWTONIAN_FORCE_BALANCE)

# ---------- ELECTROMAGNETIC WAVES ----------
in homogeneous source-free medium:

E_tt
 =
c_medium^2
*
LAPLACIAN(E)

B_tt
 =
c_medium^2
*
LAPLACIAN(B)

with divergence and Maxwell constraints

MAXWELL_EQUATIONS
 -> imply(WAVE_EQUATIONS_FOR_FIELDS)

# ---------- QUANTUM / RELATIVISTIC RELATION ----------
KLEIN_GORDON_EQUATION
 -> wave-type hyperbolic PDE_with_mass_term

SCHRODINGER_EQUATION
 -/> classical_second_order_wave_equation

WAVE_EQUATION
 -> family_relation_to(MATHEMATICAL_PHYSICS_HYPERBOLIC_MODELS)

# ---------- NUMERICAL METHODS ----------
FINITE_DIFFERENCE
 -> approximates(SPACE_AND_TIME_DERIVATIVES)

FINITE_ELEMENT
 -> spatial_weak_discretisation

SPECTRAL_METHOD
 -> global_basis_discretisation

TIME_INTEGRATION
 -> evolves(DISCRETE_WAVE_STATE)

# ---------- CFL CONDITION ----------
explicit numerical scheme

-> time step constrained by
 spatial grid and wave speed

typical form:

c*Delta_t/Delta_x
 <=
C_scheme

CFL_CONDITION
 -> numerical_domain_of_dependence_must_cover(PHYSICAL_DOMAIN_OF_DEPENDENCE)

# ---------- APPLICATIONS ----------
WAVE_EQUATION
 -> applied_to(ACOUSTICS)
 -> applied_to(STRING_VIBRATION)
 -> applied_to(ELASTICITY)
 -> applied_to(ELECTROMAGNETISM)
 -> applied_to(SEISMOLOGY)
 -> applied_to(OPTICS)
 -> applied_to(FLUID_WAVES)
 -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
WAVE_EQUATION
 -> HYPERBOLIC_PDE

INITIAL_DATA
 -> determines(TRAVELLING_WAVE_COMPONENTS)

BOUNDARY_CONDITIONS
 -> determine(NORMAL_MODES)

CHARACTERISTICS
 -> determine(DOMAIN_OF_DEPENDENCE)

FOURIER_ANALYSIS
 -> decomposes(WAVE_INTO_MODES)

ENERGY
 -> supports(STABILITY_AND_UNIQUENESS)

GREEN_FUNCTION
 -> represents(FORCED_RESPONSE)

DISPERSION_RELATION
 -> links(FREQUENCY_AND_WAVENUMBER)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "wave equation" ~= "hyperbolic PDE describing propagation of disturbances",
 "travelling wave" ~= "wave profile translated through space with time",
 "standing wave" ~= "oscillatory mode with fixed spatial node structure",
 "characteristic" ~= "curve or surface governing information propagation in a hyperbolic equation",
 "normal mode" ~= "spatial eigenfunction oscillating at a characteristic frequency",
 "wave energy" ~= "quadratic conserved quantity combining kinetic and gradient contributions in the undamped homogeneous model"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 wave_equation_solution_depends_on_initial_boundary_and_domain_data: true,
 dAlembert_formula_applies_to_specific_1D_constant_speed_whole_line_context: true,
 finite_propagation_speed_is_hyperbolic_structure_not_universal_PDE_property: true,
 energy_conservation_requires_appropriate_undamped_boundary_context: true,
 standard_linear_wave_equation_is_nondispersive_but_general_wave_systems_can_be_dispersive: true,
 electromagnetic_wave_equations_retain_Maxwell_constraints: true,
 numerical_wave_schemes_require_stability_and_CFL_analysis: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 dAlembert_relation: 0.99,
 characteristic_relation: 0.99,
 eigenmode_relation: 0.99,
 energy_relation: 0.99,
 Fourier_dispersion_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(domain_dimension_wave_speed_and_field)
 -> identify(initial_and_boundary_conditions)
 -> classify(homogeneous_forced_damped_or_variable_coefficient_problem)
 -> determine(characteristics_and_domain_of_dependence)
 -> choose(dAlembert_separation_Fourier_Green_or_energy_method)
 -> derive(travelling_standing_or_modal_solution_structure)
 -> analyse(energy_stability_and_resonance)
 -> derive(dispersion_relation_if_wave_mode_analysis_requested)
 -> select(numerical_method_if_analytic_solution_unavailable)
 -> validate(boundary_regularities_causality_and_CFL_conditions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed wave-equation semantic graph",
 primary_entity: WAVE_EQUATION,
 preserve: [
  field,
  domain,
  dimension,
  wave_speed,
  initial_conditions,
  boundary_conditions,
  forcing,
  damping,
  medium_coefficients
 ],
 infer_through: [
  characteristics,
  dAlembert_solutions,
  travelling_waves,
  standing_waves,
  eigenmodes,
  Fourier_methods,
  energy_estimates,
  Green_functions,
  dispersion_relations
 ],
 avoid: [
  using_dAlembert_formula_outside_its assumptions,
  ignoring_second_initial_condition_for second_order_time PDE,
  assuming_all_wave_equations_are nondispersive,
  claiming_energy_conservation_with damping_or energy-leaking boundaries,
  ignoring_Maxwell_constraints_in electromagnetic context,
  applying_explicit_numerics_without CFL validation
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(wave_equation_context)
 -> compare(existing_relations)
 -> preserve(domain_speed_initial_boundary_forcing_and_causality_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + WAVE_EQUATION
 + HYPERBOLIC_PDE
 + WAVE_SPEED
 + INITIAL_CONDITION
 + BOUNDARY_CONDITION
 + CHARACTERISTIC
 + D_ALEMBERT_SOLUTION
 + STANDING_WAVE
 + TRAVELLING_WAVE
 + ENERGY
 + FOURIER_METHOD
 + GREEN_FUNCTION
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> WAVE_EQUATION
 -> PROBLEM_DATA_IDENTIFICATION
 -> PROBLEM_CLASSIFICATION
 -> CHARACTERISTIC_CAUSAL_ANALYSIS
 -> METHOD_SELECTION
 -> SOLUTION_STRUCTURE_ANALYSIS
 -> ENERGY_STABILITY_ANALYSIS
 -> MODE_DISPERSION_ANALYSIS
 -> NUMERICAL_METHOD_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW