Helmholtz equation
E544889
The Helmholtz equation is a fundamental partial differential equation that describes time-harmonic wave propagation in fields such as acoustics, electromagnetism, and optics.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Helmholtz equation canonical | 2 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf | partial differential equation ⓘ |
| appliesTo | frequency-domain analysis ⓘ |
| assumes |
linear medium
ⓘ
time-harmonic dependence e^{-iωt} ⓘ time-invariant medium ⓘ |
| definedOn |
Euclidean space
NERFINISHED
ⓘ
Riemannian manifolds NERFINISHED ⓘ |
| describes | time-harmonic wave propagation ⓘ |
| domain | linear wave phenomena ⓘ |
| hasDependentVariable | field amplitude ⓘ |
| hasDimension |
can be formulated in 1D
ⓘ
can be formulated in 2D ⓘ can be formulated in 3D ⓘ |
| hasIndependentVariable | spatial coordinates ⓘ |
| hasInhomogeneousForm | ∇²u + k²u = -f ⓘ |
| hasOperator | Laplacian NERFINISHED ⓘ |
| hasParameter | wave number k ⓘ |
| hasSolutionType |
cylindrical waves
ⓘ
plane waves ⓘ spherical waves ⓘ |
| hasType | homogeneous equation ⓘ |
| models |
resonant modes in cavities
ⓘ
standing waves ⓘ wave scattering by obstacles ⓘ |
| namedAfter | Hermann von Helmholtz NERFINISHED ⓘ |
| obtainedBy | separation of variables in time from the wave equation ⓘ |
| relatedTo |
Maxwell equations
NERFINISHED
ⓘ
Schrödinger equation NERFINISHED ⓘ wave equation NERFINISHED ⓘ |
| requires |
boundary conditions
ⓘ
radiation condition at infinity ⓘ |
| solvedBy |
Fourier transform methods
ⓘ
Green function methods ⓘ boundary integral methods ⓘ finite difference methods ⓘ finite element methods ⓘ separation of variables ⓘ spectral methods ⓘ |
| specialCaseOf | Poisson equation NERFINISHED ⓘ |
| symbolicForm | ∇²u + k²u = 0 ⓘ |
| usedIn |
acoustic scattering
ⓘ
acoustics ⓘ antenna theory ⓘ electromagnetic scattering ⓘ electromagnetism ⓘ optics ⓘ quantum mechanics ⓘ seismology ⓘ vibration analysis ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Helmholtz equation Description of subject: The Helmholtz equation is a fundamental partial differential equation that describes time-harmonic wave propagation in fields such as acoustics, electromagnetism, and optics.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.