d’Alembert’s formula
E158707
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
All labels observed (1)
| Label | Occurrences |
|---|---|
| d’Alembert’s formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1380682 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: d’Alembert’s formula Context triple: [Jean d’Alembert, knownFor, d’Alembert’s formula]
-
A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
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B.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
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C.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: d’Alembert’s formula Target entity description: d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
-
A.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
B.
Laplace equation
The Laplace equation is a fundamental second-order partial differential equation widely used in physics and engineering to describe steady-state phenomena such as electrostatics, gravitation, and heat conduction.
-
C.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
D.
Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
-
E.
Euler’s formula for complex exponentials
Euler’s formula for complex exponentials is the fundamental identity \(e^{i\theta} = \cos\theta + i\sin\theta\), which links complex exponentials with trigonometric functions and underpins much of complex analysis and engineering mathematics.
- F. None of above. chosen
Statements (38)
| Predicate | Object |
|---|---|
| instanceOf |
analytical solution method
ⓘ
mathematical formula ⓘ partial differential equation solution ⓘ |
| applicableWhen | no boundary conditions or infinite domain ⓘ |
| appliesTo |
one-dimensional wave equation
ⓘ
vibrating string ⓘ |
| assumes |
constant wave speed
ⓘ
infinite string ⓘ linear wave equation ⓘ |
| captures | finite speed of disturbance propagation ⓘ |
| characteristicMethod | uses propagation along characteristics x±ct = const ⓘ |
| contrastWith |
energy method for wave equations
ⓘ
numerical methods for the wave equation ⓘ |
| domain | real-valued functions of two variables ⓘ |
| expresses | displacement of a vibrating string ⓘ |
| field |
applied mathematics
ⓘ
mathematical physics ⓘ partial differential equations ⓘ |
| generalizes | to piecewise smooth initial data ⓘ |
| historicalPeriod | 18th century ⓘ |
| mathematicalType | closed-form solution ⓘ |
| namedAfter |
Jean le Rond d’Alembert
ⓘ
linked to:
Jean d’Alembert
|
| property |
preserves finite propagation speed
ⓘ
represents superposition of left- and right-traveling waves ⓘ |
| relatedTo |
Fourier series methods for the wave equation
ⓘ
Green’s function for the one-dimensional wave equation ⓘ method of characteristics ⓘ |
| relates | solution to initial data ⓘ |
| requires | twice differentiable solution in x and t ⓘ |
| solutionForm |
u(x,t) = F(x-ct) + G(x+ct)
ⓘ
u(x,t) = \tfrac12[f(x-ct)+f(x+ct)] + \tfrac1{2c} \int_{x-ct}^{x+ct} g(s)\,ds ⓘ |
| solves | u_{tt} = c^2 u_{xx} ⓘ |
| usedFor | Cauchy problem for the 1D wave equation ⓘ |
| usedIn |
acoustics
ⓘ
theory of vibrating strings ⓘ wave propagation theory ⓘ |
| uses |
initial displacement
ⓘ
initial velocity ⓘ |
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: d’Alembert’s formula Description of subject: d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.