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Showing 181–210 of 1709 reference entries

InformationFinite graphs and trees

Graphs represent pairwise connections; trees impose acyclicity.

DefinitionInformationData-structure interfaces and operation costs

An interface defines operations; an implementation determines costs.

MethodInformationLoop invariants and total correctness

An invariant establishes correctness only with its proof obligations.

MethodInformationComparison sorting lower bound

Distinct-key comparisons need enough branches to distinguish orders.

Mathematical statementInformationBreadth-first and depth-first traversal

Graph searches visit reachable vertices using different orders.

MethodInformationShortest-path algorithm conditions

Edge weights determine which shortest-path method applies.

MethodInformationTuring machines and effective computation

Finite instructions can act on an unbounded writable tape.

DefinitionInformationRelations and tabular data

A relation is a set of tuples from declared domains.

DefinitionPhysicsCentre-of-mass dynamics

External force controls the mass-weighted centre's acceleration.

ModelPhysicsRigid-body inertia tensor

A tensor relates rotation to angular momentum.

ModelPhysicsEuler equations for a rigid body

Body-axis angular acceleration includes rotation of the axes.

ModelPhysics · Mathematics & logicEuler–Lagrange equations

Stationary action supplies coordinate equations of motion.

Mathematical statementPhysics · Mathematics & logicHamilton's canonical equations

Canonical coordinates and momenta evolve from the Hamiltonian.

Mathematical statementPhysicsFluid mass continuity

Density change balances mass flux through a region.

ModelPhysicsBernoulli's streamline relation

An ideal steady flow balances pressure, motion and height.

ModelPhysicsIncompressible Navier–Stokes equation

Fluid inertia balances pressure, viscosity and gravity.

ModelPhysicsReynolds number

Compare inertial and viscous scales of a flow.

ModelPhysicsDriven damped oscillator

A linear restoring force, damping and periodic drive determine response.

ModelPhysicsIdeal-string wave equation

A local differential equation determines small string disturbances.

ModelPhysics · Mathematics & logicWave superposition and interference

Linear disturbances add; their relative phase determines reinforcement.

ModelPhysicsFixed-end string modes

Boundary conditions select standing-wave frequencies.

ModelPhysicsSnell's refraction relation

Angles at an interface depend on refractive indices.

ModelPhysicsThin-lens image equation

A paraxial lens relates object and image distances.

ModelPhysicsSingle-slit diffraction minima

An aperture's finite width selects dark far-field directions.

ModelPhysicsLinear polarization and Malus's law

An ideal analyzer transmits the field component along its axis.

ModelPhysicsElectromagnetic potentials

Scalar and vector potentials encode electric and magnetic fields.

ModelPhysicsCapacitance

A linear two-conductor arrangement stores separated charge.

DefinitionPhysicsSelf-inductance

Changing current changes linked magnetic flux and induced emf.

ModelPhysicsElectromagnetic energy balance

The rate of field-energy change balances outward energy flow and power transferred to charges.

ModelPhysicsKirchhoff circuit balances

Lumped circuit models balance node currents and loop voltages.

Model
ModelSource checked

Fixed-end string modes

Physics

Boundary conditions select standing-wave frequencies.

Key knowledge

An ideal uniform string fixed at both ends has wavelengths λ_n = 2L/n and frequencies f_n = nv/(2L), for positive integer n.

Values and equations

λ_n = 2L/n; f_n = nv/(2L), n = 1,2,…

λ_n, L
Mode wavelength and string length m
f_n
Mode frequency Hz
v
Wave speed m s⁻¹
n
Positive integer mode index

Conditions and scope

  • Both endpoints are displacement nodes; wave speed is constant.
  • Other boundary conditions or dispersion produce other spectra.
Sources · 1

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. University Physics 1: Standing Waves and ResonanceSamuel J. Ling, Jeff Sanny and William Moebs / OpenStax, Rice UniversityTextbook chapteropenstax.orgRead source

Evidence and review

Equations 16.15–16.16 and fixed-end normal modes.

2026-10-10 · Ideal two-fixed-end string spectrum only; arbitrary cavity spectra excluded.

Related reference entries

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Showing 1–2 of 2 relationships

Scientific relationships · 2 / 2

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Derived from
    Relationship sources · 1

    Linked statements

    • An ideal uniform string fixed at both ends has wavelengths λ_n = 2L/n and frequencies f_n = nv/(2L), for positive integer n.

    Ideal uniform string with compatible fixed-end homogeneous boundary conditions.

  2. Uses definition
    Relationship sources · 1

    Linked statements

    • An ideal uniform string fixed at both ends has wavelengths λ_n = 2L/n and frequencies f_n = nv/(2L), for positive integer n.

    Each sinusoidal mode uses f = v/λ with positive wave speed and wavelength.

Entry detailsPermanent link to this entry ↗
Record identity
physics-standing-wave-modes
Revision
university-reference.1
Structured reference record
University reference · University core expansion