Physics

The registered systems of skcomplex.systems.physics. The page of each shows its factory, its component types with their states, parameters, and laws as written, and, for the forms that have them, its equations as SymbolicAnalysis returns them, so it serves as the pattern for a system of your own.

Active matter

• Active Brownian particles: Active Brownian particles in the plane, which swim at a constant speed in a diffusing direction.
• Active Brownian particles with local alignment: Active Brownian particles in a periodic box whose headings align with the neighbours within d0.
• Active Ornstein-Uhlenbeck particles: Active Ornstein-Uhlenbeck particles in the plane, driven by exponentially correlated propulsion.
• Active System B+ of motility-induced phase separation: Active System B+, the conserved field theory of the density of active particles.
• Active nematic and polar gels: An active gel, a nematic tensor or a polarisation transported in the Stokes flow of its own stresses.
• Active polymers: An active polymer, a Rouse chain of active Brownian beads that each swim along their own orientation.
• Bertin-Droz-Grégoire kinetic flocking: The kinetic theory of flocking, whose polar order grows above a transition density.
• Chiral active particles: Chiral active particles, which swim along circles of radius v / omega that drift by diffusion.
• Cucker-Smale flocking: Cucker-Smale flocking, where every agent aligns its velocity with all others, weighted by distance.
• Deformable active particles of Ohta and Ohkuma: Deformable self-propelled particles of Ohta and Ohkuma, which move straight or along circles.
• Depot-pumped Brownian particles: Brownian particles with energy depots, which start at rest and take up the speed sqrt(q/gamma - c/d).
• Dry active nematics: A dry active nematic, a conserved density coupled to a symmetric traceless alignment tensor.
• Flocking with active and passive agents: The active and passive agents of Charbonneau (2017, section 10.4) in a periodic box.
• Ginelli nematic particles: Polar particles with nematic alignment, which order along a common axis but move both ways along it.
• Grégoire-Chaté cohesive flocks: The Grégoire-Chaté system: aligning particles that also attract and repel, forming cohesive flocks.
• Inertial active Brownian particles: Active Brownian particles with inertia, which lag behind their orientation as they speed up and turn.
• Keller-Segel chemotaxis: The Keller-Segel system of cells that diffuse and climb the gradient of a chemical they secrete.
• Living crystals: Active colloids of radius 1 with phoretic attraction, which gather into clusters that slowly coarsen.
• Microscopic flocking with aligning binary collisions: Self-propelled particles that flock through aligning binary collisions, the Bertin-Droz-Grégoire system.
• Motility-induced phase separation: Repulsive active Brownian disks of unit diameter, which phase separate into clusters at high speed.
• Nonreciprocal Vicsek system: Two species of Vicsek movers in a periodic box whose alignment with each other is nonreciprocal.
• One-dimensional active viscoelastic and poroelastic gel with contractile cells: A one-dimensional active viscoelastic gel, where contractile cells concentrate by dragging the gel.
• Run-and-tumble: Run-and-tumble agents in a periodic square, which diffuse with v^2 (1 + q) / (4 (1 - q)) per step.
• Squirmers: Spherical squirmers that swim at 2 B1 / 3 and stir each other as pushers or pullers.
• Szabó cell system: The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.
• Toner-Tu hydrodynamics: The Toner-Tu hydrodynamics of a polar flock on a periodic lattice, from a disordered velocity field.
• Vicsek system with metric, topological, vision-cone, and delayed alignment: The Vicsek system: particles that move at a constant speed along the mean heading of their neighbours.

Celestial mechanics

• Bending of light by a mass: A light ray that passes a static mass and is deflected towards it.
• Frame dragging around a spinning mass: A particle without angular momentum that falls into a spinning mass and is dragged around it.
• Keplerian dumbbell and spin-orbit coupling: The Keplerian dumbbell: two masses on a rigid massless arm that orbit a primary under gravity.
• N-body and solar system: Bodies in the plane that attract each other pairwise by Newtonian gravity.
• Perihelion precession in the Schwarzschild spacetime: A massive particle on a bound orbit around a static mass, whose perihelion advances every turn.
• Relativistic orbit equation: Einstein's orbit equation u'' + u = alpha + epsilon u² of a planet, whose perihelion precesses.
• Restricted three-body problem: A light body moving in the plane of two primaries on circular orbits, in the frame that rotates with them.
• Sitnikov problem: A light body moving on the axis through the centre of mass of two equal primaries, normal to their orbit.
• Störmer charge in a dipole field: A charged particle moving in the magnetic dipole field of a planet, the Störmer problem.
• Two-body orbit: A light planet orbiting a heavy star under Newtonian gravity, as the Kepler problem of their separation.

Climate

• Budyko-Sellers latitude system: The Budyko-Sellers energy balance of one hemisphere, resolved in the sine of latitude.
• Coupled soil and deep temperature maps: The coupled maps of the soil surface temperature and the deeper soil temperature.
• Daisyworld: Daisyworld, where white and black daisies regulate the temperature of their planet through its albedo.
• Declining energy return on investment: The extraction of a finite energy reserve whose energy return on investment declines as it depletes.
• ENSO delay oscillator: The delayed oscillator of El Niño of Suarez and Schopf, dT/dt = T - T^3 - alpha T(t - delta).
• Forest dieback: The forest dieback model of the Amazon, whose cover tips to bare soil when the ambient temperature rises.
• Forest dieback with replicator dynamics of mitigators: The forest dieback model coupled with the replicator dynamics of mitigators who perceive its warming.
• Gissinger geodynamo: Gissinger's geodynamo, whose dipole field reverses at irregular times.
• Hasselmann stochastic climate: Hasselmann's stochastic climate, where fast weather noise drives a slow, damped temperature anomaly.
• Leading-following tipping cascade: A cascade of two tipping elements, a leading fold driven by a ramp and a follower forced by the leader.
• Lorenz-84: Lorenz-84, the general circulation reduced to a westerly jet and a travelling wave, chaotic for F = 8.
• Lorenz-96: Lorenz-96, dx_i/dt = (x_{i+1} - x_{i-2}) x_{i-1} - x_i + F on a ring of sites around a latitude circle.
• Network of interacting tipping elements: A network of tipping elements, where one element pushed past its tipping point can tip its neighbours.
• Population and pollution: The pollution model of Bala, Arshad, and Noh, a population that degrades its environment.
• Rothman excitable carbon cycle: Rothman's excitable carbon cycle, where a large enough perturbation of the ocean sets off an excursion.
• Saltzman-Maasch glaciation: The glacial cycles of Saltzman and Maasch, a self-sustained oscillation of ice, carbon, and ocean.
• Stochastic energy balance with bifurcation-, noise-, and rate-induced tipping: The Sutera-Fraedrich stochastic energy balance set up for bifurcation-, noise-, or rate-induced tipping.
• Stochastic fold with a ramped control parameter: A stochastic fold whose control is ramped through the tipping point of its lower branch.
• Van der Pol ice ages: Glacial cycles as a Van der Pol relaxation oscillator, optionally paced by orbital forcing.
• Zero-dimensional energy balance with ice albedo: The zero-dimensional energy balance of the Earth, bistable through the ice-albedo feedback.

Cosmology

• FLRW cosmology from the Einstein equations: A homogeneous universe whose expansion follows from the Einstein equations of the Robertson-Walker metric.
• Friedmann expansion: The scale factor of a homogeneous universe of matter and dark energy, after Friedmann and Lemaître.
• Oppenheimer-Snyder collapse of a dust ball: A ball of dust at rest that collapses under its own gravity into a black hole, as Oppenheimer and Snyder.

Criticality

• Bak-Tang-Wiesenfeld sandpile: The Bak-Tang-Wiesenfeld sandpile on a lattice with open edges, started from random stable heights.
• Drossel-Schwabl forest fire: The Drossel-Schwabl forest fire, where slow growth and rare lightning give fires of every size.
• Evolutionary Sherrington-Kirkpatrick game: The evolutionary Sherrington-Kirkpatrick game, which self-organises to the edge of ergodicity breaking.
• Fixed-energy sandpile: The fixed-energy sandpile, a Bak-Tang-Wiesenfeld pile with no drive and no loss on a periodic lattice.
• Forest fire on hexagonal lattices and graphs: The Drossel-Schwabl forest fire on a triangular lattice of hexagonal cells, or on the nodes of a graph.
• Langevin field theory of absorbing-state transitions with a conserved energy field: The Langevin field theory of an absorbing-state transition, an active density fed by an energy field.
• Manna sandpile: The Manna sandpile, a stochastic sandpile of another universality class than Bak-Tang-Wiesenfeld.
• SK game of reinforcement learners: The SK game, reinforcement learners whose rewards come from quenched Sherrington-Kirkpatrick couplings.
• Self-organized branching process: The self-organized branching process of Zapperi, Lauritsen, and Stanley, a mean-field sandpile.
• Zhang sandpile: The Zhang sandpile, a continuous sandpile whose unstable sites give away all their energy.

Fluids

• BGK: The BGK equation of the Sod shock tube over one position and one velocity axis.
• Boltzmann: The Boltzmann equation of Maxwell molecules over one position and two velocity axes.
• Burgers: The inviscid Burgers equation on a periodic interval of length 2 pi, whose sine steepens into a shock.
• Bénard-Marangoni convection with a free, thermocapillary surface: The order-parameter equation of Bénard-Marangoni convection, which forms hexagonal cells.
• Decaying turbulence: Decaying two-dimensional turbulence from a random vorticity field on a periodic square of side 2 pi.
• Discrete-velocity Boltzmann gas of Carleman: The Carleman discrete-velocity Boltzmann equation on a periodic line, from a pulse of right movers.
• Enskog equations: The local Enskog equation, the Boltzmann equation with the contact value of the pair correlation of disks.
• FHP lattice gas: The FHP-I lattice gas on a periodic triangular lattice, with a dense disc that spreads into the gas.
• HPP lattice gas: The HPP lattice gas on a periodic square lattice, whose particles move along four directions and collide.
• Kuramoto-Sivashinsky: The Kuramoto-Sivashinsky equation u_t = -u u_x - u_xx - u_xxxx, spatiotemporally chaotic.
• Rayleigh-Bénard: Rayleigh-Bénard convection between free-slip walls, from a small perturbation of the critical mode.
• Shallow-water dam break: The shallow-water dam break: still water at two depths that a removed dam separated at the middle.
• Vlasov: The Vlasov-Poisson equation of electrons over a neutralizing background, on a periodic position axis.
• Wave-number conservation and wave breaking: The kinematic wave equation of deep-water wave trains on a periodic interval of length 2 pi.

Heat

• Cooling coffee: A cup of coffee whose temperature relaxes to that of the room, T' = -k (T - T_room).
• Heat diffusion: Heat diffusing on a plate from a hot square in its middle, du/dt = D laplacian(u).
• Heat on an L-shaped domain: Heat on the L-shaped domain with fixed boundary temperatures, discretised by linear finite elements.

Mechanics

• Ballistic pendulum: A bullet that sticks in the block of a pendulum at rest, which then swings up.
• Beam-beam map: The beam-beam map of a particle of an accelerator beam that meets the opposite beam once every turn.
• Bouncing ball: A ball that falls under gravity and bounces off the floor, losing speed by a coefficient of restitution.
• Circular billiard: A particle moving freely inside a circle and reflecting specularly off its wall.
• Coupled pendulums: Two pendulums whose bobs are joined by a spring, which pass their energy back and forth.
• Double pendulum: A double pendulum, a pendulum hanging from the bob of another, written as its two Lagrange equations.
• Driven damped oscillator: A mass on a spring with damping and a periodic force, m x'' = -k x - c x' + F cos(ω t).
• Gyroscope in a gimbal: A gyroscope in a gimbal: a rotor in an inner ring that turns in an outer ring about a fixed axis.
• Harmonic oscillator: A particle on a spring, optionally damped.
• Huygens clocks and pendulums on coupled supports: Pendulum clocks on supports joined by springs, which synchronize through the motion of the supports.
• Magnetic pendulum: A magnetic bob on a string swinging above magnets on a circle, which comes to rest at one of them.
• Mass-spring oscillator with static and Coulomb friction: A mass on a linear spring with static, Coulomb, and viscous friction, m x'' + c x' + k x + f = 0.
• Newton's cradle: Newton's cradle: pendulum balls in a row that pass momentum along by elastic collisions.
• Particle in the well k x^2 e^-x: A particle moving without friction in the potential U(x) = k x² e^-x.
• Pendulum: A pendulum, whose angle follows Newton's second law m l θ'' = -m g sin θ.
• Pendulum on a sliding cart: A pendulum hanging from a cart that slides without friction along a horizontal track.
• Pendulum with a vertically or horizontally oscillating pivot: A pendulum whose suspension point oscillates vertically or horizontally, with acceleration ε g cos Ω t.
• Projectile: A particle launched under uniform gravity, optionally slowed by linear drag.
• Rotating balls on rods: Two balls on rods that turn about a vertical axis and conserve their angular momentum.
• Rotating swing: A rotating swing: chairs on chains hanging from the rods of a stand that turns about a vertical axis.
• Two-mass spring-damper between walls: Two masses between two walls, tied to the walls and to each other by springs, with dampers and forces.

Nonlinear oscillators

• Adler equation: The Adler equation Δφ' = Δω + ε sin Δφ for the phase difference of an oscillator and its drive.
• Duffing: The forced Duffing oscillator x'' = -delta x' - alpha x - beta x^3 + gamma cos(omega t), chaotic here.
• Dynamical billiards of Sinai, mushroom, and open three-disk shapes: A particle moving freely on a billiard table and reflecting specularly off its walls.
• Forced Toda oscillator: The forced Toda oscillator x'' + d x' + e^x - 1 = a sin(omega t), a passive asymmetric oscillator.
• Forced Van der Pol: The Van der Pol oscillator driven by a periodic force, x'' = μ (1 - x²) x' - x + A cos(ω t).
• Hénon-Heiles: The Hénon-Heiles system of a star in a galactic potential, whose motion is chaotic at higher energies.
• Nuclear quadrupole vibrations: The quadrupole vibrations of a nuclear surface, a Hamiltonian system of two degrees of freedom.
• Strongly nonlinear oscillator under periodic impulses: An oscillator kicked by a periodic train of impulses of alternating sign.
• Stuart-Landau: The Stuart-Landau oscillator z' = (μ + i ω) z - (1 + i β) |z|² z, the normal form of a Hopf bifurcation.
• Van der Pol: The Van der Pol oscillator x'' = μ (1 - x²) x' - x, which settles on a limit cycle for μ > 0.

Oceanography

• Quasi-geostrophic double gyre: The quasi-geostrophic double gyre, the wind-driven circulation of a closed ocean basin.
• Stommel thermohaline boxes: Stommel's two-box thermohaline circulation, bistable between a thermal and a haline mode.

Pattern formation

• Allen-Cahn non-conserved relaxation (System A): The Allen-Cahn equation, System A, the non-conserved relaxation of an order parameter in a double well.
• Cahn-Hilliard phase separation (System B): The Cahn-Hilliard equation, System B, the conserved relaxation of an order parameter in a double well.
• Complex Ginzburg-Landau: The complex Ginzburg-Landau equation A_t = A + (1 + i b) laplacian(A) - (1 + i c) |A|^2 A.
• Nonreciprocal Cahn-Hilliard: The nonreciprocal Cahn-Hilliard equations, a demixing field coupled to a diffusing one.
• Swift-Hohenberg: The Swift-Hohenberg equation u_t = r u - (1 + laplacian)^2 u - u^3, which forms rolls of length 2 pi.

Seismology

• Burridge-Knopoff spring-block fault: The Burridge-Knopoff fault, a chain of blocks pulled by a moving plate that sticks and slips.
• Olami-Feder-Christensen: The Olami-Feder-Christensen earthquake system with open edges, started from random forces.

Soft matter

• Annealed transient time motion: Annealed transient time motion, Brownian walkers whose diffusion coefficient changes at random times.
• Brownian motion of Janus, ellipsoidal, and chiral particles in three dimensions: Overdamped axisymmetric Brownian particles in three dimensions that translate and rotate.
• Brownian particle near a wall: A Brownian particle whose diffusion slows near a wall, D(z) = D0 z² / (z² + a²), as it settles on it.
• Continuous-time random walk: The continuous-time random walk, Gaussian jumps after waiting times with the tail tau^(-1 - alpha).
• Diffusion in a speckle field: Brownian particles in a random speckle field, trapped at hotspots for longer as the field grows stronger.
• Fractional Brownian motion: Fractional Brownian motion, dx = sqrt(2 D) dB_H, with <x^2> = 2 D t^(2 H) along every axis.
• Fully biased walk among quenched broad traps: Walkers that hop always to the right through quenched traps with broadly distributed trapping times.
• Hydrodynamically coupled particles in blinking traps: Two Brownian spheres in blinking optical traps, whose free motion is correlated through the fluid.
• Inertial and overdamped Brownian motion: Free Brownian particles with inertia, m x'' = -gamma x' + sqrt(2 kT gamma) W, or without it.
• Kramers escape: Particles that escape over the barrier of U = x^4 / 4 - x^2 / 2 by thermal noise of intensity D.
• Langevin equation with colored noise: Particles whose velocity dv/dt = -gamma v + xi is driven by Ornstein-Uhlenbeck noise xi.
• Lévy walk: The Lévy walk, flights at constant speed in random directions for times with the tail tau^(-1 - sigma).
• Multiplicative noise conventions: Free particles in a box with position-dependent noise, read in the Itô, Stratonovich, or anti-Itô sense.
• On-off flashing ratchet: The on-off flashing ratchet, Brownian particles in a sawtooth potential switched on and off at random.
• Optical trap including oscillating and blinking traps: Overdamped particles in a harmonic optical trap, still, oscillating, or blinking.
• Overdamped particle in a tilted periodic potential: Overdamped particles in the tilted periodic potential V(x) = U0 cos(2 pi x / L) - f x.
• Rigid Brownian and active dimers: Colloidal molecules: pairs of Brownian or active spheres joined by rigid rods, free in the plane.
• Scaled Brownian motion: Scaled Brownian motion, dx = sqrt(2 K alpha (t + t0)^(alpha - 1)) dW, with <x^2> ~ 2 K t^alpha.
• Shaken frictional spring-block chain: A chain of blocks joined by springs on a frictional substrate, shaken and relaxed in turn.
• Stochastic resonance: A particle in a double well, dx = (x - x^3 + A cos(2 pi f t)) dt + sqrt(2 D) dW.
• Superdiffusive Levy flight: The Lévy flight, a walk whose jumps have lengths with the tail l^(-1 - mu) in random directions.

Statistical mechanics

• Asymmetric exclusion process with open boundaries: The asymmetric simple exclusion process on an open line, in continuous time.
• Bouchaud trap system: The Bouchaud trap system, walkers that escape from traps of exponentially distributed depths.
• Contact process: The contact process on a periodic lattice, which survives above a critical infection rate near 1.65.
• Continuum percolation: Overlapping disks at uniform positions in a square, whose clusters percolate near a filling of 1.13.
• Diffusion-limited aggregation: Diffusion-limited aggregation on a periodic grid, where random walkers stick to a seed at the center.
• Discrete master equation: The discrete-time master equation of states on a ring, which relaxes to the Boltzmann distribution.
• Dissipative zero-range process on a tree: The dissipative zero-range process on a tree, with energy injected at the root and lost at the leaves.
• Edwards-Anderson spin glass: The Edwards-Anderson spin glass on a periodic square lattice, with quenched random bonds.
• Ehrenfest urn system of two boxes: The Ehrenfest urn system, balls moved one at a time between two boxes, which relaxes to equal counts.
• Fully connected Ising system with Glauber dynamics: The fully connected Ising system with Glauber dynamics, ordered below the critical temperature T_c = J.
• Fully connected branching and annihilation: The fully connected branching and annihilation process, active above kappa = nu until it dies out.
• Ideal gas: An ideal gas of non-interacting molecules in a reflecting box.
• Ising: The two-dimensional Ising system on a torus, ordered below the critical temperature 2 / ln(1 + sqrt 2).
• Ising on a graph: The Ising system on a graph, ordered below a critical temperature set by the degrees of the nodes.
• Landau mean-field relaxation: The relaxation of a Landau order parameter, to sqrt(a (Tc - T) / b) below Tc and to 0 above it.
• Lennard-Jones fluid: Lennard-Jones particles in a periodic square box, optionally held at a temperature by a thermostat.
• Mattis spin glass: The Mattis spin glass on a periodic square lattice, with couplings J_ij = J xi_i xi_j of a pattern xi.
• Potts: The q-state Potts system on a torus, ordered below the critical temperature 1 / ln(1 + sqrt q).
• Q2R reversible Ising automaton: The Q2R automaton, a reversible, energy-conserving Ising dynamics on a periodic lattice of even sides.
• Quantum Heisenberg system: A chain of spins one half with the XXZ Hamiltonian, a field, and optional dephasing.
• Random energy system: The random energy system, 2^N quenched Gaussian energies of variance N / 2 at a temperature.
• Sherrington-Kirkpatrick spin glass: The Sherrington-Kirkpatrick spin glass with Glauber dynamics, frozen below the temperature T_c = J.
• Site percolation: Site percolation probed by a fire lit along the top row, which spans the lattice above p_c = 0.5927.
• TASEP: The totally asymmetric simple exclusion process with parallel update, on an open line or a ring.
• Two-phase zone exchange: Independent zones of a two-phase apparatus, each exchanging particles with the rest of the apparatus.
• XY: The XY system of planar spins on a torus, with vortices that bind below the Kosterlitz-Thouless point.
• Zero-range process: The zero-range process on a ring, which condenses above the critical density 1 / (b - 2) for b > 2.

Synchronization

• Adaptive Kuramoto-Sakaguchi oscillators with frequency clusters: Globally coupled phase oscillators whose coupling weights adapt to their phase differences.
• Aggregate-averaging synchronization of logistic and delay-coupled Bernoulli maps: Two chaotic maps coupled by aggregate averaging, which synchronize their chaos for strong coupling.
• Coupled Rössler and the master stability function: Chaotic Rössler oscillators on a ring, diffusively coupled through their second variable.
• Coupled map lattices on networks: Chaotic logistic maps on the nodes of a network, each pulled towards the mean of its neighbours.
• Delayed Kuramoto: Kuramoto phase oscillators coupled all to all through signals that arrive after a delay.
• Explosive topological Kuramoto of node and link phases on a simplicial complex, with the Nodes-Links-Triangles variant: Phases on the nodes and links of a clique complex, coupled through its incidence by order parameters.
• Globally coupled Stuart-Landau oscillators with frequency spread: Globally coupled Stuart-Landau oscillators, whose oscillations all die for strong coupling and spread.
• Globally coupled systems with heterogeneous time scales: Globally coupled oscillators with heterogeneous time scales, whose fixed point the coupling restabilizes.
• Kuramoto: Kuramoto oscillators coupled all to all, theta_i' = omega_i + (K / N) sum_j sin(theta_j - theta_i).
• Kuramoto on networks: Kuramoto oscillators coupled along the edges of a network, theta_i' = omega_i + K sum_j A_ij sin(...).
• Kuramoto oscillators with three-body coupling: Kuramoto oscillators coupled in pairs and by three-body interactions on the triangles of a hypergraph.
• Kuramoto-Sakaguchi chimera: Two populations of identical phase oscillators with phase lag, where one synchronizes and one drifts.
• Mirollo-Strogatz pulse-coupled oscillators: Integrate-and-fire oscillators coupled by pulses, which end up firing in unison.
• Nonlocal ring chimera: Identical phase oscillators on a ring, each coupled to its R nearest neighbours on both sides.
• Ott-Antonsen reduction: The Ott-Antonsen equation for the order parameter of infinitely many Kuramoto oscillators.
• Rössler-driven Lorenz and generalized synchronization: A chaotic Rössler system that drives a Lorenz-63 system through its second variable.
• Swarmalators: Swarmalators, identical oscillators whose phases set their attraction and whose positions their coupling.
• Winfree system: The Winfree system, theta_i' = omega_i + (K / N) sum_j P(theta_j) R(theta_i), with the sum over all j.

Waves

• Lattice Maxwell and gauge link fields: Link variables on the cell complex of a periodic lattice, sampled or evolved in real time.
• Lattice Yang-Mills and pure lattice QCD: SU(2) or SU(3) matrices on the links of a periodic lattice, sampled by the Wilson action.
• Schrödinger equation as real and imaginary fields: A Gaussian wave packet that follows the Schrödinger equation, as its real and imaginary parts.
• Schrödinger equation minimally coupled to an electromagnetic potential: A Gaussian wave packet of a charge in a prescribed electromagnetic potential, after Schwichtenberg 4.3.
• Spring chains and guitar string: Masses joined in a row by springs of zero rest length between two anchors, a discrete string.
• Standing waves on a string: A string with fixed ends on [-length / 2, length / 2] that vibrates as a standing wave.