Standing_wave - Wikiparency

# Standing wave A **standing wave**, also called a stationary wave, is a [[Wave|wave]] whose pattern does not travel: each point of the medium [[Oscillation|oscillates]] in place with an amplitude that depends on its position. Points that never move are nodes; points that swing the most are antinodes, and the two alternate every quarter wavelength. A standing wave forms when two waves of the same frequency and amplitude travel in opposite directions through each other, most often a wave and its own reflection, or when a traveling wave moves against a current at exactly its own speed. Standing waves are the reason bounded systems ring at particular frequencies. A guitar string, the air column of a flute, a room, a lake and the whole Earth each admit only the standing waves that fit their boundaries, the [[Normal_mode|normal modes]], and those modes set the pitches of [[Musical_acoustics|musical instruments]], the [[Acoustic_resonance|resonances]] of rooms and tubes, and the sloshing of large lakes. The same mathematics describes light between mirrors, microwaves in an oven and X-rays inside a crystal. The framework microsim *Standing waves: two waves in a tube that stand still* shows a right-moving and a left-moving wave adding to a pattern that stays put inside a tube; the reader picks the mode, the tube's ends, its length and the gas, weakens the reflection, and watches cork dust heap at the nodes as in [[Kundt's_tube|Kundt's tube]]. ## Moving medium A wave does not need a reflection to stand still. If the medium flows toward the source at exactly the wave's own [[Velocity|speed]], the crests travel through the water or air but stay fixed relative to the ground. Long water waves in shallow water travel at `c = √(gh)`, where g is the acceleration of gravity and h the depth; in water 1 m deep that is about 3.1 m/s. A river running at 3.1 m/s over a 1 m deep bed can therefore hold a wave in place, and the stationary waves below rapids and weirs that kayakers and river surfers ride are waves of this kind, standing against the current of the [[Fluid_dynamics|flow]]. The [[Atmosphere_of_Earth|atmosphere]] shows the same effect on a larger scale: air flowing over a mountain range oscillates as it passes downstream, and the resulting lee waves stay locked to the terrain while the wind blows through them. Because such a wave is held by a flow rather than by two reflecting ends, it shifts when the current changes. ## Opposing waves The more common way to make a standing wave is to send two identical waves through each other in opposite directions. If one [[Sine_wave|sine wave]] is `y₁ = A sin(kx − ωt)` and the other `y₂ = A sin(kx + ωt)`, the [[Superposition_principle|superposition principle]] adds them point by point, and a trigonometric identity turns the sum into `y(x, t) = 2A sin(kx) cos(ωt)`, a shape that depends only on position multiplied by an oscillation that depends only on time.[^up16-6] The factor `sin(kx)` is zero at `x = 0, λ/2, λ, 3λ/2, …`, so those points, the nodes, never move no matter what the time is. Halfway between them, at odd multiples of λ/4, the antinodes oscillate between +2A and −2A, twice the amplitude of either traveling wave. The textbook describes the result as waves that "bounce back and forth through a particular region, effectively becoming stationary."[^up16-6] Every point between two neighboring nodes moves in step, rising and falling together; the next loop moves in the opposite direction. Twice per cycle the whole medium passes through its rest position at once, with all the energy in motion, and twice it stops at maximum displacement, with all the energy stored as strain. The [[Energy|energy]] sloshes between kinetic and potential forms within each loop but, for a perfect standing wave, never crosses a node. A musical instrument uses exactly this. In the words of one music theory text, instruments "produce pitches by trapping sound waves" between two ends, so that reflected waves reinforce new ones instead of canceling them.[^bmt3-2] If the reflection is imperfect, the two opposing waves are unequal, the nodes no longer reach zero and part of the wave travels on, a case of [[Wave_interference|wave interference]] treated in the ratio section below. ## Mathematical description ### Standing wave on an infinite length string On an unbounded string nothing selects a particular frequency: the two opposing waves can have any k and ω related by the wave speed `v = ω/k`, and the result is a standing wave with nodes spaced λ/2 apart. Boundaries turn this continuum into a discrete list of allowed [[Normal_mode|modes]]. ### Standing wave on a string with two fixed ends A string of length L fixed at both ends must have nodes at `x = 0` and `x = L`. Only whole numbers of half-wavelengths fit, so `λ_n = 2L/n` and the frequencies are `f_n = n v / (2L)`, with `v = √(F_T/μ)` for tension F_T and linear density μ.[^up16-6] The lowest, n = 1, is the [[Fundamental_frequency|fundamental]] or first harmonic; the rest are integer multiples of it, the [[Harmonic|harmonic series]] that gives a plucked or bowed string its clear pitch. The same formula holds for any symmetric pair of boundaries, two nodes or two antinodes.[^up16-6] The details for real instruments are the subject of [[String_vibration|string vibration]]. ### Standing wave on a string with one fixed end If one end is fixed (a node) and the other free (an antinode, as for a string tied to a ring sliding on a frictionless rod), the length must hold an odd number of quarter-wavelengths. Then `λ_n = 4L/(2n − 1)` and `f_n = (2n − 1) v / (4L)`: only the odd harmonics exist, and the fundamental is half that of the same string fixed at both ends. ### Standing wave in a pipe Air in a pipe behaves the same way, with one change of vocabulary. A closed end stops the air, so it is a displacement node (and a pressure antinode); an open end lets the air swing freely, so it is a displacement antinode (and very nearly a pressure node). A pipe open at both ends has symmetric boundaries and all harmonics, `f_n = n c / (2L)`; a pipe closed at one end has antisymmetric boundaries and only the odd ones, `f_n = (2n − 1) c / (4L)`.[^up17-4] Because the frequencies are proportional to the [[Speed_of_sound|speed of sound]], they rise with temperature, the textbook's explanation for why organs in unheated cathedrals go out of tune and why players warm their wind instruments before a concert.[^up17-4] A worked number from the microsim's default: a 1.80 m tube open at both ends, filled with air at 343 m/s, has a fundamental of 343 / 3.60 ≈ 95 Hz, and its third mode sits at 286 Hz with a wavelength of 1.20 m. Close one end and the fundamental halves to 48 Hz while the second, fourth and sixth harmonics disappear. The simple formulas place the open-end antinode exactly at the end of the pipe; in reality it sits slightly outside, and Levine and Schwinger's classic solution for an unflanged round pipe gives an end correction of about 0.61 times the radius.[^levine1948] The microsim omits the correction, which is small for a narrow tube. *Try: step mode n from 1 to 6 and watch the dust heaps multiply while the f_n readout climbs in equal steps; switch tube ends to open-closed and see the even-harmonic bars turn grey; change gas from air to helium and watch every frequency rise almost threefold while the node positions stay put; lower reflected amplitude r below 1 and see the nodes fill in and the dust spread out.* ### 2D standing wave with a rectangular boundary In two dimensions a rectangular membrane or plate of sides L_x and L_y, clamped at its edges, supports modes labeled by two integers. Each is a product of a standing wave across and a standing wave along, `sin(mπx/L_x) sin(nπy/L_y)`, with frequency `f_mn = (v/2) √((m/L_x)² + (n/L_y)²)`. The nodes are now lines, which cross the rectangle in a grid. [[Ernst_Chladni|Ernst Chladni]] made such nodal lines visible in 1787 by drawing a bow across sand-covered plates: the sand jumps off the moving regions and collects along the lines that stay still.[^chladni1787] The same product form, extended to three integers, gives the modes of a rectangular room in [[Room_acoustics|room acoustics]]. ## Standing wave ratio, phase, and energy transfer When the reflected wave is weaker than the incident one, with amplitude ratio r between 0 and 1, the [[Reflection_(physics)|reflection]] leaves a partial standing wave: a pure standing wave of amplitude 2rA plus a traveling wave of amplitude (1 − r)A. The envelope then swings between a maximum of A(1 + r) at the antinodes and a minimum of A(1 − r) at the nodes. The ratio of the two is the **standing wave ratio**, `SWR = (1 + r)/(1 − r)`. A perfect reflection gives an infinite SWR and true nodes; no reflection gives SWR = 1 and a pure traveling wave. The ratio measures how much energy gets through. The fraction of incident power reflected is r², so the fraction transmitted is `1 − r²`. With r = 0.5 the SWR is 3 and 75 percent of the power travels on; with r = 0.9 the SWR is 19 and only 19 percent does. Radio engineers measure the SWR on transmission lines for exactly this reason, as a quick check that an antenna is matched to the line feeding it, a routine task in [[Microwave_engineering|microwave engineering]] and [[Radar|radar]]. The same partial standing wave appears in acoustics wherever a wave meets a change of [[Acoustic_impedance|acoustic impedance]], and measuring its maxima and minima in a tube is a standard way to find how much a sample absorbs. Phase behaves differently in the two kinds of wave. In a traveling wave the phase advances steadily with position; in a pure standing wave it is the same everywhere inside one loop and jumps by half a cycle at each node. A perfect standing wave carries no net energy: the power flowing right and left cancels, and energy only sloshes within each loop. A microwave oven shows the practical side. Its magnetron works at 2,450 MHz, one of the frequencies set aside for industrial, scientific and medical equipment,[^cfr18] so the wavelength is about 12.2 cm and standing-wave hot spots in the cavity fall about 6 cm apart, which is why ovens rotate the food. ## Examples ### Acoustic resonance Almost every [[Acoustic_resonance|acoustic resonance]] is a standing wave: the air in a bottle or organ pipe, the modes of a concert hall, the hum of a duct. In 1866 August Kundt sprinkled fine powder in a horizontal glass tube, drove a standing wave in the gas and saw the powder swept off the antinodes into heaps at the nodes, half a wavelength apart.[^kundt1866] Measuring the heap spacing with a ruler and knowing the driving frequency gives the speed of sound; at 1 kHz in room-temperature air the heaps sit 17.2 cm apart. Standing waves in a tube can also be driven by heat, as in the Rijke tube, the subject of [[Thermoacoustics|thermoacoustics]]. ### Visible light Light reflected from a mirror forms a standing wave with the incoming light. Otto Wiener showed this in 1890 by tilting a very thin photographic film through the standing wave in front of a silvered mirror: the film darkened in bands at the antinodes of the electric field and stayed clear at the nodes, a direct picture of optical nodes a fraction of a micrometer apart.[^wiener1890] The resonator of a laser is the same idea between two mirrors: only wavelengths for which a whole number of half-wavelengths fits in the cavity build up. ### X-rays When X-rays are diffracted by a perfect [[Crystal_structure|crystal]] at the Bragg angle, the incident and diffracted beams overlap inside the crystal and form a standing wave with the period of the atomic planes. The treatment of this field is part of the dynamical theory of X-ray diffraction reviewed by Batterman and Cole in 1964.[^batterman1964] Sweeping the angle slides the antinodes across the planes, and the changing fluorescence of atoms in or on the crystal reveals where they sit, the X-ray standing wave method. ### Mechanical waves The strings of [[Piano_acoustics|pianos]], guitars and violins, the bars of a xylophone and the membranes of drums all vibrate in standing waves, and each instrument's [[Timbre|timbre]] depends on which modes are excited and how strongly. The textbook points out a harmful side: buildings and wide roofs whose natural frequencies match an earthquake's shaking can [[Resonance|resonate]] and collapse while neighbors of a different height survive,[^up16-6] a central concern of [[Earthquake_engineering|earthquake engineering]]. Strong ultrasonic standing waves can even hold small objects at their pressure nodes, the basis of [[Acoustic_levitation|acoustic levitation]]. ### Seismic waves A large enough earthquake sets the entire Earth ringing in standing waves, its free oscillations. After the great Chilean earthquake of May 22, 1960, long-period instruments recorded these whole-Earth modes, with periods approaching an hour for the slowest, and Benioff, Press and Smith identified them as the Earth's normal modes in 1961.[^benioff1961] Their frequencies, together with traveling [[Seismic_wave|seismic waves]], map the density and elasticity of the deep interior. ### Faraday waves Michael Faraday reported in 1831 that a thin layer of liquid on a vertically vibrating plate breaks into a regular pattern of standing ripples.[^faraday1831] These Faraday waves are parametric: the ripples respond at half the driving frequency, and the patterns they form, stripes, squares or hexagons depending on the driving, have become a standard laboratory example of [[Pattern_formation|pattern formation]]. ### Seiches A seiche is a standing wave in an enclosed or partly enclosed body of water, the whole lake sloshing from end to end with a period set by the basin rather than by the Moon and Sun that drive a [[Tide|tide]]. Its fundamental period follows from the long-wave speed: a basin of length L and mean depth h sloshes with period about `T = 2L / √(gh)`. For [[Lake_Superior|Lake Superior]], 563 km long with a mean depth of 147 m,[^mseagrant] the wave speed is about 38 m/s and the estimate gives 8.2 hours. NOAA's Great Lakes Environmental Research Laboratory puts the lake's characteristic seiche period at approximately 8 hours.[^glerl2025] The music theory text notes that such sloshing "does apparently happen very rarely in lakes, resulting in freak disasters."[^bmt3-2] ## Minnesota *This section is specific to Wikitube.* Duluth, [[Minnesota]], sits at the western end of [[Lake_Superior|Lake Superior]], at one end of the lake's longest standing wave. NOAA's Great Lakes Environmental Research Laboratory describes the fundamental seiche as water that takes "about 8 hours" to slosh back and forth across the lake, "between Duluth and Sault Ste. Marie."[^glerl2025] In the storm of June 21, 2025, a meteotsunami and seiche changed water levels around the lake within hours; the laboratory lists Duluth among the coastal communities where beaches were seen to drain and then flood again, and it measured a rise of 45 inches (about 1.1 m) in Whitefish Bay in less than 2.5 hours.[^glerl2025] The uniform-channel estimate above lands within a few percent of that period: the half-wavelength rule for an open pipe and the sloshing of Lake Superior are the same physics. ## See also - [[Kundt's_tube]] - [[Wave_interference]] - [[String_vibration]] - [[Wave]] - [[Acoustic_resonance]] - [[Thermoacoustics]] - [[Normal_mode]] ## References [^up16-6]: OpenStax (2016). *University Physics Volume 1*. Ling, S. J.; Sanny, J.; Moebs, W. Rice University. Chapter 16 "Waves," §16.6 "Standing Waves and Resonance," Eq. 16.14–16.16, pp. 781–790. https://openstax.org/details/books/university-physics-volume-1 (Portal Books 077). [^up17-4]: OpenStax (2016). *University Physics Volume 1*. §17.4 "Normal Modes of a Standing Sound Wave," Eq. 17.13–17.16, pp. 826–833. Portal Books 077. [^bmt3-2]: Schmidt-Jones, Catherine; Jones, Russell (2013). *Understanding Basic Music Theory*. OpenStax CNX. §3.2 "Standing Waves and Musical Instruments," pp. 100–103. Portal Books 092. [^levine1948]: Levine, Harold; Schwinger, Julian (1948). "On the radiation of sound from an unflanged circular pipe." *Physical Review* 73 (4): 383–406. https://doi.org/10.1103/PhysRev.73.383 [^chladni1787]: Chladni, Ernst Florens Friedrich (1787). *Entdeckungen über die Theorie des Klanges*. Leipzig: Weidmanns Erben und Reich. (Publisher as recalled; the title and year are standard in histories of acoustics.) [^cfr18]: U.S. Code of Federal Regulations, Title 47, §18.301 "Operating frequencies" (industrial, scientific and medical equipment; 2,450 MHz ± 50 MHz). https://www.ecfr.gov/current/title-47/chapter-I/subchapter-A/part-18/subpart-C/section-18.301 [^kundt1866]: Kundt, August (1866). "Ueber eine neue Art akustischer Staubfiguren und über die Anwendung derselben zur Bestimmung der Schallgeschwindigkeit in festen Körpern und Gasen." *Annalen der Physik und Chemie* 127: 497–523. (Volume and pages as recalled; bibliographic record: https://www.semanticscholar.org/paper/1a4f26ef1b3d5d35d56b2459b5b7a945375e6f18) [^wiener1890]: Wiener, Otto (1890). "Stehende Lichtwellen und die Schwingungsrichtung polarisirten Lichtes." *Annalen der Physik und Chemie* 40: 203–243. (Volume and pages as recalled; bibliographic record: https://zs.thulb.uni-jena.de/receive/jportal_jparticle_00122846) [^batterman1964]: Batterman, B. W.; Cole, H. (1964). "Dynamical diffraction of X rays by perfect crystals." *Reviews of Modern Physics* 36 (3): 681–717. https://doi.org/10.1103/RevModPhys.36.681 [^benioff1961]: Benioff, Hugo; Press, Frank; Smith, Stewart (1961). "Excitation of the free oscillations of the Earth by earthquakes." *Journal of Geophysical Research* 66 (2): 605–619. https://doi.org/10.1029/JZ066i002p00605 [^faraday1831]: Faraday, Michael (1831). "On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces." *Philosophical Transactions of the Royal Society of London* 121: 299–340. https://doi.org/10.1098/rstl.1831.0018 [^mseagrant]: Michigan Sea Grant. "Lake Superior" (Great Lakes Fast Facts): length 350 mi (563 km), breadth 160 mi (257 km), average depth 483 ft (147 m). https://www.michiganseagrant.org/topics/great-lakes-fast-facts/lake-superior/ [^glerl2025]: NOAA Great Lakes Environmental Research Laboratory, GLERL Communications Team (July 18, 2025). "June 21, 2025 Storm Causes Significant Meteotsunami and Seiche on Lake Superior." https://www.glerl.noaa.gov/blog/2025/07/18/june-21-2025-storm-causes-significant-meteotsunami-and-seiche-on-lake-superior/  **Microsim — three.js (Wikitube framework):** *Standing wave* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Standing_wave.html" data-title="Standing wave"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Standing_wave.json`; part of the [[PORTAL_Acoustics|Acoustics portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Standing_wave) : [Wikitube](https://en.wikitube.io/wiki/Standing_wave) - skeleton pinned to revision 1319585291 (2026-09-11).