14.6 Wave Interference and Standing Waves

Wave interference is what happens when two or more waves overlap and you add their displacements point-by-point (the superposition principle). If the overlapping displacements point the same way you get constructive interference (bigger amplitude); if they point opposite ways you get destructive interference (smaller or zero amplitude). Important: waves travel through each other—they don’t bounce—and the resultant at any instant equals the algebraic sum of individual displacements (CED 14.6.A.1–A.4). Two slightly different frequencies produce beats: the loudness oscillates at the beat frequency |f1 − f2| (CED 14.6.A.6). Standing waves form when two equal waves travel opposite directions and interfere; they have nodes (always zero) and antinodes (maximum amplitude), and λ = 2 × (distance between adjacent nodes) (CED 14.6.B). These ideas show up on the AP exam in conceptual and free-response items—practice sketching wave overlays and identifying nodes/antinodes. Review the Topic 14.6 study guide on Fiveable (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and try practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Waves don’t “bounce off” each other because they’re disturbances in a medium (or in a field) rather than solid objects. The medium’s particles just oscillate locally and don’t carry the whole wave with them, so when two disturbances meet their displacements add—that’s the superposition principle (CED 14.6.A.3). If the displacements point the same way you get constructive interference; if opposite, destructive interference (14.6.A.4). For most linear waves the individual waves pass through and keep their shape after overlapping because the wave equation is linear, so the interaction only changes the net displacement while they overlap, not the waves themselves (14.6.A.2). Standing waves are a special case where two opposite-traveling waves interfere to produce nodes and antinodes (14.6.B.1). Want practice or a quick refresher? Check the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and the unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-14). For lots of problems, use the practice set (https://library.fiveable.me/practice/ap-physics-2-revised).

Think of interference as simple addition of displacements (the superposition principle from the CED). When two (or more) waves meet, you add their instantaneous displacements point-by-point. - Constructive interference: the waves’ displacements are in the same direction (they’re “in phase” at that location). Their amplitudes add, so the resultant wave has larger amplitude (peak + peak → bigger peak). This is what the CED calls constructive (14.6.A.4.i). - Destructive interference: the waves’ displacements are opposite (out of phase). They subtract, so amplitude is reduced and can even go to zero if they’re equal and opposite (node). That’s destructive interference (14.6.A.4.ii). Use sketches to decide results—draw the two wave shapes and add them point-by-point (CED 14.6.A.5). Related ideas: standing waves form when opposite-traveling waves interfere, producing nodes (zero amplitude) and antinodes (max amplitude) (CED 14.6.B.1). For more examples and exam-style practice, see the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and thousands of practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Superposition just means you add the displacements of overlapping waves point-by-point to find the net disturbance. If two waves meet at the same place and time, take each wave’s vertical displacement (signed: up positive, down negative) and add them. If the signs are the same you get constructive interference (bigger amplitude); if opposite you get destructive interference (smaller or zero amplitude). For sinusoidal waves, add amplitudes algebraically taking phase into account; two waves of equal frequency give simple constructive/destructive results when they’re in phase or 180° out of phase. Slightly different frequencies produce beats with beat frequency |f1 − f2|. On the AP exam you’re expected to describe this and use sketches or algebra to show the resultant (CED 14.6.A: net disturbance = sum of individual displacements). Practice adding pulses and sinusoids with sketches—see the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and try more problems at (https://library.fiveable.me/practice/ap-physics-2-revised).

Use superposition: add the displacements of the two waves at the same location and time. If the individual displacements point the same way (both positive or both negative) they add to a larger displacement—that’s constructive interference. If they point opposite ways (one positive, one negative) they partially or completely cancel—that’s destructive interference. Practically, check the phase difference Δφ or path difference Δx between the waves: - Constructive when Δφ = 0, 2π, 4π... (i.e., Δx = mλ, m an integer) → maxima (antinodes for standing waves). - Destructive when Δφ = π, 3π, 5π... (i.e., Δx = (m + 1/2)λ) → minima (nodes in standing waves). On the exam you may be given wave sketches—just compare sign/direction at the overlap point or compute phase/path difference. For extra practice and quick review, see the Topic 14.6 study guide (Fiveable) here: (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and try practice problems at (https://library.fiveable.me/practice/ap-physics-2-revised).

Beats happen when two sound waves of nearly the same frequency overlap and you add their displacements using superposition. Sometimes the waves line up (constructive interference) so the loudness is bigger; sometimes they’re out of step (destructive interference) so the loudness drops. Because the two frequencies slip in and out of phase, the resulting amplitude rises and falls periodically—that pulsing you hear. The beat frequency equals the difference between the two frequencies: |f_beat| = |f1 − f2|. For example, a 440 Hz and a 442 Hz tuning fork produce beats at 2 Hz (two loud pulses per second). Tuning forks are a classic demo of this (CED 14.6.A.6). If you want a quick review, see the Topic 14.6 study guide on Fiveable (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and try practice problems (https://library.fiveable.me/practice/ap-physics-2-revised)—beats show up often on AP-style questions.

Beats come from superposing two nearly equal frequencies. Use the CED formula: |f_beat| = |f1 − f2|. Steps: identify the two frequencies (f1 and f2), subtract them, and take the absolute value so beats are positive. The beat rate you hear (pulses per second) equals that difference. Quick example: if one tuning fork is 440 Hz and another is 442 Hz, f_beat = |440 − 442| = 2 Hz—you’ll hear the loudness vary twice per second. Beats are useful on the AP exam for describing interference and real devices like tuning forks (CED 14.6.A.6.i–iii). Want more practice? Review Topic 14.6 on Fiveable (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and try extra problems at Fiveable Practice (https://library.fiveable.me/practice/ap-physics-2-revised).

Nodes are points on a standing wave that never move—their displacement is always zero because the two opposite-traveling waves cancel there (destructive interference). Antinodes are halfway between nodes and have the largest oscillation amplitude because the two waves add there (constructive interference). Key facts from the CED you should remember: the wavelength of the traveling wave is twice the distance between adjacent nodes (or antinodes), standing waves form from two waves traveling in opposite directions, and whether an end of a string or pipe is a node or antinode affects which harmonics can exist (e.g., a node at one end and an antinode at the other allows only odd harmonics). On the AP exam you’ll often be asked to identify nodes/antinodes on diagrams, use node spacing to find λ or f, or reason about boundary conditions for harmonics. For a clear review and practice, see the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and try more problems on Fiveable’s practice page (https://library.fiveable.me/practice/ap-physics-2-revised).

Think of a standing wave as two identical traveling waves moving opposite ways and adding by superposition. A convenient math form is y(x,t) = 2A sin(kx) cos(ωt), where k = 2π/λ. Nodes are points that never move, so they satisfy sin(kx) = 0 → kx = nπ. That gives x = nπ/k = n(λ/2). Adjacent nodes have n and n+1, so their separation is Δx = (n+1)(λ/2) − n(λ/2) = λ/2. Rearrange: λ = 2(Δx). So the wavelength is twice the distance between neighboring nodes. Physically: one half-wavelength fits between two successive nodes (node → antinode → node), so two node spacings equal a full wavelength. This is exactly the CED idea in 14.6.B.1.ii (nodes, antinodes, standing waves). For more worked examples and practice, check the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Standing waves form when two waves of the same frequency and amplitude travel in opposite directions in a confined region (like a string with fixed ends or a pipe) and overlap. By the superposition principle you add their displacements at each point; where the two waves are always in phase you get constructive interference (antinodes, max amplitude), and where they are always out of phase you get destructive interference (nodes, zero amplitude). Because the interfering waves have the same frequency and a fixed phase relationship at each location, the pattern of nodes and antinodes doesn’t move—the result is a stationary (standing) pattern. The distance between adjacent nodes (or antinodes) is λ/2, so allowed standing modes (fundamental and harmonics) depend on the boundary conditions (e.g., fixed-fixed or open-closed). This is exactly what the CED calls for in 14.6.B (nodes/antinodes, harmonic series). For a quick review, see the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and more unit review (https://library.fiveable.me/ap-physics-2-revised/unit-14). Practice problems are at (https://library.fiveable.me/practice/ap-physics-2-revised).

The fundamental frequency (first harmonic) is the lowest frequency standing wave that fits in a region—it has the longest wavelength and the simplest pattern (one antinode between two nodes for a fixed-fixed string). Harmonics are the higher-frequency standing modes whose wavelengths are shorter integer fractions of the fundamental. For a string or an open-open pipe: fn = n·f1 where n = 1,2,3,... and λn = (2L)/n. For an open-closed pipe only odd harmonics occur (n = 1,3,5,...) because one end must be a node and the other an antinode (CED 14.6.B.2). Nodes are points of zero amplitude; antinodes are max amplitude. On the AP exam you should be able to relate length, wavelength, frequency, and harmonic number (CED 14.6.B.1–B.3). Want more examples and quick practice? Check the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and the unit review (https://library.fiveable.me/ap-physics-2-revised/unit-14). For lots of practice problems go here: (https://library.fiveable.me/practice/ap-physics-2-revised).

Because of the boundary conditions at the ends. A pipe with one closed end must have a displacement node at the closed end (air can’t move there) and a displacement antinode at the open end. Those end conditions force standing-wave patterns whose wavelengths satisfy L = (λ/4), (3λ/4), (5λ/4), … so λ = 4L/n with n = 1, 3, 5…—only odd n work, so only odd harmonics (f = n·v/4L, n odd). A string fixed at both ends has nodes at both ends, so allowed standing wavelengths are λ = 2L/n with n = 1, 2, 3, … giving every harmonic (f = n·v/2L, all n). This is just superposition of two opposite-traveling waves producing node/antinode patterns; the end constraints pick which integer n satisfy the geometry (see CED 14.6.B.1–B.3 and 14.6.B.2). For a quick refresher, check the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and more practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).

Doable in one lab period. Set a string fixed at both ends (or one fixed, one loose for odd-only harmonics), attach one end to a vibration driver or a mechanical oscillator (or a tuning fork/phone app + bridge), and vary frequency or tension until you see a pattern that doesn’t travel—only nodes (no motion) and antinodes (max motion). Measure nodes by: - Count adjacent nodes; distance between adjacent nodes = λ/2 so λ = 2(dnode). - For a string of length L with fixed ends: allowed modes satisfy λn = 2L/n and fn = n(v/2L). - Use v = sqrt(T/μ) if you need wave speed (T = tension, μ = linear mass density). Practical tips: use chalk/ tape to mark node positions, dim lights or use strobe/slow-motion video to freeze motion, measure multiple node spacings and average, and record tension and string mass to compute μ. This matches CED ideas about nodes/antinodes, harmonics, and using measurements for f, λ, v (Topic 14.6.B). For a quick review see the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and more practice at (https://library.fiveable.me/practice/ap-physics-2-revised).

Short answer: the physics is the same (v = fλ and nodes/antinodes from superposition), but the boundary conditions change the allowed wavelengths and thus the formulas. Key formulas to memorize: - String (both ends fixed) or pipe open at both ends: standing modes have wavelengths λn = 2L/n and frequencies fn = n·v/(2L) for n = 1,2,3,... - Pipe open at one end and closed at the other: only odd harmonics. Wavelengths λn = 4L/n and frequencies fn = n·v/(4L) with n = 1,3,5,... Quick tips: - Fixed end = node; open end = antinode. Distance between adjacent nodes (or antinodes) = λ/2 (CED 14.6.B.1.ii). - Always get v from the medium (e.g., speed of sound in air) and use v = fλ to switch between f and λ. - For AP problems, draw node/antinode diagrams to pick n and check whether n must be odd (open-closed). Want more practice and a short topic review? See the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9) and thousands of practice questions (https://library.fiveable.me/practice/ap-physics-2-revised).

Noise-canceling headphones are a real-world application of wave interference and the superposition principle (CED 14.6.A). Tiny microphones pick up ambient sound and the electronics generate a sound wave that’s the same amplitude but 180° out of phase with that noise. When the original and the generated wave overlap, their displacements add and cancel (destructive interference), reducing the net pressure variations you hear. This works best for lower-frequency, continuous sounds (engine hums) because it’s easier to match phase and amplitude in real time; high-frequency, impulsive sounds are harder to cancel and are often handled by passive insulation (ear pads). On the AP side, knowing superposition, constructive vs. destructive interference, and phase differences is exactly what the CED expects (14.6.A.3–4). If you want a concise refresher or practice problems about interference and standing waves, check the Topic 14.6 study guide (https://library.fiveable.me/ap-physics-2-revised/unit-6/6-wave-interference-and-standing-waves/study-guide/3twkmfrYKPDOuep9), the unit overview (https://library.fiveable.me/ap-physics-2-revised/unit-14), and Fiveable’s practice problems (https://library.fiveable.me/practice/ap-physics-2-revised).