Superposition Standing Waves Beats & Doppler Superposition, Standing Waves, Beats, and Doppler When two or more waves meet, they do not collide like balls; they add. This simple rule is called superposition: the net displacement is the algebraic sum of individual displacements at that instant and point. The magic begins when waves have the same frequency and comparable amplitude. Sometimes they reinforce (constructive interference), sometimes they cancel (destructive interference), and often they create striking patterns. In strings and air columns, reflections at boundaries make forward and backward waves overlap, producing a stationary pattern of nodes (no motion) and antinodes (maximum motion). That is a standing wave, the basis of musical notes and resonance in instruments. If two sounds of slightly different frequencies arrive together, the intensity rises and falls periodically; our ears perceive this slow variation as beats. The beat rate equals the difference in frequencies, a powerful tool to tune instruments with high precision. Finally, if the source or the listener moves, the time interval between successive crests at the observer changes even though the source’s own emission rate stays the same. This kinematic shift of the observed frequency is the Doppler effect—the reason a siren’s pitch seems higher as it approaches and lower as it recedes. Across all four ideas, the common thread is phase: where each wave’s crest and trough land in time and space. Get the phase right, and you predict whether the sound is loud or soft, the string is still or shaking, the pitch is high or low. Get the limits right, and you avoid classic traps: superposition requires linear media, standing waves trap energy locally rather than carrying it along, beats need close frequencies, and the Doppler formula for sound demands a stationary medium and careful signs for moving source and observer. Everyday picture: Two people pushing a swing in step make it soar (constructive). Out of step, they spoil each other’s effort (destructive). A rope shaken at just the right rhythm forms fixed loops (standing waves). A passing ambulance’s wail rises then falls in pitch (Doppler). remember In a linear medium, the net displacement is the algebraic sum of individual wave displacements at each point and instant. Superposition Principle The pattern resulting from superposition of two (or more) coherent waves; constructive when in phase, destructive when out of phase. Interference Sources with a constant phase difference and the same frequency; necessary for stable interference patterns. Coherent Sources Node A fixed point in a standing wave where displacement is always zero. A point in a standing wave where displacement amplitude is maximum. Antinode Standing Wave A non-propagating wave pattern formed by two identical waves traveling in opposite directions, showing fixed nodes and antinodes. The rate at which intensity rises and falls when two close frequencies superpose: f b = |f 1 - f 2| . Beat Frequency Doppler Effect (Sound) The change in observed frequency due to relative motion of source and observer with respect to the medium. Harmonics and Overtones Allowed standing-wave frequencies are integer multiples of the fundamental (harmonics). The first overtone is the second harmonic for strings and open pipes, but the third harmonic for a closed pipe. The superposition principle applies when the medium responds linearly: double the input, double the response. In that regime, combining two sinusoidal waves is easiest in phase language. If two waves have equal frequency f and amplitudes A , but a phase difference , their resultant amplitude at a point is 2A ( 2 ) . Thus, = 0, 2 , gives maximum reinforcement; = , 3 , gives cancellation. Phase difference connects directly to path difference via = 2 x . That is why path difference conditions decide bright and dark regions in interference—and where nodes and antinodes sit in standing waves. Sum-to-product identity for two sines Key identity behind beats and standing waves formation. Standing waves arise naturally when a traveling wave reflects from a boundary and returns with the same speed and frequency. With a rigidly fixed end (string tied to a wall), the reflected wave inverts in phase, ensuring a node at the boundary. With a free end (loosely supported string tip or an open pipe end for pressure nodes), the reflection occurs without inversion for displacement. The forward and backward waves superpose to form a time-oscillating but space-stationary pattern: the hallmark of a standing wave. Equation of a standing wave Standing wave from two counter-propagating waves of equal A, , k . Resultant of y 1 = A (kx- t) and y 2 = A (kx+ t) ; nodes at kx = n and antinodes at kx=(2n+1) 2 . tip Superposition boundaries: requires linear, time-invariant media and small amplitudes. Nonlinear media (very large amplitudes, shocks in gases) break simple addition. Node and antinode locations follow directly from the spatial factor (kx) . Nodes occur where (kx)=0 kx = n x n = n 2 . Antinodes occur where | (kx)| is maximum: kx = (2n+1) 2 x anti = (2n+1) 4 . Consecutive nodes or consecutive antinodes are /2 apart; a node to its nearest antinode is /4 apart. Fixed-node lattice for a standing wave along x. Node and antinode positions This formula defines the fixed positions of nodes (zero displacement) and antinodes (maximum displacement) along a standing wave pattern, ty On a string with both ends fixed (nodes at x=0 and x=L ), only those wavelengths fit for which an integer number of half-wavelengths equals L : L = n n 2 n = 2L n . The corresponding allowed frequencies are f n = v n = n v 2L , where v = T is the wave speed on the string with tension T and linear mass density . The fundamental (first harmonic) corresponds to n=1 . String harmonics Allowed modes for a string with both ends fixed. This set of formulas applies to standing waves (harmonics) formed by transverse vibrations on a string fixed at both ends. Air columns behave similarly but swap displacement and pressure nodes at open/closed ends. An open end is a displacement antinode (pressure node); a closed end is a displacement node (pressure antinode). Thus, an open–open pipe supports all harmonics like a string. A closed–open pipe supports only odd harmonics because one end is a node and the other an antinode, forcing an odd number of quarter wavelengths inside the tube. Organ Pipe Harmonics Pipe Type Boundary Conditions Frequency Formula Harmonics Present Tone Quality Open pipes provide 'All' (Even and Odd) harmonics with 2L , while Closed pipes are 'Odd' only with 4L . Open Organ Pipe (Both Ends Open) Antinodes ( A ) at both ends f n = n ( v 2L ) for n = 1, 2, 3, All harmonics present (Both Even and Odd: 1f 1, 2f 1, 3f 1, ) Richer quality due to full harmonic series Closed Organ Pipe (One End Closed) Node ( N ) at closed end; Antinode ( A ) at open end f n = (2n-1) ( v 4L ) for n = 1, 2, 3, Only Odd harmonics present ( 1f 1, 3f 1, 5f 1, ) Distinctive mellow or 'hollow' tone quality organ pipe harmonics Closed–open pipes have only odd harmonics (n = 1, 3, 5, ...). Do not insert n = 2 or 4 into f n = n·v/(4L). That is a frequent trap. neet-alert First overtone is always the second harmonic. For strings and open–open pipes, yes. For a closed–open pipe, the first overtone is the third harmonic because even harmonics are missing. When two sounds of close frequencies f 1 and f 2 reach your ear, the pressure variations add. The amplitude becomes slowly modulated by a cosine envelope at frequency |f 1-f 2| 1 , while the rapid oscillation is near the average frequency f 1+f 2 2 . Your ear senses intensity (amplitude) 2 , which peaks twice per envelope cycle. The perceived beat frequency equals |f 1 - f 2| , not the average f 1+f 2 2 . Beat envelope Sum of equal-amplitude tones f 1 and f 2 ( f 1 f 2 ). Beats are heard at f b = |f 1 - f 2| for close frequencies of comparable amplitude. Beat frequency f b = |f 1 - f 2| Use the identity in Eq. (1). Slowly varying envelope. Two intensity maxima per envelope period. Maxima spacing in intensity occurs each 1/ f s. f b = |f 1 - f 2| Linear superposition; equal amplitudes A for clarity Frequencies are close: |f 1 - f 2| f 1, f 2 Ear perceives intensity amplitude 2 f 1 = 256 Hz f 2 = 260 Hz Beat frequency and perceived pitch (approximate) Hz easy Two tuning forks of frequencies 256 Hz and 260 Hz are sounded together. How many beats per second are heard, and what is the approximate pitch perceived? Use f b = |f 1 - f 2| ; pitch is heard near the average frequency. The Doppler effect for sound links purely to kinematics in a stationary medium: motion alters the spacing or arrival rate of crests. If the observer moves towards the source, the arrival rate increases; if the source moves towards the observer, the emitted crests are compressed, decreasing the effective wavelength in the medium. These effects combine into a single compact formula when motion is collinear and speeds are much less than the wave speed in the medium. v : speed of sound in medium; v o : observer velocity towards source (negative if away); v s : source velocity towards observer (negative if away). Doppler formula (sound, standard sign convention) Observed frequency increases if the observer moves towards ( v o>0 ) or source moves towards ( v s>0 ). Careful with signs. tip Sign convention used: velocities are positive if directed towards the other party. Then use f obs = f s , v + v o v - v s . If you choose another convention, rewrite consistently; never mix conventions. Medium is stationary Non-relativistic speeds: v s, v o v Source emits at frequency f s continuously Collinear motion along source–observer line f obs = f s , v + v o v - v s Doppler effect for sound: f obs = f s , v + v o v - v s If source moves towards observer ( v s>0 ), crests are closer. If observer moves towards source ( v o>0 ), wavefronts are met sooner. L = 1.2 m T = 120 N = 0.010 kg/m f 3 Hz A string of length L = 1.2 m is fixed at both ends. Tension T = 120 N and linear density μ = 0.010 kg/m. Find the frequency of the third harmonic. medium Use v = T/ and f n = n ,v/(2L) . custom Spatial standing-wave envelope (snapshot) on a fixed–fixed string for harmonic n. Position along string x control control arbitrary Displacement amplitude Node Antinode (n=1) Max L/2 Node Standing wave profile for mode n, with nodes at x = 0, L and internal nodes at x = L·k/n (k = 1..n-1). Antinodes midway between nodes. Standing waves carry energy from one end to the other like traveling waves. In a standing wave, energy oscillates locally between kinetic and potential near nodes/antinodes; there is no net energy transport along the medium. Intensity peaks twice per envelope cycle, so the beat frequency is |f 1 - f 2| , not half of it. Beat frequency equals the frequency of the slow cosine envelope. Beats count per second equals |f 1 - f 2| only when both tones are simultaneously audible at the same point with comparable amplitudes. If one fades or reflections shift phase, counts may mislead. neet-alert Reflections and phase flips: At a rigidly fixed end of a string, the transverse displacement is forced to zero, so the reflected wave inverts in phase (node). At a free end, slope is forced to zero, the displacement antinode appears, and there is no inversion for displacement. In air columns, think in terms of pressure and displacement: open ends are pressure nodes (displacement antinodes), closed ends are pressure antinodes (displacement nodes). Matching the correct boundary condition is the key to writing allowed wavelengths. TOPS: Towards Observer → Plus in numerator (v + v o ); Towards Source → Plus for v s in your mind, but the formula needs (v − v s ) in the denominator. Think “source squeezes crests,” so subtract its towards-speed. Use with the convention: velocities positive towards the other party. Use f obs = f s , v + v o v - v s with the stated signs. hard A siren of frequency 500 Hz moves towards a listener at 20 m/s. The listener moves away from the siren at 10 m/s. Speed of sound v = 340 m/s. What frequency does the listener hear? Hz f obs f s = 500 Hz v = 340 m/s v s = +20 m/s (towards observer) v o = -10 m/s (away from source) How to solve standing-wave questions (strings/pipes) Identify boundary conditions at each end: node or antinode (for displacement). Write the fitting condition for L in terms of λ (e.g., fixed–fixed: L = n·λ/2; closed–open: L = (2n−1)·λ/4). Relate f, λ, v: f = v/λ. For strings, v = √(T/μ); for air, use given speed of sound. Pick allowed n (integers for strings/open pipes; odd only for closed–open). Compute the required quantity (f, λ, n, or L) and check units. Interference conditions with path difference: If two coherent waves of wavelength meet with path difference x , the phase difference is = 2 x . Constructive interference occurs for x = m (in phase), and destructive for x = (m+ 1 2 ) (out of phase), where m is any integer. These same rules decide the stationarity points of standing waves because the forward and reflected paths impose definite x at each position along the medium. Phase–path relation Bridge between geometry (path) and phase (interference). This relationship links the physical path difference between two waves to the resulting phase difference at any point. Applications at a glance Tuning instruments via beats: adjust until beats vanish. Measuring string tension or linear density by resonant modes. Resonance tubes (Kundt’s tube) to measure speed of sound or frequency. Doppler radar and medical ultrasound (conceptually similar effect; electromagnetic and high-frequency sound require appropriate formulas). Speed detection and flow measurements (echolocation, sonar; non-relativistic Doppler for sound). Wind in Doppler (sound): replace v by the effective speed along the propagation path: use v + u w if wind aids wave travel from source to observer, or v − u w if it opposes. Apply the same modified v in both numerator and denominator with consistent direction. tip Edge cases and limits: For beats, if |f 1-f 2| is large (say > 15 Hz), the ear no longer hears discrete beats; it hears a rough or fused tone. For standing waves, if the drive frequency is off-resonance, the pattern is weak or absent because boundary conditions cannot sustain clean nodes/antinodes. For Doppler in sound, if the source speed approaches the speed of sound ( v s v ), the denominator tends to zero and the simple formula signals a blow-up; physically, the source piles up wavefronts forming a shock (sonic boom), outside this syllabus’ linear regime. neet-alert Do not use the sound Doppler formula for light. Light requires the (relativistic) Doppler formula because no medium is involved and speeds are comparable to c in many contexts. Worked reasoning example (qualitative Doppler): If both source and observer move towards each other, the numerator increases (observer meets crests faster) and the denominator decreases (source squeezes crests), so f obs must exceed f s . If they move apart, the reverse happens and the pitch drops. Always predict qualitatively before plugging numbers; it helps catch sign mistakes. Practical tuning via beats: Suppose a 440 Hz reference (A4) and your instrument’s note give 3 beats/s. If turning the peg to tighten the string (raises frequency) reduces beats to 1/s, you were initially below 440 Hz; keep tightening until beats vanish. If tightening increases beats, you started above 440 Hz; loosen instead. The beat method is robust because you only count slow pulsations. Amplitude vs intensity: In superposition problems, you may compute a resultant amplitude A r . Remember that loudness relates to intensity I A r 2 for sound (pressure amplitude squared) and energy density for strings. Doubling amplitude quadruples intensity. This matters for beats: the loudest moments are when the instantaneous amplitudes add to 2A , giving four times the intensity of a single tone. Measuring μ or T with standing waves: A standard sonometer experiment uses a known driving frequency f from a tuning fork and adjusts the string length L until a strong resonance is observed (say at fundamental). Then f = v/(2L) with v = T/ gives = T/(4f 2L 2) . Alternatively, by changing tension and counting harmonic numbers, you can determine relationships like f T at fixed L, . Amplitude affects loudness, not frequency. Doppler shift in sound is due to relative motion only, with the medium’s speed setting the scale. Changing the loudness (amplitude) of a siren changes the observed frequency. Detecting closed vs open end in questions: If the end is thermally and mechanically free to move, it is an open end (displacement antinode). If it is sealed or rigid, it is closed (displacement node). In sketches, an open end often shows maximum oscillation of air; a closed end shows a point of zero displacement. Translate the picture into node/antinode labels before writing formulas. Instantaneous displacement x = A ( t + ) . Serves as the local oscillation law at any fixed point in a standing wave. At a fixed location in a standing wave, motion is SHM with a = - 2 x ; acceleration is largest in magnitude at antinodes. Distance between adjacent nodes (or adjacent antinodes) is exactly /2 . From a node to the nearest antinode is /4 . remember Multiple resonances in pipes: In a closed–open pipe, the allowed lengths for a fixed frequency f are L = (2n-1) 4 with n=1,2,3, . Thus, successive resonances occur at odd multiples of the quarter wavelength. This is why the difference between successive resonance lengths equals /2 , a common measurement trick in labs. Small-angle approximation not needed: Unlike the simple pendulum where SHM needs , standing waves, beats, and Doppler rely on linear wave equations and kinematics. Their formulas remain valid without small-angle approximations, as long as the medium stays linear and the motion is collinear (for the Doppler formula used here). Frequency locking in resonance: When a string is driven at frequency f , the system selects the harmonic whose natural frequency is closest to f . The length L and tension T can be tuned so that one of the allowed f n equals f , maximizing amplitude. Off-resonance, the amplitude is small, and the pattern is not a perfect standing wave. Qualitative beat puzzles: If two instruments produce 5 beats/s and after slightly tightening one string you hear 3 beats/s, the changed instrument moved closer to the other’s pitch. If tightening increases beats to 7/s, you moved away. This logic resolves the common ambiguity in f b = |f 1 - f 2| where both f 1 = f 2 f b are possible. Superposition Linear addition of displacements. Result of superposition; constructive or destructive. Interference Zero displacement point in a standing wave. Node Antinode Maximum displacement point in a standing wave. Harmonic Integer multiple of the fundamental frequency. Magnitude of frequency difference: |f 1 - f 2| . Beat frequency Frequency shift due to relative motion in a medium. Doppler effect Wave speed (string) v = T/ Key terms recap