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Superposition of Waves

Ch. 6Std 12

Easy Overview

Waves are everywhere — sound waves carrying music to your ears, light waves bringing images to your eyes, water waves rippling across a pond. One of the most remarkable things about waves is what happens when they meet: they simply add together. This is the principle of superposition, and it leads to some of the most beautiful and useful phenomena in physics — interference, beats, standing waves, and diffraction. When two waves meet, they pass right through each other as if the other were not there, and at each point the net displacement is the sum of the individual displacements. The principle of superposition is simple to state but rich in consequences. When waves add constructively, crest meets crest and trough meets trough, resulting in a larger wave. When they add destructively, crest meets trough and they cancel. Between these extremes, any intermediate combination is possible. This superposition is not a special property of waves — it is a mathematical consequence of linear differential equations. Interference is the most direct consequence of superposition. When two coherent sources produce waves, the resulting pattern has regions of constructive and destructive interference. The condition for constructive interference is path difference = nλ. For destructive: path difference = (n + ½)λ. Thomas Young's double-slit experiment (1801) was the classic proof that light is a wave. Standing waves are another beautiful consequence — when a wave reflects from a boundary and interferes with itself, the result is a stationary pattern with nodes and antinodes. The Doppler effect — the change in frequency due to relative motion — applies to all waves and explains why an ambulance siren sounds different as it passes. From musical harmony to medical ultrasound, from the colours of soap bubbles to the design of concert halls, the principles of wave superposition shape our world.

The Principle of Superposition

The principle of superposition states that when two or more waves overlap, the resultant displacement is the algebraic sum of the individual displacements: y_resultant = y₁ + y₂ + y₃ + ... This applies to all types of waves — mechanical and electromagnetic. The principle holds because the wave equation is linear. After the waves pass through each other, each continues exactly as before. This is uniquely wavelike — particles scatter, but waves pass through unchanged.

Coherent Sources and Interference Conditions

For stable interference, sources must be coherent — constant phase relationship. Two independent bulbs are not coherent. Coherent sources are produced by splitting a single wavefront (Young's double slits) or splitting amplitude (thin film interference). Conditions: (1) Sources must be coherent. (2) Same frequency and wavelength. (3) Comparable amplitudes. (4) Path difference ≤ coherence length. Constructive: path diff = nλ. Destructive: path diff = (n + ½)λ.

Young's Double-Slit Experiment

Young's double-slit experiment provided the first experimental proof that light is a wave. Light passes through two narrow parallel slits, creating two coherent sources. The waves interfere on a screen, producing alternating bright and dark fringes. Fringe width β = λD/d. The intensity follows I = 4I₀ cos²(πd sinθ/λ). The central fringe is bright. The experiment also works with electrons and neutrons, demonstrating universal wave-particle duality.

Conditions for Sustained Interference

For sustained interference: (1) Sources must be coherent. (2) Same frequency. (3) Amplitudes comparable. (4) Narrow sources. (5) Source separation comparable to wavelength. (6) Monochromatic light preferred — white light produces coloured fringes that overlap. In practice, coherence is achieved using lasers or by dividing the wavefront from a single source using slits.

Thin Film Interference

Light reflects from top and bottom surfaces of a thin film (soap bubble, oil slick). The path difference between reflected waves is 2μt cos r. An additional π phase change occurs when light reflects from a denser medium. Constructive interference: 2μt cos r = (n + ½)λ. The rainbow colours of soap bubbles arise because different wavelengths interfere constructively at different thicknesses.

Beats — Interference in Time

Beats occur when two waves of slightly different frequencies are superposed. The resultant amplitude varies slowly: y = 2A cos((ω₁ − ω₂)t/2) sin((ω₁ + ω₂)t/2). Beat frequency f_beat = |f₁ − f₂|. Applications: tuning musical instruments (beats disappear when in tune), detecting small frequency differences, heterodyne detection in radio.

Standing Waves on a String

When a wave reflects from a fixed end, it is inverted (π phase change). Incident and reflected waves interfere to form a standing wave: y = 2A sin(kx) cos(ωt). Nodes (sin(kx) = 0) are permanently at rest; antinodes (|sin(kx)| = 1) have maximum displacement. For a string fixed at both ends: L = nλ/2, f_n = nv/(2L) = nf₁. The harmonic series: f₁, 2f₁, 3f₁, ...

Vibrations of Air Columns — Open and Closed Pipes

Open pipe (both ends open): both ends are antinodes. L = nλ/2, f_n = nv/(2L) — all harmonics present. Closed pipe (one end closed): closed end is node, open is antinode. L = (2n−1)λ/4, f_n = (2n−1)v/(4L) — only odd harmonics. The fundamental of a closed pipe is half that of an open pipe of the same length. End correction: effective length = L + 0.6R.

Harmonics and Overtones

The fundamental (first harmonic) is the lowest frequency. Higher frequencies are overtones. In a string or open pipe, the nth harmonic has frequency n× fundamental. In a closed pipe, only odd harmonics exist — first overtone is third harmonic. The timbre of a musical sound depends on which overtones are present. A flute sounds different from a clarinet due to different harmonic content.

Resonance in Air Columns

An air column resonates when driving frequency matches a natural frequency. In a resonance tube experiment, first resonance at L₁ = λ/4, second at L₂ = 3λ/4. λ = 2(L₂ − L₁), speed of sound v = fλ = 2f(L₂ − L₁). With end correction: L₁ + e = λ/4, L₂ + e = 3λ/4. Speed of sound in air at 0°C is about 331 m/s, increasing by 0.6 m/s per °C.

Characteristics of Musical Sound

Musical sounds are characterised by: (1) Pitch — determined by fundamental frequency (20 Hz to 20,000 Hz for human ear). (2) Loudness — related to intensity in decibels: β = 10 log(I/I₀), where I₀ = 10⁻¹² W/m². (3) Quality (timbre) — determined by overtones. (4) Duration. These characteristics let us distinguish different instruments playing the same note.

Doppler Effect in Sound

The Doppler effect is the apparent change in frequency due to relative motion. f_obs = f_src(v ± v_obs)/(v ∓ v_src). Approaching source → higher frequency. Receding source → lower frequency. When source speed exceeds wave speed, a sonic boom occurs (Mach cone, sinθ = 1/M). The Doppler effect also applies to light (redshift/blueshift) and is used to measure the expansion of the universe.

Wave Speed on a Stretched String

The speed of a transverse wave on a stretched string is v = √(T/μ), where T is tension and μ = mass per unit length. Doubling tension increases speed by √2. Heavier strings (higher μ) produce lower notes. The fundamental frequency f₁ = v/(2L) = (1/2L)√(T/μ). Tightening a guitar string raises the pitch.

Reflection and Transmission of Waves at Boundaries

When a wave encounters a boundary, part is reflected and part transmitted. Reflection at a fixed end is inverted (π phase change); at a free end, it is not inverted. For sound, reflection at a closed end occurs without phase change for pressure waves. The amplitude of the reflected wave depends on impedance mismatch between the two media.

Energy Transport and Intensity of Waves

Waves transport energy. Power on a string: P = ½μω²A²v. Intensity I = P/A. For spherical waves from a point source: I ∝ 1/r², A ∝ 1/r. Decibel scale: β = 10 log(I/I₀) dB. A 10 dB increase = 10× intensity. Normal conversation: 60 dB. Threshold of pain: 120 dB. Prolonged exposure above 85 dB can damage hearing.

Key Points

  • •Principle of superposition: resultant displacement = sum of individual displacements.
  • •Coherent sources maintain constant phase difference.
  • •Young's double-slit: fringe width β = λD/d.
  • •Intensity in double-slit: I = 4I₀ cos²(πd sinθ/λ).
  • •Thin film interference: path difference = 2μt cos r.
  • •Beats: f_beat = |f₁ − f₂|.
  • •Standing waves: y = 2A sin(kx) cos(ωt).
  • •String fixed at both ends: L = nλ/2, f_n = nv/(2L).
  • •Open pipe: all harmonics. Closed pipe: only odd harmonics.
  • •End correction: ~0.6R for open ends.
  • •Wave speed on string: v = √(T/μ).
  • •Resonance occurs at natural frequencies.
  • •Doppler effect: f_obs = f_src(v ± v_obs)/(v ∓ v_src).
  • •Sonic boom: Mach cone, sinθ = 1/M.
  • •Intensity: I ∝ 1/r² for spherical waves.
  • •Decibel scale: β = 10 log(I/I₀), I₀ = 10⁻¹² W/m².

Practice Questions

  • State the principle of superposition. Explain how it leads to beats. Derive beat frequency.
  • Describe Young's double-slit experiment. Derive fringe width. Slits 0.1 mm apart, screen 1 m away, third bright fringe at 1.8 cm from centre. Find wavelength.
  • Distinguish between progressive and stationary waves. Explain standing waves on a string fixed at both ends. Derive frequencies of different modes.
  • Explain resonance tube experiment. A tuning fork of 512 Hz gives first resonance at 16.5 cm and second at 50.5 cm. Find speed of sound and end correction.
  • What is the Doppler effect? Derive apparent frequency when source moves towards stationary observer. Train approaching at 30 m/s sounds whistle at 500 Hz. Frequency heard? (v = 340 m/s)
  • Compare open and closed organ pipes. Why do closed pipes produce only odd harmonics? Effect on sound quality?
  • Explain thin film interference. Why do soap bubbles show colours? Find minimum thickness of soap film (μ = 1.33) appearing yellow (λ = 580 nm) in reflected light.
  • Derive wave speed on a stretched string. String of length 1 m, mass 5 g, tension 200 N. Find fundamental frequency. Tension needed to double this frequency?