3cSuperposition and Interference
Apply the principle of superposition, distinguish constructive from destructive interference, and analyze standing waves, nodes, and resonance.
Superposition and interference explain musical instrument resonance, noise-canceling headphones, radio signal interference, and the quantum behavior of particles — it is one of the most far-reaching principles in physics.
What happens when two waves occupy the same space at the same time — and how do standing waves form?
Lesson Overview
The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements. Constructive interference occurs when waves add together to produce a larger amplitude; destructive interference occurs when they cancel. Standing waves form when two identical waves travel in opposite directions, creating fixed nodes and antinodes at resonance frequencies.
Key Concepts
Principle of Superposition
Resultant displacement = algebraic sum of individual wave displacements at each point
Constructive Interference
Waves in phase add together; resultant amplitude = A₁ + A₂
Destructive Interference
Waves out of phase cancel; resultant amplitude = |A₁ − A₂|
Standing Wave
Pattern formed by two identical waves traveling in opposite directions; appears stationary
Nodes & Antinodes
Nodes: points of zero displacement; antinodes: points of maximum displacement
Resonance Frequency
fₙ = nv/(2L) for a string fixed at both ends; n = 1, 2, 3, … (harmonics)
Worked Examples
Two waves with amplitudes 4 cm and 6 cm meet in phase (constructive interference). What is the resultant amplitude?
Two waves with amplitudes 5 cm and 3 cm meet completely out of phase (destructive interference). What is the resultant amplitude?
A string of length 0.80 m is fixed at both ends. The wave speed on the string is 160 m/s. Find the fundamental frequency (first harmonic) and the second harmonic.
Explain why noise-canceling headphones work using the principle of superposition.
A standing wave on a 1.20 m string has 3 antinodes. Find the wavelength and the harmonic number.
Guided Problems
Two waves of amplitude 7 cm each interfere constructively. What is the resultant amplitude? What if they interfere destructively?
Hint: Constructive: add amplitudes. Destructive: subtract amplitudes (take absolute value).
A guitar string is 0.65 m long and has a wave speed of 390 m/s. Find the fundamental frequency.
Hint: For a string fixed at both ends: f₁ = v/(2L).
Describe the difference between a node and an antinode in a standing wave.
Hint: Think about which points never move and which points have maximum displacement.
Two identical waves travel in opposite directions on a string. Under what condition do they form a standing wave?
Hint: They must have the same frequency, wavelength, and amplitude, traveling in opposite directions.
A standing wave has nodes at x = 0, 0.30 m, 0.60 m, and 0.90 m. What is the wavelength?
Hint: The distance between adjacent nodes is λ/2.
Key Vocabulary
Principle of Superposition
When two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements.
Example: Two water waves crossing each other momentarily add their heights at the point of overlap.
Constructive Interference
Interference in which waves are in phase and their amplitudes add together, producing a larger resultant amplitude.
Example: Two speakers playing the same tone in phase produce a louder sound at points of constructive interference.
Destructive Interference
Interference in which waves are out of phase and their amplitudes subtract, producing a smaller (or zero) resultant amplitude.
Example: Noise-canceling headphones use destructive interference to reduce unwanted sound.
Standing Wave
A wave pattern formed when two identical waves travel in opposite directions; it appears stationary with fixed nodes and antinodes.
Example: A vibrating guitar string forms standing waves at its resonance frequencies.
Interactive Practice — 5 Questions
The principle of superposition states that when two waves overlap, the resultant displacement is:
Two waves of amplitude 8 cm interfere constructively. The resultant amplitude is:
In a standing wave, a node is a point of:
The fundamental frequency of a string fixed at both ends is given by:
Resonance occurs when a system is driven at:
Independent Practice
State the principle of superposition and explain the difference between constructive and destructive interference.
A string 1.00 m long fixed at both ends has a wave speed of 200 m/s. Find the frequencies of the first three harmonics.
Two waves have amplitudes 10 cm and 6 cm. Find the resultant amplitude for (a) constructive and (b) destructive interference.
Explain what a standing wave is and describe the conditions needed to form one on a string.
★ A pipe open at both ends has a length of 0.85 m. The speed of sound is 340 m/s. Find the fundamental frequency and the first three harmonics. Then explain how closing one end changes the resonance frequencies.
ChallengeCommon Mistakes
Thinking destructive interference destroys energy.
Energy is conserved — destructive interference redistributes energy to other locations. The total energy of the system is unchanged.
Confusing nodes and antinodes — "nodes are where the wave is biggest."
Nodes are points of zero displacement (no motion). Antinodes are points of maximum displacement.
Thinking standing waves travel through the medium.
Standing waves do not propagate — they appear stationary. The pattern is the result of two traveling waves moving in opposite directions.
Math Tips
For a string fixed at both ends: fₙ = nv/(2L) where n = 1, 2, 3, … The nth harmonic has n antinodes and (n+1) nodes including the endpoints.
Distance between adjacent nodes = λ/2. If you know the node spacing, you can find the wavelength and then the frequency using f = v/λ.