03Wave Motion
Describe wave properties, calculate wave speed, and analyze reflection, superposition, standing waves, resonance, and the Doppler effect for mechanical waves.
Waves carry energy without transporting matter. Understanding wave behavior is the foundation for studying sound, light, and quantum mechanics.
How do waves transfer energy through a medium, and what determines the speed, frequency, and behavior of a wave as it travels and interacts with other waves?
A wave is a disturbance that transfers energy from one place to another without permanently displacing the medium. Mechanical waves require a medium (solid, liquid, or gas), while electromagnetic waves can travel through a vacuum. In this lesson we classify wave types, define the key measurable properties of a wave, derive the wave-speed equation, explore what happens when two waves occupy the same space simultaneously (superposition), and apply the Doppler effect to situations where the source or observer is moving.
Wave Types
Transverse Wave
Particle motion is perpendicular to the direction of wave propagation.
Examples: light (EM waves), waves on a string, seismic S-waves.
Key features: crests (peaks) and troughs; amplitude A measured from equilibrium to crest.
Longitudinal Wave
Particle motion is parallel to the direction of wave propagation.
Examples: sound waves in air, seismic P-waves, compression waves in a spring.
Key features: compressions (high pressure) and rarefactions (low pressure).
Standing Waves — Harmonics (String Fixed at Both Ends)
For a string of length L with wave speed v, only certain frequencies produce standing waves. The condition is that an integer number of half-wavelengths fits in the string length.
| Harmonic (n) | Name | Wavelength λₙ | Frequency fₙ | Nodes / Antinodes |
|---|---|---|---|---|
| 1 | Fundamental | 2L | v / 2L | 2 / 1 |
| 2 | 2nd harmonic (1st overtone) | L | 2v / 2L = v/L | 3 / 2 |
| 3 | 3rd harmonic (2nd overtone) | 2L/3 | 3v / 2L | 4 / 3 |
| 4 | 4th harmonic (3rd overtone) | L/2 | 4v / 2L = 2v/L | 5 / 4 |
General formula: fₙ = nv / 2L, where n = 1, 2, 3, … Nodes are points of zero displacement; antinodes are points of maximum displacement.
Key Equations
Wave speed
v = fλ
v in m/s, f in Hz, λ in m
Period ↔ frequency
T = 1 / f
T in seconds, f in Hz
Wave speed on string
v = √(T / μ)
T = tension (N), μ = linear density (kg/m)
Harmonics (fixed ends)
fₙ = nv / 2L
n = 1, 2, 3, …; L = string length (m)
Doppler effect
f' = f · (v ± v₀) / (v ∓ vₛ)
Upper signs: observer/source approaching
Worked Examples
A wave has a wavelength of λ = 0.4 m and a frequency of f = 25 Hz. Calculate the wave speed.
Identify the known values: λ = 0.4 m, f = 25 Hz.
Apply the wave speed equation: v = fλ.
Substitute: v = 25 Hz × 0.4 m.
Calculate: v = 10 m/s.
A guitar string has a tension T = 80 N and a linear density μ = 0.005 kg/m. Find the wave speed on the string.
Identify knowns: T = 80 N, μ = 0.005 kg/m.
Apply the wave speed on a string formula: v = √(T / μ).
Substitute: v = √(80 / 0.005) = √16 000.
Calculate: √16 000 = 126.49 m/s ≈ 126.5 m/s.
A string of length L = 0.6 m is fixed at both ends. The wave speed on the string is v = 240 m/s. Find the fundamental frequency f₁ and the next two harmonics f₂ and f₃.
Use the harmonic formula: fₙ = nv / 2L.
Fundamental (n = 1): f₁ = (1 × 240) / (2 × 0.6) = 240 / 1.2 = 200 Hz.
2nd harmonic (n = 2): f₂ = (2 × 240) / 1.2 = 480 / 1.2 = 400 Hz.
3rd harmonic (n = 3): f₃ = (3 × 240) / 1.2 = 720 / 1.2 = 600 Hz.
Notice each harmonic is an integer multiple of f₁: f₂ = 2f₁, f₃ = 3f₁.
An ambulance siren emits a frequency of f = 800 Hz. The ambulance moves toward a stationary observer at vₛ = 30 m/s. The speed of sound is v = 340 m/s. What frequency does the observer hear?
Identify: f = 800 Hz, vₛ = 30 m/s (source approaching), v₀ = 0 (observer stationary), v = 340 m/s.
Apply the Doppler formula for approaching source: f' = f × v / (v − vₛ).
Substitute: f' = 800 × 340 / (340 − 30) = 800 × 340 / 310.
Calculate numerator: 800 × 340 = 272 000.
Divide: 272 000 / 310 ≈ 877.4 Hz.
Two waves travel along the same string. Each has amplitude A = 3 cm. Describe the resultant amplitude when they interfere (a) constructively and (b) destructively. What phase difference produces each case?
Apply the superposition principle: the resultant displacement is the algebraic sum of the individual displacements.
(a) Constructive interference: waves are in phase (phase difference = 0°, 360°, 720°, … or path difference = 0, λ, 2λ, …).
Resultant amplitude = A₁ + A₂ = 3 cm + 3 cm = 6 cm.
(b) Destructive interference: waves are exactly out of phase (phase difference = 180°, 540°, … or path difference = λ/2, 3λ/2, …).
Resultant amplitude = |A₁ − A₂| = |3 − 3| = 0 cm (complete cancellation).
Note: partial interference occurs for all other phase differences, giving amplitudes between 0 and 6 cm.
Guided Practice
A water wave has a frequency of 3 Hz and a wavelength of 2.5 m. Calculate the wave speed.
Hint: Use v = fλ. Multiply frequency (Hz) by wavelength (m) to get speed in m/s.
A violin string has tension T = 50 N and linear density μ = 0.002 kg/m. What is the wave speed on the string?
Hint: Use v = √(T/μ). Divide tension by linear density first, then take the square root.
A string of length L = 0.8 m fixed at both ends has a wave speed of 160 m/s. What is the frequency of the 4th harmonic?
Hint: Use fₙ = nv/2L with n = 4. Substitute L = 0.8 m and v = 160 m/s.
A train horn emits 600 Hz. The train moves away from a stationary observer at 20 m/s. Speed of sound = 340 m/s. What frequency does the observer hear?
Hint: Source is moving away, so use f' = f × v/(v + vₛ). The denominator increases because the source is receding.
Two waves on a string have amplitudes of 5 cm and 2 cm. What are the maximum and minimum possible resultant amplitudes when they overlap?
Hint: Maximum (constructive): add the amplitudes. Minimum (destructive): subtract the smaller from the larger.
Key Vocabulary
Transverse wave
A wave in which the particle displacement is perpendicular to the direction of wave propagation.
Example: A wave on a guitar string; light waves.
Longitudinal wave
A wave in which the particle displacement is parallel to the direction of wave propagation, creating compressions and rarefactions.
Example: Sound waves traveling through air.
Wavelength (λ)
The distance between two consecutive points that are in phase (e.g., crest to crest or compression to compression). Measured in metres.
Example: A wave with λ = 0.4 m has crests 0.4 m apart.
Frequency (f)
The number of complete wave cycles passing a point per second. Measured in hertz (Hz = cycles/s).
Example: A 440 Hz sound wave completes 440 cycles every second.
Wave speed (v)
The speed at which the wave pattern (phase) travels through the medium. v = fλ.
Example: Sound travels at ~340 m/s in air at room temperature.
Superposition
The principle that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the individual displacements.
Example: Two pulses meeting on a string add together momentarily.
Standing wave
A wave pattern formed by the superposition of two identical waves traveling in opposite directions, producing fixed nodes and antinodes.
Example: A vibrating guitar string forms a standing wave between its fixed ends.
Resonance
The phenomenon where a system vibrates with maximum amplitude when driven at one of its natural (resonant) frequencies.
Example: Blowing across a bottle opening excites its resonant frequency.
Workbook Check
Interactive Practice — 5 Questions
A wave has frequency 10 Hz and wavelength 3 m. What is its speed?
In a longitudinal wave, how do particles move relative to the wave direction?
A string fixed at both ends has length L = 1 m and wave speed v = 200 m/s. What is the fundamental frequency?
Two waves with equal amplitudes interfere destructively. What is the resultant amplitude?
A source emitting 500 Hz moves toward a stationary observer at 17 m/s. Speed of sound = 340 m/s. What does the observer hear?
Independent Practice
A wave has a frequency of 440 Hz and a wavelength of 0.78 m. Calculate the wave speed.
A string fixed at both ends has a length of 0.60 m. Find the wavelengths of the first three harmonics.
A transverse wave has amplitude 0.04 m, frequency 5.0 Hz, and wavelength 2.0 m. Write the wave equation and find the wave speed.
Two waves on a string have the same frequency and amplitude but are 180° out of phase. Describe the resulting interference pattern and calculate the resultant amplitude.
★ Two speakers emit sound at 680 Hz in phase. They are placed 2.0 m apart. A listener stands 4.0 m directly in front of one speaker. (a) Calculate the path difference. (b) Determine whether the listener hears constructive or destructive interference. (c) Find the nearest position along the line parallel to the speakers where destructive interference occurs. (v_sound = 340 m/s)
ChallengeCommon Mistakes
Thinking the medium moves in the direction of wave propagation for all waves
Transverse waves: medium oscillates perpendicular to propagation (e.g., light, string waves). Longitudinal waves: medium oscillates parallel to propagation (e.g., sound)
Confusing the speed of the wave with the speed of the medium particles
Wave speed v = fλ is the speed of energy propagation. The medium particles oscillate back and forth but don't travel with the wave
Forgetting that wave speed depends on the medium, not the frequency
In a given medium, v is fixed. If frequency changes, wavelength adjusts: λ = v/f. Frequency is set by the source, not the medium
Applying the standing wave formula without identifying whether the ends are nodes or antinodes
Fixed ends → nodes at both ends: λ_n = 2L/n. Open ends → antinodes at both ends: same formula. Mixed (one fixed, one open): λ_n = 4L/(2n−1), odd harmonics only
Math Tips
Wave equation: v = fλ. Rearrange as needed: f = v/λ, λ = v/f. Period T = 1/f. These four quantities are always linked
Standing waves on a string (fixed both ends): f_n = nv/2L for n = 1, 2, 3, ... The fundamental (n=1) has the lowest frequency; harmonics are integer multiples
Wave speed on a string: v = √(F_T/μ) where F_T is tension and μ = m/L is linear mass density. Higher tension or lower mass density → faster wave
Superposition: when two waves overlap, displacements add algebraically. Constructive interference: amplitudes add (path difference = nλ). Destructive: amplitudes cancel (path difference = (n+½)λ)