Difficulty: Intermediate | Prerequisites: SHM study notes, trigonometric identities (sum-to-product), basic algebra
This material extends SHM from a single oscillator to disturbances that travel through a medium. Waves on strings are the starting model, and every principle here (superposition, reflection, standing waves) applies again when the course moves to sound and light. If you are comfortable with how a single mass on a spring moves sinusoidally, the leap is understanding that a wave is many such oscillators coupled together. You need to be solid on angular frequency, wavelength, and the basic wave equation before tackling interference and the Doppler effect.
Waves carry energy through a medium without transporting matter. Their speed is set by the medium's properties, while frequency is set by the source. When waves overlap they superpose, and two identical counter-travelling waves produce a standing wave. Beats arise from two slightly different frequencies, and the Doppler effect shifts the observed frequency when source or observer moves.
Transverse wave
A wave in which the particles of the medium oscillate perpendicular to the direction of wave travel. Waves on a string are transverse.
Longitudinal wave
A wave in which the particles oscillate parallel to the direction of wave travel. Sound waves in air are longitudinal.
Wavelength (λ)
The distance between two consecutive points in phase (e.g. crest to crest). In simple terms, it is the spatial length of one full cycle of the wave.
Wave speed (v)
The speed at which the wave pattern moves through the medium: v = λf. For a string, v = √(T/μ), where T is tension and μ is linear mass density. The wave speed depends only on the medium, not on the source.
Superposition
When two or more waves overlap, the resulting displacement is the sum of the individual displacements. Think of it as: waves just add up wherever they meet, then carry on unchanged.
Standing wave
A pattern formed by two identical waves travelling in opposite directions. The result oscillates in place with nodes (zero displacement) and antinodes (maximum displacement). There is no net energy transport.
Node
A point on a standing wave that never moves. Nodes are spaced half a wavelength apart.
Antinode
A point on a standing wave with maximum oscillation amplitude, located midway between nodes.
Constructive interference
When two waves arrive in phase and their amplitudes add together, producing a larger displacement.
Destructive interference
When two waves arrive out of phase and their amplitudes partially or fully cancel.
Beat frequency
The number of amplitude pulsations per second when two waves of slightly different frequencies overlap: f_beat = |f₁ - f₂|. In simple terms, beats are the slow "wah-wah" you hear when two similar notes are slightly out of tune.
Doppler effect
The shift in observed frequency caused by relative motion between source and observer. Moving towards each other raises the frequency; moving apart lowers it.
Fixed end (reflection)
When a wave pulse hits a boundary that cannot move (e.g. a wall), it reflects inverted (flipped vertically) and reversed in time (flipped horizontally).
Free end (reflection)
When a wave pulse hits a boundary that is free to move (e.g. a ring on a frictionless rod), it reflects without vertical inversion, only reversed left to right.
The source determines the period (and therefore the frequency) of the wave. The medium determines the wave speed.
Wave speed on a string: v = √(T/μ), where T is tension and μ is linear mass density (mass per unit length). This depends only on the string's properties.
The maximum transverse speed of a particle on the string (v_max = Aω) depends on both the source (which sets A and ω) and the medium (which, together with ω, sets the wavelength).
Since v = λf, and f is fixed by the source, a change in wave speed (different medium) changes the wavelength.
Fixed end: the reflected pulse is inverted vertically (flipped upside down) and reversed left to right (travels back the way it came). Both the spatial shape and the vertical displacement are flipped.
Example: a pulse with a hump travelling rightward toward a wall comes back as a dip travelling leftward.
Free end: the reflected pulse is reversed left to right only. There is no vertical inversion.
Example: a hump travelling rightward returns as a hump travelling leftward.
To determine the boundary type from before-and-after snapshots, check whether the reflected pulse is inverted relative to the incident one. If inverted, it is a fixed end. If the same orientation, it is a free end.
A standing wave forms when two waves with the same amplitude, frequency, and wavelength travel in opposite directions.
The mathematical requirement: two terms like A sin(kx - ωt) + A sin(kx + ωt). These must have the same k, the same ω, and the same amplitude.
A sin(3x - 2t) + A sin(3x + 2t) produces a standing wave (same k = 3, same ω = 2, same amplitude).
A sin(3x - 2t) - A sin(3x + 2t) also works, since a sign flip is equivalent to a phase shift of π.
Waves with different k values, different ω values, or different amplitudes do not produce a standing wave.
Using the trig identity: A sin(kx - ωt) + A sin(kx + ωt) = 2A sin(kx) cos(ωt). This is a standing wave because the spatial part sin(kx) and the temporal part cos(ωt) are separated.
When a wave passes from one medium to another (e.g. a light string spliced to a heavy string), the frequency stays the same.
The wave speed changes because the medium changes. Since v = λf and f is constant, the wavelength must change in proportion to the speed.
Example: if the wave speed in the heavier string is half that in the lighter string, the wavelength in the heavier string is also half.
Part of the wave is transmitted and part is reflected at the boundary.
When a spring is stretched, its tension increases and its linear mass density decreases (same total mass spread over a longer length).
If equilibrium length is L and you stretch to length nL, the tension is proportional to the elongation (n - 1), and the mass density becomes μ₀/n (where μ₀ is the original mass per unit length at natural length).
The wave speed on the stretched spring: v = √(T/μ). Both T and μ change with stretching.
The time for a pulse to traverse the spring: t = length / v.
Example: stretch to 2L gives T = T₀, μ = μ₀/2. Stretch to 3L gives T = 2T₀, μ = μ₀/3. Computing the ratio t_B/t_A = √(3/2).
Beats occur when two waves of slightly different frequencies overlap. The combined wave oscillates at the average frequency with an amplitude that modulates at the beat frequency.
Beat frequency: f_beat = |f₁ - f₂|.
Given the average frequency and the beat frequency, you can solve for both individual frequencies.
Example: average = 3500 Hz, 4 beats every 2 seconds means f_beat = 2 Hz. So f₁ - f₂ = 2 and f₁ + f₂ = 7000. Solving: f₁ = 3501 Hz, f₂ = 3499 Hz.
For a stationary source and moving observer, or vice versa, the observed frequency shifts.
For a source at rest and observer (reflector) moving away at speed v, the frequency received by the moving object is f₁ = f₀(c - v)/c.
If that object reflects the wave back, it acts as a new source moving at speed v, and the frequency received back at the original source is f₂ = f₁ × c/(c + v).
Combined (echo Doppler): f_detected = f₀(c - v)/(c + v).
Example: sonar at 120 kHz, submarine moving away at 12.0 m/s, sound speed in water 1480 m/s. f = 120 × (1480 - 12)/(1480 + 12) ≈ 118.1 kHz.
When two pulses meet on a string, their displacements add at every point (superposition principle). After passing through each other, they continue unchanged.
If the pulses are on opposite sides of the string (one up, one down), they destructively interfere where they overlap. A pulse extending upward and one extending downward can cancel at the overlap region.
Both pulses travel at the same speed if the medium is uniform, so they meet at the midpoint between their starting positions if launched simultaneously (or at a predictable location given the timing).
v = \lambda f = \frac{\omega}{k}v = \sqrt{\frac{T}{\mu}} \quad \text{(wave on a string)}y(x,t) = A\sin(kx - \omega t) \quad \text{(travelling wave, +x direction)}A\sin(kx - \omega t) + A\sin(kx + \omega t) = 2A\sin(kx)\cos(\omega t) \quad \text{(standing wave)}f_{\text{beat}} = |f_1 - f_2|f_{\text{detected}} = f_0 \frac{c \pm v_o}{c \mp v_s} \quad \text{(Doppler effect, general form)}f_{\text{echo}} = f_0 \frac{c - v}{c + v} \quad \text{(echo Doppler, target moving away)}Beats are how piano tuners work: they play a reference pitch alongside the string and listen for the slow pulsation to disappear, which means the frequencies have matched. The Doppler effect is the principle behind speed cameras, weather radar, and medical ultrasound (measuring blood flow velocity). Standing waves on strings are what make musical instruments produce specific notes, with the string's length and tension setting the harmonic series.
Students often think the frequency changes when a wave enters a new medium. It does not. Frequency is set by the source and stays constant across media boundaries. Only wavelength and speed change.
Students often think a standing wave requires two waves with opposite signs (subtraction). Addition of two counter-travelling waves also works. Both A sin(kx - ωt) + A sin(kx + ωt) and A sin(kx - ωt) - A sin(kx + ωt) give standing waves.
Students often think a fixed-end reflection only inverts the pulse vertically. It also reverses the pulse in time (horizontally). So an asymmetric pulse changes shape as well as flipping.
Students often confuse beat frequency with the number of beats. If you hear 4 beats in 2 seconds, the beat frequency is 2 Hz, not 4 Hz.
⚠️ Wave speed depends on the medium only (tension and mass density for a string). The source sets frequency and amplitude.
⚠️ Frequency is conserved across a boundary between two media. This is tested frequently. If speed halves, wavelength halves.
⚠️ For standing waves, check that both waves have the same k, same ω, and same amplitude. Different k or ω values do not produce a standing wave.
⚠️ Fixed-end reflection inverts both vertically and horizontally. Free-end reflection inverts only horizontally (left-right flip). Be able to draw the reflected pulse.
⚠️ Doppler problems involving reflection (sonar, radar) require two applications of the Doppler formula: once for the signal reaching the target, once for the echo returning.
⚠️ Beat problems: "4 beats every 2 seconds" means f_beat = 2 Hz, not 4. Read the wording carefully.
True or false: when a wave passes from a light string to a heavy string, the frequency changes. (False)
Fill in the blank: a pulse reflecting off a fixed end is inverted ______ and ______. (vertically and horizontally)
True or false: 4 sin(3x - 2t) + 8 sin(3x + 2t) produces a standing wave. (False, different amplitudes)
Fill in the blank: the beat frequency of two tones at 440 Hz and 443 Hz is ______ Hz. (3)
True or false: in a Doppler echo problem, you apply the Doppler formula twice. (True)
Q: A harmonic wave travels along a light string and enters a heavier string where the speed is one third of the original. What happens to the frequency and wavelength in the heavier string?
A: The frequency stays the same. The wavelength becomes one third of its original value, since v = λf and v has been reduced to one third.
Q: Two fire alarms produce sound waves. You hear 6 beats every 3 seconds, and the average of the two frequencies is 2000 Hz. What are the two frequencies?
A: Beat frequency = 6/3 = 2 Hz. So f₁ - f₂ = 2 and f₁ + f₂ = 4000. Solving: f₁ = 2001 Hz, f₂ = 1999 Hz.
Q: A sonar gun emits sound at 100 kHz toward a submarine moving away at 10 m/s. The speed of sound in water is 1500 m/s. What frequency does the sonar gun detect in the returning echo?
A: f = f₀(c - v)/(c + v) = 100 × (1500 - 10)/(1500 + 10) = 100 × 1490/1510 ≈ 98.7 kHz.
Q: Which of the following produces a standing wave: 3 sin(2x - 4t) + 3 sin(2x + 4t), or 3 sin(2x - 4t) + 3 sin(4x + 4t)?
A: Only the first one. Both waves must have the same wave number k and the same angular frequency ω. In the second expression, the wave numbers differ (k = 2 vs k = 4).
Q: A pulse travels rightward along a spring. At t = 0 the pulse is a triangle pointing up. At t = 3 s, the pulse has reflected and is a triangle pointing up travelling leftward. Is the boundary a fixed end or a free end?
A: Free end. The pulse was not inverted vertically, only reversed left to right. A fixed end would have inverted the pulse (triangle pointing down).
The superposition principle here is the same one that governs interference in optics (double-slit experiment, thin-film interference). The Doppler effect extends to electromagnetic waves and underpins everything from redshift in astronomy to radar speed guns. Standing waves on strings lead directly into resonance in air columns (open and closed pipes), which is the next unit in most PHYS 123 courses.
Waves on a string, transverse wave, longitudinal wave, wave speed, wavelength, frequency, period, wave number, superposition principle, constructive interference, destructive interference, standing wave, node, antinode, fixed end reflection, free end reflection, pulse reflection, pulse inversion, beats, beat frequency, Doppler effect, Doppler shift, sonar, echo Doppler, wave transmission, boundary conditions, linear mass density, tension, medium properties, PHYS 123, UW physics