Difficulty: Intermediate | Prerequisites: SHM fundamentals (period, frequency, amplitude), trigonometry, logarithms.
Waves and sound build directly on SHM. Once you understand how a single point oscillates, waves describe how that oscillation travels through a medium. This section covers wave speed on strings, the travelling wave equation, standing waves, sound intensity and decibels, the Doppler effect, beats, and pulse propagation. These topics make up roughly half the first midterm for PHYS 116 at UW.
Wave speed on a string depends on tension and linear mass density. Standing waves only form at specific frequencies determined by string length and boundary conditions. Sound intensity falls off with the square of the distance, and decibels are a logarithmic scale. The Doppler effect shifts frequency when source or observer moves, but only source motion changes the wavelength.
Wave speed (v)
The speed at which a wave disturbance travels through a medium. On a string, v = √(T/μ), where T is tension and μ is linear mass density. Not the same as the speed of the particles in the medium.
Wavelength (λ)
The spatial distance between two consecutive points in phase (e.g. crest to crest), measured in metres.
Standing wave
A wave pattern that results from two identical waves travelling in opposite directions on the same medium. The pattern appears stationary, with fixed nodes and antinodes.
Harmonic / mode number (m or n)
The integer that labels which standing wave pattern is present. The mode number equals the number of antinodes on the string.
Node
A point on a standing wave that remains at zero displacement at all times.
Antinode
A point on a standing wave where the displacement reaches its maximum value. In simple terms, it is the point that oscillates the most.
Beats
The periodic variation in loudness heard when two sounds of slightly different frequencies are played together. The beat frequency equals the absolute difference of the two frequencies: f_beat = |f₁ – f₂|.
Sound intensity (I)
Power per unit area carried by a sound wave, measured in W/m². It decreases with the square of the distance from the source.
Sound intensity level (β)
A logarithmic measure of sound intensity relative to a reference: β = (10 dB) log₁₀(I/I₀), where I₀ = 1 × 10⁻¹² W/m². Think of it as a way to compress the enormous range of human hearing into a manageable number.
Doppler effect
The shift in observed frequency (or wavelength) caused by relative motion between a source and an observer. A moving source changes the wavelength of the emitted sound; a moving observer changes the frequency they detect but not the wavelength in the medium.
v = √(T/μ), where T is tension (in newtons) and μ is linear mass density (in kg/m).
To find the ratio of linear mass densities between two strings, rearrange: μ = T/v².
If string A has tension 3T and speed v, and string B has tension T and speed 4v, then μ_A/μ_B = (3T/v²) / (T/(4v)²) = (3T/v²) × (16v²/T) = 48.
For a string fixed at both ends, the allowed wavelengths are λ_n = 2L/n, where n is the mode number (n = 1, 2, 3...).
The mode number equals the number of antinodes visible on the string.
To identify the harmonic from a diagram, count the antinodes (the bumps). Five antinodes on a 2.0 m string: n = 5, so λ = 2(2.0)/5 = 0.80 m.
General form: y(x, t) = A cos(2πx/λ ± 2πt/T).
Use the minus sign when the wave travels in the +x direction; plus when it travels in the –x direction.
For a boat riding a wave, the boat's own motion changes its effective position over time: substitute x = v_boat × t for the boat's horizontal position.
Worked example: a wave with λ = 60 m, v_wave = 7.5 m/s, boat at the crest moving left at 2.0 m/s. After 7.0 s, the argument becomes (2π/60)(–2.0)(7.0) – (2π/60)(7.5)(7.0) = –6.96 rad, and y = 1.2 cos(–6.96) = 0.93 m.
Intensity falls off with distance squared: I₁/I₂ = r₂²/r₁².
To find intensity at a new distance, use I₂ = (r₁²/r₂²) × I₁.
Sound intensity level in decibels: β = (10 dB) log₁₀(I/I₀), where I₀ = 1 × 10⁻¹² W/m².
Worked example: a jackhammer at 25 m gives I = 14.0 × 10⁻³ W/m². At 50 m, I = (25²/50²)(14.0 × 10⁻³) = 3.5 × 10⁻³ W/m². Then β = 10 log₁₀(3.5 × 10⁻³ / 10⁻¹²) = 95 dB.
A moving observer changes the detected frequency but not the wavelength in the medium.
A moving source changes the wavelength in the medium.
If a student perceives the wavelength to be 1.08λ (larger than emitted), the source must be moving away from the student.
The key test: does the perceived wavelength change? If yes, the source is moving.
Beat frequency = |f₁ – f₂|.
If a violin string is too tight and produces 3 beats/s with a 440 Hz tuning fork, the string frequency is 443 Hz (higher tension means higher frequency).
Period of the violin string: T = 1/443 = 2.26 ms.
A pulse moves at the wave speed without changing shape (in a non-dispersive medium).
When two pulses approach each other on the same string, the displacements add (superposition principle). After they pass through each other, each pulse continues unchanged.
Halving the wave speed (by reducing tension) means the pulse travels half as far in the same time, but its shape and duration stay the same.
The time for a pulse to pass a fixed point equals the time it took to create the pulse, regardless of wave speed. To reduce the passing time, reduce the creation time.
v = \sqrt{\frac{T}{\mu}}\mu = \frac{T}{v^2}\lambda_n = \frac{2L}{n}y(x,t) = A\cos\!\left(\frac{2\pi}{\lambda}\,x \pm \frac{2\pi}{T}\,t\right)\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}\beta = (10\;\text{dB})\,\log_{10}\!\left(\frac{I}{I_0}\right)f_{\text{beat}} = |f_1 - f_2|Guitar strings produce different notes because each string has a different linear mass density and tension, which sets the wave speed and therefore the fundamental frequency. Tuning a guitar means adjusting the tension until the string vibrates at the correct frequency.
The inverse-square law for sound intensity explains why a concert sounds much quieter from the back of a large venue. Doubling your distance from the speakers cuts the intensity to one quarter.
Students often confuse wave speed with particle speed. Wave speed is how fast the disturbance travels along the medium. Particle speed is how fast an individual point in the medium moves up and down.
Students assume that changing wave speed changes the time for a pulse to pass a point. It does not. The passing time depends on how long it took to create the pulse.
Students mix up which motion (source or observer) changes wavelength. Only source motion changes the wavelength in the medium. Observer motion changes the detected frequency but not the wavelength.
Students count nodes instead of antinodes when identifying the harmonic number. The mode number equals the number of antinodes, not nodes.
⚠️ Linear mass density ratio problems are a favourite. Set up μ = T/v² for each string and divide carefully.
⚠️ Standing wave problems require you to read a diagram. Count antinodes for the mode number, then apply λ = 2L/n.
⚠️ The travelling wave equation is used with specific numbers. Know which sign to use and how to substitute for a moving observer or object.
⚠️ Decibel problems involve two steps: first find the intensity at the new distance (inverse-square law), then convert to dB. Do not skip the first step.
⚠️ Doppler questions test whether you know that a change in perceived wavelength means the source is moving, not the observer.
True or false: doubling the tension on a string doubles the wave speed. (False, it increases it by a factor of √2)
Fill in the blank: the mode number of a standing wave equals the number of ______. (antinodes)
True or false: moving the observer changes the wavelength of a sound wave in the medium. (False)
Fill in the blank: doubling the distance from a sound source reduces the intensity by a factor of ______. (4)
True or false: the time for a pulse to pass a point depends on the wave speed. (False, it depends on the time to create the pulse)
Q: String A has tension 3T and wave speed v. String B has tension T and wave speed 4v. What is μ_A/μ_B?
A: μ = T/v². So μ_A/μ_B = (3T/v²)/(T/16v²) = (3T/v²)(16v²/T) = 48.
Q: A 2.0 m string shows 5 antinodes. What is the harmonic number and wavelength?
A: m = 5 (count the antinodes). λ = 2L/n = 2(2.0)/5 = 0.80 m.
Q: A jackhammer produces 14.0 × 10⁻³ W/m² at 25.0 m. What is the sound level at 50.0 m?
A: I at 50 m = (25²/50²)(14.0 × 10⁻³) = 3.5 × 10⁻³ W/m². β = 10 log₁₀(3.5 × 10⁻³/10⁻¹²) = 95 dB.
Q: A violin string is too tight and produces 3 beats/s with a 440 Hz fork. What is the period of the string's oscillation?
A: Too tight means higher frequency: 440 + 3 = 443 Hz. T = 1/443 = 2.26 ms.
Q: A source emits wavelength λ. An observer perceives 1.08λ. Which is possible: source moving away, source moving toward, observer moving away, or observer moving toward?
A: The source is moving away from the observer. A change in perceived wavelength requires source motion, and the wavelength is longer (red-shifted), so the source moves away.
Q: A boat at the crest of a 60 m wavelength wave (v = 7.5 m/s) is 1.2 m above equilibrium, moving left at 2.0 m/s. What is the boat's vertical position after 7.0 s?
A: y = 1.2 cos((2π/60)(–2.0)(7.0) – (2π/60)(7.5)(7.0)) = 1.2 cos(–6.96 rad) = 0.93 m.
Q: A pulse takes time Δt to pass point P. Which of the following changes would decrease Δt: increasing spring tension, decreasing linear mass density, or decreasing the time to create the pulse?
A: Only decreasing the time to create the pulse. The passing time equals the creation time and is independent of wave speed (and therefore independent of tension and linear mass density).
Standing waves on strings connect directly to musical acoustics and the physics of wind instruments, where the same harmonic series appears but with different boundary conditions (open vs closed ends).
The inverse-square law for sound intensity is the same mathematical relationship that governs gravitational and electric field strength. If you understand one, you understand the pattern in all three.
The Doppler effect extends to electromagnetic waves (light), where it explains the redshift of distant galaxies, a cornerstone observation in cosmology.
Wave speed, string waves, tension, linear mass density, standing waves, harmonics, mode number, nodes, antinodes, wavelength, travelling wave equation, superposition, pulse propagation, sound intensity, inverse-square law, decibel, dB, sound level, Doppler effect, Doppler shift, wavelength shift, beats, beat frequency, tuning fork, violin string, PHYS 116, University of Washington, general physics, midterm review