Difficulty: Intermediate | Prerequisites: Oscillations and SHM (see companion study notes), basic trigonometry, logarithms
Waves carry energy from one place to another without transporting matter. This unit extends SHM to systems where the oscillation propagates through a medium, whether that medium is a stretched string, a column of air, or the surface of the ocean. Understanding wave speed, superposition, and standing waves gives you the tools to analyse musical instruments, architectural acoustics, and medical ultrasound. You will also meet the decibel scale and the Doppler effect, both of which appear in everyday life and on exams with reliable frequency.
Wave speed on a string depends on tension and linear mass density. Standing waves form at specific frequencies determined by the length of the string and its boundary conditions. Sound intensity drops with the square of the distance, the decibel scale is logarithmic, and the Doppler effect shifts observed frequency when source or observer moves.
Wavelength (lambda)
The distance between two consecutive points in phase (crest to crest, or trough to trough), measured in metres.
Wave speed (v)
The speed at which a wave crest (or any point of constant phase) moves through the medium. For a string: v = sqrt(T_s / mu), where T_s is tension and mu is linear mass density.
Linear mass density (mu)
Mass per unit length of a string, in kg/m. Think of it as how heavy the string is per metre. A heavier string carries waves more slowly.
Standing wave
A wave pattern that forms when two identical waves travel in opposite directions and interfere. The result looks like it is vibrating in place rather than travelling.
Node
A point on a standing wave that never moves (zero displacement at all times).
Antinode
A point on a standing wave where the displacement is maximum.
Harmonic / mode number (m)
The number of half-wavelengths that fit along the string (for a string fixed at both ends). The first harmonic (m = 1) is the fundamental.
Beat frequency
The number of beats per second heard when two waves of slightly different frequencies overlap. f_beat = |f_1 - f_2|. In simple terms, beats are the slow pulsing you hear when two notes are almost but not quite the same pitch.
Doppler effect
The change in observed frequency (or wavelength) when the source, the observer, or both are in motion relative to the medium.
Sound intensity (I)
Power per unit area, in W/m^2. It decreases with the square of the distance from a point source.
Sound intensity level (beta)
A logarithmic measure of sound intensity in decibels (dB): beta = 10 log_10(I / I_0), where I_0 = 1.0 x 10^-12 W/m^2.
The speed of a wave on a stretched string is:
v = sqrt(T_s / mu)
where T_s is the tension in the string and mu is the linear mass density. Increasing tension makes waves faster. Increasing mu (heavier string) makes them slower.
When you know the tension and wave speed for two different strings, you can find the ratio of their linear mass densities.
From v = sqrt(T_s / mu), solve for mu: mu = T_s / v^2.
Ratio: mu_A / mu_B = (T_A / v_A^2) / (T_B / v_B^2)
Example: String A has tension 3T and speed v. String B has tension T and speed 4v.
mu_A / mu_B = (3T / v^2) / (T / 16v^2) = (3T / v^2)(16v^2 / T) = 48.
A sinusoidal travelling wave moving in the positive x-direction is described by:
y(x, t) = A sin(kx - omega t + phi) or equivalently y(x, t) = A cos(kx - omega t + phi)
where k = 2 pi / lambda is the wave number and omega = 2 pi f.
The wave speed relates wavelength and frequency: v = lambda f = omega / k.
To find the vertical position of a boat or particle on a wave at a later time, you need the full wave function. Find the initial phase from the initial conditions (position and velocity at t = 0), then evaluate at the later time.
When a wave reflects off a fixed boundary and interferes with the incoming wave, a standing wave can form. For a string of length L fixed at both ends:
lambda_m = 2L / m, where m = 1, 2, 3, ...
f_m = m v / (2L)
The mode number m equals the number of antinodes (or the number of half-wavelengths fitting on the string).
To identify the harmonic from a picture of a standing wave, count the number of antinodes (the bulges). That count is the mode number m.
Example: a 2.0 m string shows 5 antinodes. So m = 5, and lambda = 2L/m = 2(2.0)/5 = 0.80 m.
Be careful: some students count nodes instead of antinodes. For a string fixed at both ends, the number of nodes is m + 1 (including the two endpoints).
For a point source radiating uniformly, intensity falls off with distance squared:
I = P / (4 pi r^2)
If you know I at distance r_1, then at distance r_2: I_2 = I_1 (r_1 / r_2)^2.
Example: I = 14.0 x 10^-3 W/m^2 at 25.0 m. At 50.0 m: I = (14.0 x 10^-3)(25/50)^2 = 3.50 x 10^-3 W/m^2.
beta = 10 log_10(I / I_0), where I_0 = 1.0 x 10^-12 W/m^2.
Example: I = 3.50 x 10^-3 W/m^2. beta = 10 log_10(3.50 x 10^-3 / 10^-12) = 10 log_10(3.50 x 10^9) = 10 (9.544) = 95.4 dB, so approximately 95 dB.
Doubling the distance from a point source reduces intensity by a factor of 4, which corresponds to a drop of about 6 dB.
The perceived wavelength or frequency changes when the source, observer, or both move relative to the medium.
Observer moving away from a stationary source: the observed wavelength increases (lambda_obs = lambda + v_obs / f = lambda (1 + v_obs / v_sound)).
Source moving away from a stationary observer: the observed wavelength increases (lambda_obs = lambda + v_source / f).
Source moving toward a stationary observer: the observed wavelength decreases.
For a perceived wavelength of 1.08 lambda (8% longer), the source must be moving away from the observer. The source velocity that produces this: v_source = 0.08 v_sound. With v_sound = 343 m/s, that gives v_source = 0.08 (343) = 27.4 m/s. Among typical answer choices, 25 m/s is closest (the source moving away).
When two waves with frequencies f_1 and f_2 are played together:
f_beat = |f_1 - f_2|
If a violin string is "too tightly stretched" and beats at 3.00 Hz with a 440 Hz tuning fork, the string frequency is 443 Hz (above the fork, since the string is too tight).
The period of the string oscillation: T = 1/443 = 2.26 x 10^-3 s = 2.26 ms.
When two pulses or waves occupy the same region of a medium, the total displacement at each point is the sum of the individual displacements. After the pulses pass through each other, they continue unchanged in shape and speed.
If two pulses travel toward each other on the same string:
At any instant, add the displacements point by point.
After they pass through each other, each pulse continues as if the other were never there.
If one pulse is upright and the other is inverted, they partially or fully cancel during overlap.
If you create the same hand motion but on a string with lower wave speed (lower tension or higher linear mass density):
The pulse travels more slowly, so at the same time t_3 it has not moved as far.
The pulse is shorter (compressed along the direction of travel) because the hand moved at the same rate but the wave carried each part of the pulse away more slowly.
The amplitude (height) stays the same, because the hand's displacement was the same.
The time for a pulse to pass a fixed point P equals the pulse length divided by the wave speed.
Increasing spring tension increases wave speed. The pulse also gets longer (it is stretched out along the string). Both the length and the speed increase in the same proportion, so the passage time does not change.
Decreasing linear mass density increases wave speed with the same stretching effect, so again the passage time is unaffected.
Decreasing the time to create the pulse (moving the hand faster) makes the pulse shorter without changing the wave speed. This decreases the passage time. This is the quantity the student should change.
v = \sqrt{\frac{T_s}{\mu}}v = \lambda f = \frac{\omega}{k}\lambda_m = \frac{2L}{m}, \quad f_m = \frac{m v}{2L}I = \frac{P}{4\pi r^2}\beta = 10 \log_{10}\!\left(\frac{I}{I_0}\right), \quad I_0 = 1.0 \times 10^{-12}\;\text{W/m}^2f_{\text{beat}} = |f_1 - f_2|\lambda_{\text{obs}} = \lambda \left(1 \pm \frac{v_s}{v_{\text{sound}}}\right)The Doppler effect is the reason an ambulance siren changes pitch as it passes you. Sound intensity and decibel calculations are used in workplace safety regulations to determine acceptable noise exposure times.
Students often confuse nodes and antinodes. A node is where displacement is always zero. An antinode is where it is maximum. For a string fixed at both ends, both endpoints are nodes.
Students sometimes think that doubling the distance halves the sound intensity. It reduces intensity by a factor of four (inverse-square law), not two.
Students often assume the Doppler effect depends on who is moving in the same way regardless of whether it is the source or the observer. It does not: the formulas differ for a moving source and a moving observer, even at the same speed.
Students sometimes count nodes instead of antinodes to find the harmonic number. For a string fixed at both ends, the harmonic number equals the number of antinodes, not the number of nodes.
⚠️ Linear mass density ratio problems are a staple. Set up mu = T/v^2 for each string and take the ratio.
⚠️ Standing wave identification from a diagram appears regularly. Count the antinodes to get m, then use lambda = 2L/m.
⚠️ Decibel calculations require comfort with logarithms. Remember that I_0 = 10^-12 W/m^2.
⚠️ Beat problems often ask for the period (not the frequency) of the higher-frequency source. Do not forget to take 1/f after finding the beat-shifted frequency.
⚠️ Pulse superposition and propagation problems test whether you understand that pulses pass through each other unchanged and that changing wave speed compresses or stretches the pulse shape.
True or false: wave speed on a string increases when you increase the tension. (True.)
Fill in the blank: the harmonic number of a standing wave on a string fixed at both ends equals the number of ____. (Antinodes.)
True or false: doubling the distance from a point sound source reduces the decibel level by 3 dB. (False, it reduces it by about 6 dB.)
Fill in the blank: beat frequency equals ____. (The absolute difference of the two frequencies.)
True or false: after two pulses pass through each other, their shapes are permanently changed. (False, they continue unchanged.)
Q: String A has tension 3T and wave speed v. String B has tension T and wave speed 4v. What is mu_A / mu_B?
A: mu = T/v^2. mu_A/mu_B = (3T/v^2) / (T/16v^2) = 48.
Q: A 2.0 m string carries a standing wave with 5 antinodes. What is the harmonic and wavelength?
A: m = 5, lambda = 2L/m = 2(2.0)/5 = 0.80 m.
Q: Sound intensity is 14.0 x 10^-3 W/m^2 at 25.0 m. What is the sound intensity level at 50.0 m?
A: At 50.0 m: I = (14.0 x 10^-3)(25/50)^2 = 3.50 x 10^-3 W/m^2. beta = 10 log(3.50 x 10^9) = 95 dB.
Q: A violin string is too tight and produces beats at 3.00 Hz with a 440 Hz tuning fork. What is the period of the string's oscillation?
A: f_string = 440 + 3 = 443 Hz. T = 1/443 = 2.26 ms.
Q: A sound source emits wavelength lambda. A student perceives 1.08 lambda. Which scenario is possible: (a) student moving away at 24 m/s, (b) student moving toward at 22 m/s, (c) source moving away at 25 m/s, (d) source moving toward at 26 m/s?
A: The perceived wavelength is longer, so either the observer moves away or the source moves away. For a source moving away: lambda_obs = lambda (1 + v_s/v_sound). 1.08 = 1 + v_s/343, so v_s = 27.4 m/s. The closest answer choice is 25 m/s (option C).
Standing waves on strings lead directly into sound waves in pipes (open and closed tubes), which follow the same harmonic structure with different boundary conditions. The Doppler effect connects to later topics in electromagnetic waves and the redshift of light from distant galaxies. Superposition is the foundation for interference and diffraction, which you will study when the course moves to optics.
Wave speed, wavelength, frequency, wave number, travelling wave, transverse wave, longitudinal wave, linear mass density, string tension, standing wave, node, antinode, harmonic, mode number, fundamental frequency, overtone, superposition, constructive interference, destructive interference, pulse, sound intensity, inverse-square law, decibel, dB, sound intensity level, Doppler effect, Doppler shift, beat frequency, beats, tuning fork, PHYS 116 midterm 1, UW physics 116