Theory — Standing Waves on a String
Waves on a String
A disturbance sent along a stretched string travels at a wave speed v that depends on how tightly the string is stretched. A tighter (higher-tension) string carries waves faster. The wave speed links the frequency f (how often the string vibrates) and the wavelength λ (the length of one full wave).
v = wave speed (m/s), f = frequency (Hz), λ = wavelength (m)
T = tension (N), μ = mass per unit length (kg/m)
Standing Waves and Resonance
When a string is fixed at both ends and driven at just the right frequency, the wave reflected from the far end overlaps the incoming wave so that the pattern appears to stand still. Points that never move are nodes; points that swing with the largest amplitude are anti-nodes. This only happens at special resonant frequencies; at any other frequency the reflections do not line up and the string looks messy.
Harmonics
For a string of length L fixed at both ends, a whole number of half-wavelengths must fit between the ends. This gives a series of allowed patterns called harmonics. The n-th harmonic has n anti-nodes.
Frequency: f_n = n · v / (2L) = n · f₁
n = 1, 2, 3, … (harmonic number = number of anti-nodes)
Because f_n = n·f₁, the resonant frequencies are evenly spaced: the second harmonic is twice the first, the third is three times the first, and so on. A graph of frequency against harmonic number is therefore a straight line through the origin whose slope is the fundamental frequency f₁.
Off resonance
If the frequency is slightly wrong, reflections don't line up — the pattern blurs and shrinks instead of standing still.
Higher harmonic
More anti-nodes, shorter wavelength (λ = 2L/n), higher frequency (f = n·f₁).
More tension
Faster waves (v = √(T/μ)), so every resonant frequency rises.
| Quantity | Relationship | Notes |
|---|---|---|
| Wave speed | v = fλ = √(T/μ) | same for all harmonics on one string |
| Wavelength of harmonic n | λ_n = 2L/n | 2nd harmonic: λ = L |
| Frequency of harmonic n | f_n = n·v/(2L) | f_n = n·f₁ |
| Anti-nodes | n | humps in the pattern |
| Nodes (incl. ends) | n + 1 | points that stay still |
Instructions — Running the Virtual Experiment
The Pulse tab lets you measure the wave speed by timing a single pulse; the Harmonics tab lets you drive the string and lock in standing-wave patterns. Record every reading in your lab report, with screenshots of each pattern.
Simulation — Waves on a String
Tension & pulse
Drive the string
| Harmonic n | Anti-nodes | Resonant f (Hz) | Wavelength λ = 2L/n (m) | v = fλ (m/s) |
|---|---|---|---|---|
| Lock onto each harmonic (or use the buttons) to fill this table. | ||||
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission, and remember to include labelled screenshots of each standing-wave pattern.
Exploring Standing Waves and String Harmonics
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To measure the speed of a wave on a string by timing a single pulse, to find the first three harmonics of the string fixed at both ends, to measure the frequency and wavelength of each harmonic, and to verify the relationship v = fλ. The experiment also examines how string tension affects the wave speed and the resonant frequencies.
Theory
A wave on a string travels at speed v = √(T/μ) and obeys v = fλ. For a string of length L fixed at both ends, standing waves occur only at resonant frequencies for which a whole number of half-wavelengths fits on the string, giving λ_n = 2L/n and f_n = n·v/(2L) = n·f₁. The n-th harmonic has n anti-nodes.
λ_n = 2L/n · f_n = n·v/(2L) = n·f₁ (L = 1.00 m)
Part A — Wave Speed (Low tension)
Measurement: pulse travel time t = 0.0833 s over length L = 1.00 m
Wave speed: v = L/t = 1.00 / 0.0833 = 12.0 m/s (3 s.f.)
Uncertainty: with a reaction-time uncertainty of about ±0.02 s in the stopwatch reading, v is uncertain by roughly ±3%.
Part B — Harmonics (Results Table, L = 1.00 m, Low tension)
| Harmonic n | Anti-nodes | Frequency f (Hz) | Wavelength λ = 2L/n (m) | v = fλ (m/s) |
|---|---|---|---|---|
| 1 | 1 | 6.00 | 2.00 | 12.0 |
| 2 | 2 | 12.0 | 1.00 | 12.0 |
| 3 | 3 | 18.0 | 0.667 | 12.0 |
A graph of frequency against harmonic number was a straight line through the origin with slope f₁ = 6.00 Hz.
Discussion
The product fλ was 12.0 m/s for all three harmonics, matching the wave speed measured from the pulse in Part A and confirming v = fλ. The resonant frequencies were in the ratio 1 : 2 : 3, so frequency was directly proportional to harmonic number, with slope equal to the fundamental frequency. The second-harmonic wavelength equalled the string length, as expected from λ = 2L/n.
Slightly off a resonant frequency, the reflected and incoming waves no longer reinforced consistently, so the pattern blurred and shrank rather than standing still — only at the resonant frequencies do the reflections superimpose to give stable nodes and anti-nodes. When the tension was raised to High, the wave speed increased (v = √(T/μ)), so a higher frequency was needed to reach the same harmonic. Of the three measured quantities, time was the hardest to measure accurately because of reaction-time error with the stopwatch; frequency and length were comparatively easy to read.
Conclusion
The wave speed on the string was 12.0 m/s, consistent between the pulse method and the v = fλ check across all three harmonics. The harmonics followed f_n = n·f₁ and λ_n = 2L/n, and raising the tension raised the resonant frequencies, confirming v = √(T/μ).
Practice Questions
Show all work and include units. Assume a string fixed at both ends unless told otherwise.