Theory — Wavelength and Wave Energy
Waves Carry Energy, Not Matter
A wave transfers energy through the oscillation of a medium without transporting the medium itself. On a string, each piece moves up and down while the wave pattern travels along — or, for a standing wave, stays fixed in place with stationary nodes and antinodes.
Wavelength of a Standing Wave
The wavelength λ is the distance between two consecutive crests or troughs — equivalently, twice the distance between adjacent nodes. For a string of length L fixed at both ends, only certain wavelengths "fit," set by the number of segments n.
L = length of the string (m)
n = number of segments (loops) along the string
v = f · λ (wave speed relates frequency and wavelength)
Energy of a Wave
The energy a wave carries depends on the amplitude, the angular frequency, the wavelength, and the medium's linear mass density. The energy rises with the square of both the amplitude and the frequency.
μ = linear mass density (kg/m)
A = amplitude (m)
ω = angular frequency = 2πf (rad/s)
λ = wavelength (m)
Amplitude
K ∝ A². Doubling the amplitude multiplies the wave energy by four.
Frequency
K ∝ ω² = (2πf)². Doubling the frequency multiplies the energy by four — and shortens the wavelength.
Wavelength
λ = 2L/n. More segments → shorter wavelength → higher frequency for the same wave speed.
| Quantity | Relationship | Notes |
|---|---|---|
| Wavelength | λ = 2L/n | n = number of segments |
| Angular frequency | ω = 2πf | rad/s |
| Wave speed | v = f·λ | set by tension & density |
| Wave energy | K = ¼μA²ω²λ | ∝ A², ∝ f² |
Instructions — Running the Virtual Experiment
The Wavelength tab drives the string into a standing wave so you can measure λ with the ruler and check it against 2L/n; the Wave Energy tab lets you set amplitude and frequency and computes K. Record every reading in your lab report with screenshots.
Simulation — Wave on a String
Drive the string
Amplitude & frequency
| Scenario | A (m) | f (Hz) | ω (rad/s) | λ (m) | μ (kg/m) | K (J) |
|---|---|---|---|---|---|---|
| Set A, f, n and click "Record" to log a row. | ||||||
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission, and include labelled screenshots of the standing wave and ruler for each scenario.
Wavelength and Energy of a Wave
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To measure the wavelength of a standing wave on a string with the ruler, compare it with the theoretical value λ = 2L/n, and calculate the energy the wave carries using K = ¼μA²ω²λ for two scenarios with different amplitude and frequency.
Theory
A standing wave on a string of length L has wavelength λ = 2L/n, where n is the number of segments. The angular frequency is ω = 2πf, and the energy of the wave is K = ¼μA²ω²λ, with μ the linear mass density. The energy is proportional to the square of the amplitude and the square of the frequency.
μ = 0.0015 kg/m, L = 2.0 m
Results
| Scenario | A (m) | f (Hz) | ω (rad/s) | λₑ (m) | λₜ (m) | μ (kg/m) | Kₑ (J) | Kₜ (J) |
|---|---|---|---|---|---|---|---|---|
| 1 | 0.0050 | 1.25 | 7.85 | 2.00 | 2.00 | 0.0015 | 1.16×10⁻⁶ | 1.16×10⁻⁶ |
| 2 | 0.0075 | 2.25 | 14.14 | 2.00 | 2.00 | 0.0015 | 8.43×10⁻⁶ | 8.43×10⁻⁶ |
Sample calc, Scenario 1: ω = 2π(1.25) = 7.85 rad/s; K = ¼(0.0015)(0.0050)²(7.85)²(2.00) = 1.16×10⁻⁶ J.
Discussion
The measured wavelengths matched the theoretical λ = 2L/n closely (2.00 m for n = 2 in both scenarios), with any small differences coming from reading the ruler and judging the exact crest positions on screen. From Scenario 1 to Scenario 2 the amplitude rose by a factor of 1.5 and the frequency by 1.8; the energy increased from 1.16×10⁻⁶ J to 8.43×10⁻⁶ J — a factor of about 7.3, which is close to (1.5)²×(1.8)² = 7.3, confirming that the energy grows with the square of both amplitude and frequency. Higher tension produced cleaner, more stable standing waves, while increasing damping reduced the amplitude over time and made the pattern harder to read.
Conclusion
The experimental and theoretical wavelengths agreed, validating λ = 2L/n, and the calculated wave energy rose with the square of amplitude and frequency, consistent with K = ¼μA²ω²λ. Small differences arose from simulation resolution and human measurement error.
Practice Questions
Show all work and include units. Use λ = 2L/n, ω = 2πf, K = ¼μA²ω²λ, μ = 0.0015 kg/m unless told otherwise.