C.4 · IB Physics
Standing waves on a string
A string fixed at one or both ends can only vibrate in certain patterns. Choose the harmonic, the length and the wave speed, and see where the nodes and antinodes sit and which frequency each pattern needs.
Predict first
Start from the default settings (fixed at both ends, n = 2, v = 40 m s⁻¹, L = 1.0 m). If you raise the mode n from 2 to 4, what happens to the frequency?
For a string fixed at both ends f = nv ÷ 2L, so with v and L unchanged the frequency is proportional to n. Going from n = 2 to n = 4 doubles it, from 40 Hz to 80 Hz.
Check it in the model: Move the Mode, n slider from 2 to 4 and compare the Frequency readout.
string wave moving right wave moving left · nodes (dots)
fixed–free: λ = 4L ÷ (2n − 1), f = (2n − 1)v ÷ 4L
v = fλ, T = 1 ÷ f
y = A sin(kx) cos(ωt), k = 2π ÷ λ
Nodes (including fixed ends):
Antinodes:
Wavelength:
Frequency:
Period:
Try this
- With the fixed–fixed string at L = 1.0 m and v = 40 m s⁻¹, step n from 1 to 4. How do the frequency and the number of antinodes change?
- Tick "Show the two travelling waves" and pause at different moments. When are the two travelling waves in step at an antinode, and when do they cancel?
- Switch to fixed–free with n = 1 and compare its frequency with the fixed–fixed fundamental of the same L and v. Then list the first three frequencies of the fixed–free case.
Common error: describing a standing wave as a wave that "stays in one place" with energy trapped at the nodes. It is the superposition of two identical waves travelling in opposite directions; the nodes never move, and no net energy is carried along the string.