Demonstrating stationary waves on a stretched string
A string runs from a mechanical vibrator over a pulley to a hanging mass that sets the tension. A node exists at the pulley because the string is fixed there, and approximately at the vibrator because its amplitude of vibration is very small compared with that of an antinode. Observable stationary waves only appear at resonance, when L = n(λ/2) = n(v/2f), so you must adjust either the frequency of the vibrator or the length of the string.
L = n(λ/2) = n v/(2f) with v = √(T/μ) fixed by the loadWhat this actually means
The pulley node answer is short: the string cannot move where it is held, so its amplitude there is zero. Do not overwrite it.
The vibrator node answer needs the comparison. The vibrator does move, so it is not a true node. Say its amplitude is very small compared with the antinode amplitude, so it is effectively a node.
For the adjust f or L question the logic is: the load fixes the tension, so v is constant; a stationary wave needs a whole number of half wavelengths to fit the length; λ = v/f; therefore only certain (f, L) combinations satisfy the condition. Off resonance the reflected waves do not reinforce and the amplitude stays tiny.
Changing the frequency does not change the wave speed on the string. Only changing the tension or swapping the string for a different mass per unit length does that. Students routinely get this backwards.
The tell-tale in the lab is that the amplitude grows dramatically as you creep up on a resonant frequency and collapses again as you pass it. That is resonance, not a change in wave speed.
Saying the wave speed on the string changes when you change the vibrator frequency. Tension and mass per unit length set the speed.
Prove it — watch it be true
- Select the string system and step the mode selector across n = 1 to n = 5.
- Confirm that only these discrete modes give a clean node-and-loop pattern with a node forced at each end.
- Read the frequency for each mode and confirm it goes as n v/2L, so a fixed length only resonates at a discrete set of frequencies.