PhysicsWavesClass 11

Standing waves on a string: interactive simulation

A standing wave forms when two identical waves travel in opposite directions and overlap. Some points never move (nodes) and some swing the most (antinodes).

Useful for: JEE Main, NEET

Try this

Only certain wavelengths fit on a string of fixed length. Change the harmonic number and watch a new loop appear; change the tension and see the frequency rise while the shape stays the same.

  1. 1

    Set T = 100 N, μ = 10 g/m, L = 1 m, both ends fixed.

    What you should see

    v = √(100 / 0.01) = 100 m/s, so the fundamental is 50 Hz and the 3rd harmonic is 150 Hz.

  2. 2

    Make the tension four times bigger.

    What you should see

    v and every frequency double, but the wavelengths do not change — they are set by the length.

  3. 3

    Switch to one end free.

    What you should see

    The free end is always an antinode, and only odd multiples of the fundamental are allowed.

  4. 4

    Count the nodes for n = 4 with both ends fixed.

    What you should see

    5 nodes (including both ends) and 4 antinodes.

Formulas behind it

Wave speed on a string

v=Tμv = \sqrt{\frac{T}{\mu}}

Both ends fixed

fn=nv2L,n=1,2,3,…f_n = \frac{n v}{2L},\quad n = 1, 2, 3, \ldots

One end free

f=(2n−1)v4Lf = \frac{(2n-1) v}{4L}

Node to next node

λ2\frac{\lambda}{2}

Common mistakes

  • Using μ in g/m without converting to kg/m.
  • Saying a string with one free end has a 2nd harmonic. Only odd harmonics exist there.
  • Calling the distance between two nodes λ instead of λ/2.
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FAQ

Does energy travel along a standing wave?

No net energy flows. Energy sloshes back and forth between kinetic and potential inside each loop.

Why do points in the same loop move together?

All points between two neighbouring nodes are in the same phase. Points in neighbouring loops are opposite in phase.

How is this related to organ pipes?

An open pipe behaves like a string fixed at both ends (all harmonics); a pipe closed at one end behaves like one free end (odd harmonics only).

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