Standing Waves on a String

What you are seeing: the first five normal modes of a string fixed at both ends. The displacement is $y_n(x, t) = A \sin(n\pi x / L) \cos(2\pi f_n t)$ , with mode frequency $f_n = n c / (2L)$ . Mode $n$ has $n$ antinodes (places of maximum motion) and $n - 1$ interior nodes (places where the string stays at rest). The fundamental is mode 1; modes 2, 3, ... are the harmonics that give a plucked string its characteristic timbre.

The traces overlay all selected modes' instantaneous profiles together with the envelope $\pm A \sin(n\pi x / L)$ for each (dashed). The fastest mode (highest n) oscillates n times faster than mode 1.

Figure 1. Standing-wave modes on a fixed-end string of length L = 1 with wave speed c = 1. Method: closed-form

\sin(n\pi x/L) \cos(2\pi f_n t)

mode n1

speed2

WHAT TO TRY

Step the mode number n: the string fixed at both ends fits exactly n half-wavelengths, with n-1 nodes held still while the antinodes swing.
Watch the middle panel: the standing wave is the sum of two travelling waves of equal amplitude running in opposite directions. They slide through each other while their sum only breathes in place, which is what a standing wave is.
The frequencies climb a harmonic ladder f_n = n f_1, the same overtone series that gives a plucked string its pitch and timbre. Click any bar in the spectrum to jump straight to that harmonic.
Slow it down and watch one mode: every point moves in phase and crosses the flat line at the same instant, the signature of a normal mode.