Standing wave (string)

What is a standing wave?

When a wave traveling along the string reflects off a fixed end, it superposes with the incoming wave. The result is a pattern that does not travel along the string: in some places the string does not move at all (nodes), and in others it vibrates with the largest amplitude (antinodes). The whole string oscillates in place — the envelope stands still.

Harmonic frequency:

— $v$ is the wave speed on the string (depends on tension and thickness), $L$ is the string length, and $n$ is the harmonic number. The string holds $n$ half-waves: $\lambda = 2L/n$.

Why does a short string sound high?

Because the frequency is inversely proportional to length: $f \propto 1/L$. Halve the string — the frequency doubles (a sound one octave higher). That is how a guitar neck works: pressing the string shortens it and raises the pitch.

What are harmonics?

They are the allowed vibration “modes”: $n=1$ — the fundamental (one antinode), and higher $n$ are its multiples $f_n = n\,f_1$ (two, three… antinodes). Their mix gives the instrument its timbre.

Why you need this for the exam:

The formula $f_n = n\,v/(2L)$ together with $\lambda = 2L/n$ solves problems about strings and air columns. Trick: count the nodes/antinodes, match $n$, write down what is constant ($v$ or $L$).

Remember: a standing wave is the superposition of the incoming and reflected wave — the nodes and antinodes stay in place, and a shorter string (or higher n) sounds higher: f = n·v/(2L).