Oscillations and waves

Standing waves

When an incident wave interferes with its reflection, fixed nodes and maximally oscillating antinodes appear — the principle behind every string instrument.

A standing wave appears when two waves of equal frequency and amplitude travel in opposite directions through the same medium (typically an incident wave meeting its reflection) and interfere. The result is a pattern that oscillates in place: points that never move (nodes) alternate with points of maximum oscillation amplitude (antinodes).

Select 'Standing wave' mode to see the fixed, non-oscillating nodes — quite different from the three traveling-wave modes in the same lab.

Condition for standing waves on a string fixed at both ends

On a string of length $L$ fixed at both ends (a fixed end is always a node), only wavelengths that 'fit' the string length produce a stable standing wave — specifically when $L$ equals an integer number of half-wavelengths:

L = n\frac{\lambda_n}{2} \quad\Rightarrow\quad \lambda_n = \frac{2L}{n}, \qquad n = 1, 2, 3, \ldots

Each $n$ corresponds to a harmonic (a normal mode), with frequency $f_n = v/\lambda_n = nv/(2L)$ , where $v$ is the wave speed on the string. The $n=1$ harmonic is the fundamental (the lowest possible frequency); $n=2,3,\ldots$ are higher harmonics. This is exactly the operating principle of every string instrument (guitar, violin) and of air columns in flutes and horns.

Definition: Nodes and antinodes

A node is a point of absolute rest (zero oscillation amplitude) — at a node, the incident and reflected waves always cancel. An antinode is a point of maximum oscillation amplitude (twice each component wave's amplitude) — at an antinode, the two waves always reinforce. The distance between consecutive nodes (or consecutive antinodes) is always $\lambda/2$ .

Example: Fundamental frequency of a string

A guitar string is 0.6 m long, with a wave speed of 240 m/s on it. Find the fundamental frequency the string produces.

Solution

For $n=1$ : $\lambda_1 = 2L = 1.2$ m. Frequency $f_1 = v/\lambda_1 = 240/1.2 = 200$ Hz.

Quick check

Higher modes have more antinodes and frequencies increase by integer order $n$ when wave speed is fixed. Plucking position and technique change the relative strength of harmonics, shaping timbre. Air columns have boundary conditions unlike strings: a closed end is a displacement node and an open end is approximately an antinode, so a tube closed at one end supports only compatible modes.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear.

The distance between two consecutive nodes of a standing wave on a string equals:

At a node of a standing wave, the oscillation amplitude is:

References

Young, Freedman (2019). University Physics