Standing Waves

A standing wave is a pattern that does not travel along the medium; it sits in place with fixed nodes and antinodes. On a string fixed at both ends, only certain wavelengths and frequencies are allowed, and computing them comes down to two compact formulas tied to the harmonic number $n$.

The Harmonic Formulas And Their Symbols

For an ideal string of length $L$ fixed at both ends, the allowed wavelengths and frequencies are

Here $L$ is the string length, $v$ is the wave speed on the string, and $n$ labels the harmonic. The full displacement of the standard ideal pattern can be written

where the time factor $\cos(\omega t)$ keeps the medium oscillating and the spatial factor $\sin(kx)$ fixes where the amplitude is always zero (nodes) and where it is largest (antinodes).

Why Only Certain Frequencies Are Allowed

The pattern forms when two waves of equal frequency and amplitude move in opposite directions and interfere, which on a string happens when a wave reflects from a boundary and overlaps the incoming wave. The boundary condition does the rest: with both ends fixed at zero displacement, only patterns that place a node at each end can survive. Fitting a whole number of half-wavelengths into the length $L$ gives $L = n\lambda_n/2$, which rearranges directly to $\lambda_n = 2L/n$. Combining that with $v = f\lambda$ yields $f_n = nv/(2L)$. This is why standing waves on a string appear only at discrete normal modes, or harmonics, rather than at any frequency you like. The formulas depend on the setup; they hold for an ideal string fixed at both ends, not for every standing-wave system.

Worked Example: Third Harmonic On A Fixed String

A string fixed at both ends has length $L = 0.60\ \mathrm{m}$ and supports waves with speed $v = 120\ \mathrm{m/s}$. Find the third-harmonic frequency.

Using $n = 3$,

So the third harmonic is $300\ \mathrm{Hz}$. The shape confirms it: the third harmonic fits three half-wavelengths into the string, giving nodes at both ends and two interior nodes, with neighbouring nodes separated by $L/3 = 0.20\ \mathrm{m}$.

Try It Yourself

Recompute the same string for the first and second harmonics by changing only $n$. As a check, $f_1 = (1)(120)/1.20 = 100\ \mathrm{Hz}$ and $f_2 = (2)(120)/1.20 = 200\ \mathrm{Hz}$. Notice that the harmonics are evenly spaced at $100\ \mathrm{Hz}$ apart and that the number of interior nodes equals $n - 1$. Changing only $n$ is the quickest way to see how wavelength, node pattern, and frequency move together.

Pitfalls In Standing-Wave Calculations

• Using $f_n = nv/(2L)$ without stating the boundary condition. The formula assumes a string fixed at both ends; a different boundary set changes it.
• Calling any oscillating shape a standing wave. The defining feature is fixed nodes, points that stay at zero displacement at all times.
• Thinking the medium is motionless. The pattern stays put, but most points still oscillate; a traveling wave carries crests along the medium, while interference here locks the node and antinode positions in place.
• Assuming any reflected wave makes a clean standing wave. The cleanest case needs matching frequency, opposite travel directions, and the right boundary conditions.
• Swapping nodes and antinodes. Nodes have zero displacement; antinodes reach maximum amplitude.

Where Standing Waves Show Up

Standing waves matter in strings, air columns, musical instruments, microwave cavities, and many resonance problems across physics and engineering. They are useful precisely because the allowed modes are discrete: once the boundaries are set, only certain patterns fit, and that is what gives harmonics their structure. Standing waves arise from interference, so once these formulas feel routine it is worth comparing the boundary-driven picture here with the geometry-driven one in interference and diffraction.