Standing Waves

Harmonics on a String

fitting half-wavelengths

A string fixed at both ends fits a whole number of half-wavelengths, so the n-th harmonic has wavelength twice the length over n.

Example

A string fixed at both ends fits a whole number of half-wavelengths, so the n-th harmonic has wavelength twice the length over n. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The fundamental fits one half-wavelength

A string fixed at both ends must have a node at each end. The simplest fit is a single bulge — half a wavelength spanning the whole length. So the length equals half the wavelength, making the wavelength twice the length: 2 times 6, or 12 metres.

L=λ2  ⇒  λ=2L=12 mL = \tfrac{\lambda}{2} \;\Rightarrow\; \lambda = 2L = \hl{12}\ \text{m}
The fundamental: one half-wavelengthThe lowest standing wave on the string: a single bulge with a node at each fixed end.

Higher harmonics fit more half-wavelengths

The next patterns fit 2, then 3 half-wavelengths into the same length. In general the n-th harmonic fits n half-wavelengths, so the length is n times half the wavelength, and the wavelength is twice the length over n.

L=n λn2  ⇒  λn=2Ln=12n mL = n\,\tfrac{\lambda_n}{2} \;\Rightarrow\; \lambda_n = \frac{2L}{n} = \frac{12}{n}\ \text{m}
The second harmonic: two half-wavelengthsThe second standing wave: two bulges with a node in the middle as well as at each end.

Each allowed wavelength still fills the same string

Read the fit directly. One, two, and three half-wavelengths all land exactly on the fixed ends, so each row gives back the same string length.

nλnLfit112 m6 m26 m6 m34 m6 m\begin{array}{c|c|c}n & \lambda_n & L_{\text{fit}} \\ \hline 1 & 12\ \text{m} & 6\ \text{m} \\ 2 & 6\ \text{m} & 6\ \text{m} \\ 3 & 4\ \text{m} & 6\ \text{m}\end{array}
The third harmonic: three half-wavelengthsThe third standing wave: three bulges with nodes at the ends and two nodes inside.
waves With L = 6 m the harmonics have wavelengths 12, 6, 4 m — clean for n = 1, 2, 3.