Standing Waves & Resonance Cheat Sheet

Key Facts

• Wave speed, frequency, and wavelength are related by $v=f\lambda$.
• For a string fixed at both ends, the allowed wavelengths are $\lambda_n=\frac{2L}{n}$, where $n=1,2,3,\ldots$.
• For a string fixed at both ends, the resonant frequencies are $f_n=\frac{nv}{2L}$.
• For an open-open pipe, the resonant frequencies are $f_n=\frac{nv}{2L}$, where $n=1,2,3,\ldots$.
• For a closed-open pipe, only odd harmonics occur, so $f_n=\frac{nv}{4L}$ for $n=1,3,5,\ldots$.
• Adjacent nodes or adjacent antinodes are separated by $\frac{\lambda}{2}$.
• A node and the nearest antinode are separated by $\frac{\lambda}{4}$.
• For a stretched string, wave speed is $v=\sqrt{\frac{T}{\mu}}$, where $T$ is tension and $\mu$ is linear mass density.

Vocabulary

Standing wave

A wave pattern that appears stationary because two equal waves traveling in opposite directions interfere.

Node

A point on a standing wave where the medium has zero displacement.

Antinode

A point on a standing wave where the medium has maximum displacement.

Resonance

A large amplitude vibration that occurs when a system is driven at one of its natural frequencies.

Harmonic

A resonant frequency that fits an allowed standing wave pattern in a system.

Fundamental frequency

The lowest resonant frequency of a system, usually labeled $f_1$.