Standing Waves on Strings and in Pipes Worked Examples Cheat Sheet

Key Facts

• For any wave, the speed equation is $v=f\lambda$, so frequency can be found with $f=\frac{v}{\lambda}$.
• A string fixed at both ends has nodes at both ends, and its allowed wavelengths are $\lambda_n=\frac{2L}{n}$ for $n=1,2,3,\ldots$.
• The harmonic frequencies for a string fixed at both ends are $f_n=\frac{nv}{2L}$, where $n$ is the harmonic number.
• An open pipe has antinodes at both ends, and its allowed wavelengths are $\lambda_n=\frac{2L}{n}$ for $n=1,2,3,\ldots$.
• The harmonic frequencies for an open pipe are $f_n=\frac{nv}{2L}$, using the sound speed in air for $v$.
• A pipe closed at one end has a node at the closed end and an antinode at the open end.
• A pipe closed at one end supports only odd harmonics, so $n=1,3,5,\ldots$ and $f_n=\frac{nv}{4L}$.
• The fundamental frequency is the lowest allowed frequency, so it is $f_1=\frac{v}{2L}$ for strings and open pipes, but $f_1=\frac{v}{4L}$ for a closed pipe.

Vocabulary

Standing wave

A wave pattern formed by interference in which nodes and antinodes stay in fixed positions.

Node

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

Antinode

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

Harmonic

An allowed standing-wave frequency that fits the boundary conditions of the string or pipe.

Fundamental frequency

The lowest allowed frequency of a standing wave system, also called the first harmonic.

Boundary condition

A required behavior at the end of a string or pipe, such as a node at a fixed end or an antinode at an open end.