C.4.3—Strings and pipes

Start with the boundary conditions

A fixed end of a string is a displacement node; a free end is a displacement antinode. For air displacement in a pipe, a closed end is a displacement node and an open end is a displacement antinode. These end conditions determine which standing-wave patterns are allowed.

Use the string patterns

For a string fixed at both ends, or with two free ends, the nth harmonic has $n$ half-wavelengths in length $L$: $\lambda_n=2L/n$ and $f_n=nv/(2L)$. For one fixed and one free end, the allowed patterns contain an odd number of quarter-wavelengths: $\lambda_n=4L/(2n-1)$ and $f_n=(2n-1)v/(4L)$, with $n=1,2,3,\ldots$.

Apply the same geometry to pipes

An open pipe has displacement antinodes at both ends and follows the two-open-end pattern. A closed pipe has a displacement node at the closed end and an antinode at the open end, so only the odd sequence of harmonics is allowed. Use $v=f\lambda$ after finding the wavelength from the boundary pattern. End corrections for open pipes are not required.

Common trap

Do not use the closed-pipe formula for an open pipe, and do not count pressure nodes or pressure antinodes here: the syllabus asks for air-displacement nodes and antinodes. Also use “first harmonic” for the lowest-frequency mode; the syllabus does not require the terms fundamental or overtone.