PROBLEM$-$SOLVING STRATEGY $16\mathrm{.}1$ Standing waves

We can use the same general strategy for any standing wave.

PREPARE

For sound waves, determine what sort of pipe or tube you have: open$-$open, closed$-$closed, or open$-$closed.

For string or light waves, the ends will be fixed points.

For other types of standing waves, such as electromagnetic or water waves, the mode diagram will be similar to one of the cases outlined in Synthesis $16\mathrm{.}1($Page$).$ You can then work by analogy with waves on strings or sound waves in tubes.

Determine known values: length of the tube or string, frequency, wavelength, and positions of nodes or antinodes.

Sketch a visual overview, including a picture of the relevant mode.

SOLVE Once you have determined the mode diagram, you can use the appropriate set of equations in Synthesis $16\mathrm{.}1($Page$)$ to find the wavelength and frequency. These equations work for any type of wave.

${}_m=\frac{2L}{m}$ and $f_m=\frac{v}{{}_m}=m(\frac{v}{2L}),$ where $m=1,2,3,4,$dots

For sound waves in an open$-$open or closed$-$closed tube, the allowed frequencies and wavelengths are given by

${}_m=\frac{2L}{m}$ and $f_m=m(\frac{v}{2L})=m{f}_1,$ where $m=1,2,3,4,$dots

For sound waves in a open$-$closed tube, the allowed frequencies and wavelengths are given by

${}_m=\frac{4L}{m}$ and $f_m=m(\frac{v}{4L})=m{f}_1,$ where $m=1,3,5,7.$ What is the length L of the pipe? Express your answer in centimeters