Problem: The figure shows a standing wave on a string.Draw the standing wave that occurs if the string tension is quadrupled while the frequency is held constant.

FREE Expert Solution

Frequency of oscillation for a stretched string:

$\overline{){\mathbf{f}}{\mathbf{=}}\frac{\mathbf{n}}{\mathbf{2}\mathbf{L}}{\mathbf{·}}\sqrt{\frac{\mathbf{T}}{\mathbf{\mu }}}}$

Solving for n:

$\overline{){\mathbf{n}}{\mathbf{=}}{\mathbf{2}}{\mathbf{L}}{\mathbf{f}}{\mathbf{·}}\sqrt{\frac{\mathbf{\mu }}{\mathbf{T}}}}$

We have the initial nodes, ni = 2 and initial tension, Ti.

The final tension is 4Ti.

We'll solve for new nodes, nfinal, as follows:

${\mathbit{n}}_{\mathbf{i}}\mathbf{=}\mathbf{2}\mathbit{L}{\mathbit{f}}_{\mathbf{1}}\mathbf{·}\sqrt{\frac{\mathbf{\mu }}{{\mathbf{T}}_{\mathbf{i}}}}$

${\mathbit{n}}_{\mathbf{f}}\mathbf{=}\mathbf{2}\mathbit{L}{\mathbit{f}}_{\mathbf{1}}\mathbf{·}\sqrt{\frac{\mathbf{\mu }}{\mathbf{4}{\mathbf{T}}_{\mathbf{i}}}}$

Problem Details

The figure shows a standing wave on a string.

Draw the standing wave that occurs if the string tension is quadrupled while the frequency is held constant.