Wave Speed on a Stretched String — Question 8

Question 8

Changing drive frequency changes wavelength rather than string speed

A string has constant tension $F=18\,\mathrm{N}$ and density $\mu=.020\,\mathrm{kg/m}$. A source changes frequency from $15$ to $25\,\mathrm{Hz}$ without changing the string state. Treat the wave as linear and nondispersive.

1. Find the speed and the two wavelengths.

2. If the goal instead is to keep the old wavelength at $25\,\mathrm{Hz}$, find the required tension with unchanged density.

Question 8 - Solution

Changing drive frequency changes wavelength rather than string speed

Method and mathematical working

Determine the speed from the mechanical properties. A changed driving frequency changes wavelength unless the tension is adjusted as well.$$\begin{aligned}v&=\sqrt{18/0.020}=30\,\mathrm{m/s},\\\lambda&=\frac vf,\qquad F_{\mathrm{required}}=\mu(f\lambda)^2.\\[4pt]\lambda_{15}&=30/15=2.0\,\mathrm m,\quad\lambda_{25}=30/25=1.2\,\mathrm m,\\F_{\lambda=2,f=25}&=(0.020)(25\cdot2)^2=50\,\mathrm N.\end{aligned}$$

(a) Calculation and conclusion

$v=\sqrt{18/.020}=30\,\mathrm{m/s}$. The wavelengths are $30/15=2.0\,\mathrm{m}$ and $30/25=1.20\,\mathrm{m}$. Frequency is imposed by the source; speed is set by this medium model.

(b) Interpretation and limits

Keeping $\lambda=2.0\,\mathrm{m}$ at $25\,\mathrm{Hz}$ requires $v=50\,\mathrm{m/s}$ and $F=\mu v^2=50\,\mathrm{N}$. The required tension ratio is $(25/15)^2$, not $25/15$.

Physical check

At fixed density and target wavelength, tension must scale as frequency squared. The factor $(25/15)^2$ changes $18$ N to $50$ N, consistent with the direct calculation.