Energy and Power of a Wave Traveling Along a String — Question 4

Question 4

Increasing tension changes flux even if local motion is unchanged

A source maintains fixed amplitude $A$ and frequency $f$ while string tension changes from $F$ to $4F$. The linear density stays fixed and reflection is absent.

1. Find the ratios of wave speed, mean power and mean energy density.

2. Explain how the power can change without a change in the local oscillation amplitude or frequency.

Question 4 - Solution

Increasing tension changes flux even if local motion is unchanged

Method and mathematical working

First determine how tension changes speed. Substitute that scaling into power, while checking which factors control energy density.$$\begin{aligned}\frac{v_2}{v_1}&=\sqrt{\frac{F_2}{F_1}}=2,\\\overline P&=v\overline{\mathcal E},\qquad \overline{\mathcal E}=\tfrac12\mu A^2\omega^2.\\[4pt]\frac{\overline P_2}{\overline P_1}&=2,\qquad\frac{\overline{\mathcal E}_2}{\overline{\mathcal E}_1}=1,\qquad\frac{\lambda_2}{\lambda_1}=2.\end{aligned}$$

(a) Calculation and conclusion

Speed doubles because $v\propto\sqrt F$. With $A,\omega,\mu$ fixed, $\overline u=\mu\omega^2A^2/2$ is unchanged, while $\overline P=v\overline u$ doubles.

(b) Interpretation and limits

The same energy per unit length is transported twice as quickly. The wavelength doubles, so the slope amplitude halves; the fourfold tension compensates its squared decrease in the elastic energy density. The source must supply the increased mean power.

Physical check

The same energy per unit length is transported twice as quickly, producing twice the mean power. Wavelength doubles because the driving frequency is held fixed.