Energy and Power of a Wave Traveling Along a String — Question 3

Question 3

A fixed power budget fixes particle velocity amplitude on one string

On the same uniform string, two sinusoidal waves are generated separately with equal mean powers. The second has frequency three times the first.

1. Compare displacement amplitudes and maximum transverse velocities.

2. Compare wavelengths and maximum slopes. State the assumptions behind the cancellation.

Question 3 - Solution

A fixed power budget fixes particle velocity amplitude on one string

Method and mathematical working

With the string properties fixed, equal power fixes the product $A\omega$. Track wavelength and slope using the same unchanged wave speed.$$\begin{aligned}\overline P&=\tfrac12\mu v(A\omega)^2,\\A_2\omega_2&=A_1\omega_1,\qquad kA=\frac{\omega A}{v}.\\[4pt]\omega_2=3\omega_1&\Rightarrow A_2=A_1/3,\quad\lambda_2=\lambda_1/3,\quad A_2k_2=A_1k_1.\end{aligned}$$

(a) Calculation and conclusion

Since $\overline P\propto\omega^2A^2$ at fixed $\mu,v$, $A_2=A_1/3$. Consequently $\omega_2A_2=\omega_1A_1$, so the maximum transverse velocities are equal.

(b) Interpretation and limits

$\lambda_2=\lambda_1/3$ and $k_2=3k_1$. Therefore $k_2A_2=k_1A_1$, so the maximum slopes are equal as well. The result assumes a nondispersive linear string with unchanged tension, density and traveling-wave character.

Physical check

The smaller displacement amplitude does not imply smaller material velocity: the oscillation is faster by exactly the compensating factor. The maximum slope also remains unchanged.