If the amplitude of a sinusoidal string wave doubles while everything else stays the same, by what factor does the average power change?

A string wave has \(\mu=0.010\,\mathrm{kg\,m^{-1}}\), \(A=0.020\,\mathrm{m}\), \(\omega=50\,\mathrm{rad\,s^{-1}}\), and \(v_w=30\,\mathrm{m\,s^{-1}}\). Find the average power.

Two otherwise identical sinusoidal waves have amplitudes \(A\) and \(2A\). Compare their average energy densities.

If a sinusoidal string wave keeps the same amplitude and speed but its frequency doubles, by what factor does the average power change?

A string wave has average power \(8.0\,\mathrm{W}\), \(\mu=0.020\,\mathrm{kg\,m^{-1}}\), \(\omega=40\,\mathrm{rad\,s^{-1}}\), and \(v_w=25\,\mathrm{m\,s^{-1}}\). Find the amplitude.

Explain why a wave can transport energy along a string even though each small part of the string oscillates about an equilibrium position instead of drifting steadily along the string.

A sinusoidal wave on a string keeps the same amplitude and frequency while the tension is increased by a factor of \(9\). The linear density is unchanged. By what factor does the average power change?

A string wave must carry the same average power while the wave speed stays fixed. Derive how the amplitude must scale with angular frequency. Then determine the required amplitude factor if \(\omega\) is tripled.