Suppose string 2 is embedded in a viscous medium (such as

Chapter 9, Problem 7P

(choose chapter or problem)

Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed:

\(\Delta F_{\text {drag }}=-\gamma \frac{\partial f}{\partial t} \Delta z\)

(a) Derive the modified wave equation describing the motion of the string.

(b) Solve this equation, assuming the string vibrates at the incident frequency \(\omega\). That is, look for solutions of the form \(\tilde{f}(z, t)=e^{i \omega t} \tilde{F}(z) .\)

(c) Show that the waves are attenuated (that is, their amplitude decreases with increasing z). Find the characteristic penetration distance, at which the amplitude is 1/e of its original value, in terms of \(\gamma, T, \mu\), and \(\omega\).

(d) If a wave of amplitude \(A_{I}\), phase \(\delta_{I}=0\), and frequency \(\omega\) is incident from the left (string 1), find the reflected wave’s amplitude and phase.