If friction is present, the wave equation takes the form utt − c2 uxx = −r...

If friction is present, the wave equation takes the form utt − c2 uxx = −r ut, where the resistance r > 0 is a constant. Consider a periodic source at one end: u(0, t) = 0, u(l, t) = Aeiωt .

(a) Show that the PDE and the BC are satisfied by u(x, t) = Aeiωt sin βx sin βl , where β2c2 = ω2 − irω.

(b) No matter what the IC, u(x, 0) and ut(x, 0), are, show that u(x, t) is the asymptotic form of the solution u(x, t) as t → ∞.

(c) Show that you can get resonance as r → 0 if ω = mπc/l for some integer m.

(d) Show that friction can prevent resonance from occurring.

Expert Solution

genius_generous answered 2 years ago

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