A rod of uniform elastic material of length 1/2 lies along the X axis with its left end fixed at x=0. At time t=0, an identical rod hits the riht end of the first rod with a speed of v. The second rod is thereafter kept alongside the first rod, and neither end is fixed. If the Young's Modulus and the density of the rods are such taht c=1, and if the displacement u is a generalized solution of the wave equation, find u(x,t) for t>0. Sketch the x-t diagram(adsbygoogle = window.adsbygoogle || []).push({});

im guessing the conditions look like

g(x) = [tex] \frac{\partial^2 u}{\partial t^2} - c^2 \frac{\partial^2 u}{\partial x^2} = 0[/tex]

[tex] u(0,t) = 0, for t>0 [/tex]

[tex] \frac{\partial u}{\partial t} (1,t) = 0 [/tex]

[tex] \frac{\partial u}{\partial t} (\frac{1}{2},t) = v [/tex]

well f(x) =0 because it is not stated otherwise

so g(x) = -v for 0<x<1/2

and g(x) = v for 1/2<x<1

the left hand side rod would go downward after its right hand side was hit. Also the right hand rod would go upward because of the impulse of the left side rod.

First of all am i right? Also is there any derivation of sorts that i need to put down... because this question is actually part 2 of a question of similar kind - however of the first part involved the right side's right end to be fixed. There the part between 0 and 1/2 was zero and the 1/2 to 1 interval was -v.

Also would the x-t graph be a wave that looks like a negative sine wave??

Any kind of help would be appreciated! Thank you!

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# Homework Help: Diff Eqs

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