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Unformatted text preview: Phys 2214 Homework #4 Solutions September 30, 2011 1. For a long, thin rod, the compressions created by a longitudinal wave traveling along the rod will change the length to a greater extent than the diameter. So, we can replace the equation for the bulk modulus, p = B ( V/V ), by the equation for Youngs modulus, F/A = Y ( x/x ). s(x+x,t) x+x x s(x,t) (a) In general, the change in length of the segment is L = [ x + x + s ( x + x,t )] [ x + s ( x,t )] x = s ( x + x,t ) + x + x x x s ( x,t ) = s ( x + x,t ) s ( x,t ) . 1 (b) The linear stressstrain relation is F A = Y L L = Y s ( x,t ) s ( x + x,t ) x , where the fact that L is negative has been used and the original length is x . (c) The force is given by F ( x,t ) = lim x Y A s ( x,t ) s ( x + x,t ) x = Y A s x . (d) The net force acting on the segment is F ( x,t ) F ( x + x,t ). Then Newtons 2 nd law gives Y A " s ( x,t ) x s ( x + x,t ) x # = A x 2 s t 2 . Dividing both sides by x and taking the limit as x 0 gives lim x h s ( x,t ) x s ( x + x,t ) x i x = 2 s x 2 . We then get the wave equation 2 s x 2 = Y 2 s t 2 , which, upon comparison with the general wave equation, gives v = q Y/ ....
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This note was uploaded on 01/02/2012 for the course PHYSICS 2214 at Cornell University (Engineering School).
 '11
 DAVIS,J.C.
 Work, Waves And Optics

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