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Longitudinal Wave Equation from Transverse One

  1. Apr 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Please see attached.
    Part ii)
    2. Relevant equations


    3. The attempt at a solution
    So I try to conserve volume as it suggests in the hint. I take the initial volume of the region to be given by:
    $$ h \times \delta x \times l = (\delta x + \eta) (h+\Psi) l $$
    Where l is just some fixed, constant length which can immediately be canceled. Expanding:
    $$ \Psi \delta x = - \eta (\psi + h) $$
    But ## h>> \psi \implies (\psi+h) \approx h ##
    $$ \Psi \delta x = - \eta h $$
    For small values of ## \eta ## (which is implied by the fact that ## \psi ## is small? ) we can make the statement:
    $$ \eta \approx \frac{\partial \eta}{ \partial x} \delta x $$
    So that:
    $$ \Psi \approx - h\frac{\partial \eta}{\partial x} $$

    Well, I got to the result, but I'm just not sure that this approach is correct - for instance should I not put ## \delta \Psi## instead of ## \Psi## - but then I have second differentials and I get the wrong answer... - also not entirely sure how to justify the assumption that if ## h >> \psi ## then ## \eta ## must be small... or is that okay because we are just approximating?

    Thanks!
     

    Attached Files:

  2. jcsd
  3. Apr 26, 2015 #2
    I think you shouldnt use ##\delta \psi## cause the change in the transverse direction of the volume is simply ##\psi(x)## however you should use ##\delta \eta## cause the change in the longitudinal direction of the volume is ##\eta(x+\delta x)-\eta(x)=\delta \eta##.
     
    Last edited: Apr 26, 2015
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