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Mechanical wave reflection at a boundary

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  1. Mar 1, 2017 #1
    1. The problem statement, all variables and given/known data
    Capture.jpg

    2. Relevant equations
    The right hand section (A) has an incident and reflected wave
    $$y_1=Ae^{i(kx+\omega t)} +A'e^{i(-kx+\omega t)} $$

    The middle section (B) has a transmission reflected wave

    $$y_2=Be^{i(k_2x+\omega t)} +B'e^{i(-k_2x+\omega t)}$$

    Section (C) just has the transmission wave:
    $$y_3=Ce^{i(kx+\omega t)}$$

    where ##k=2\pi / \lambda## and ##\omega = 2\pi \upsilon ##. The actual wave is the real part of the complex exponential.

    We have the boundary conditions:

    $$(1)y_3(0,t)=y_2(0,t) \text{ and } (2)\frac{\partial y_3(0,t)}{\partial x}=\frac{\partial y_2(0,t)}{\partial x}$$
    and
    $$(3) y_1(a,t)=y_2(a,t) \text{ and } (4) \frac{\partial y_1(a,t)}{\partial x}=\frac{\partial y_2(a,t)}{\partial x}$$
    3. The attempt at a solution

    by applying condition 1:
    $$C=B+B'$$
    condition 2:
    $$Ck=(B-B')k_1 $$
    condition 3:
    $$Ae^{i(2\pi n k/k_1)}+A'e^{i(2\pi n k/k_1)}=B+B'$$
    condition 4:
    $$Ake^{i(2\pi n k/k_1)}-A'ke^{i(2\pi n k/k_1)}=k_1(B-B')$$


    We get ##A'=0## but that is the only progression I can make.

    Please help, thank you x
     
  2. jcsd
  3. Mar 1, 2017 #2

    haruspex

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    Two of your equations have B+B' on the right hand side. Match those up.
     
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