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Waves: Transmission/Reflection Coefficients/Energy Conservation

  1. Mar 13, 2005 #1
    Hi

    This isn't really homework but still...

    x = 0 represents the junction between two media which can support travelling waves over two strings (specifically the junction at x = 0 is a "knot" which separates the two strings--the "media"). If a travelling wave (traveling to the right) of amplitude [itex]A_{I}[/itex] is incident at the junction from the medium on the left, given its frequency [itex]\omega[/itex] and angular wavenumber [itex]k_{1}[/itex] we can write the wave as

    [tex]Y_{I} = A_{I}\sin (-k_{1}x+\omega t)[/tex]

    If the transmitted and reflected waves have amplitudes [itex]A_{T}[/itex] and [itex]A_{R}[/itex] respectively (and the angular wavenumber of the transmitted wave is [itex]k_{2}[/itex]) then,

    [tex]Y_{R} = A_{R}\sin (k_{1}x+\omega t)[/tex]
    [tex]Y_{T} = A_{T}\sin (-k_{2}x+\omega t)[/tex]

    we know that the boundary conditions are:

    1. The frequencies of the incident, transmitted and reflected waves be equal. (This has already been accounted for, in the equations above).
    2. The slope of the resultant wave in medium 1 = slope of resultant wave in medium 2, at the junction, i.e.
    [tex]\frac{\partial(Y_{I}+Y_{R})}{\partial x}|_{x=0} = \frac{\partial Y_{T}}{\partial x}|_{x=0} [/tex]

    From these boundary conditions,

    [tex]A_{R} = \frac{k_1 - k_2}{k_1+k_2}A_{I}[/tex] (equation 1)
    [tex]A_{T} = \frac{2k_{1}}{k_1 + k_2}A_{I}[/tex] (equation 2)

    Now, it can be shown using the conservation of energy at the junction,

    [tex]P_{incident} = P_{transmitted} + P_{reflected}[/tex]

    that equations 1 and 2 are true.

    Now this is where my question begins:

    Suppose energy conservation does not hold at the junction (i.e. at x = 0). For now, let us assume that some energy from the incident wave is lost to the surroundings (perhaps as heat energy or due to disspitative forces). Then we will have

    [tex]P_{incident} \neq P_{transmitted} + P_{reflected}[/tex]

    In other words,

    [tex]P_{incident} = P_{transmitted} + P_{reflected} + P_{lost}[/tex]

    Using this "modified energy conservation" constraint, it can be shown that equations 1 and 2 will no longer hold. But equations 1 and 2 have been derived using just boundary conditions at the junction which are obvious, and nothing else. So the boundary condition method will not agree with energy conservation at the junction.

    (a) What is the solution to this dilemma?

    (b) If in a situation, I am given that some percentage of the incident energy is lost (and therefore, only a fraction of it is now available), what do I do? That is, can I apply equations 1 and 2 to find the amplitudes or is there a modification?

    I think I am making a fundamental mistake in my reasoning. I would be very grateful if someone could explain it to me.

    Thanks and cheers
    Vivek
     
    Last edited: Mar 13, 2005
  2. jcsd
  3. Mar 14, 2005 #2
    To the Moderator: Please shift this to the right forum.

    Thanks and cheers
    Vivek
     
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