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Math Help - Partial Differential Equations Rectangular Membrane Problem

  1. #1
    busundi
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    Partial Differential Equations Rectangular Membrane Problem

    Hi,

    I am not able to figure out the following membrane problem. If anyone could help that would be greatly appreciated. Thanks.

    (Minimum Property) Show that among all rectangular membranes of the same area
    A = ab and the same c the square membrane is that for which u11 has the lowest frequency.

    The frequency of the membrane is given by umn = lamda/2*pi
    where lamda = c*pi*sqrt( (m^2)/(a^2) + (n^2)/(b^2) ) where m = 1,2,...,
    where n = 1,2,...,
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  2. #2
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    Quote Originally Posted by busundi View Post
    Hi,

    I am not able to figure out the following membrane problem. If anyone could help that would be greatly appreciated. Thanks.

    (Minimum Property) Show that among all rectangular membranes of the same area
    A = ab and the same c the square membrane is that for which u11 has the lowest frequency.

    The frequency of the membrane is given by umn = lamda/2*pi
    where lamda = c*pi*sqrt( (m^2)/(a^2) + (n^2)/(b^2) ) where m = 1,2,...,
    where n = 1,2,...,
    I am not sure what the question is exactly.

    But I can say the following. How to do find the minimum (in a physics class)? You find the partial derivatives and make them zero.

    Hence the equation you gave me, you find the partial derivatives I am assuming a,b are the variables and solve the system of equations. And when you solve them you will find the a=b to get the minimum frequency.
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