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Simplifying Additive Exponential Terms

  1. Mar 5, 2013 #1
    1. The problem statement, all variables and given/known data

    I have developed an equations for calculating some pollutant concentration as a function of x and y. I'm trying to simplify the problem so that I can write the equation for wp as a function of x. All variables except x and wp are known.


    2. Relevant equations

    [itex]c(x,y=w_{p})=0.01 \cdot c_{peak} = c_{peak} \cdot \left[ exp \left( -\frac{ u \left( w_{p} \right)^{2}}{ 4 E_{y}x}\right) + exp \left( -\frac{ u \left( w_{p} - y_{1} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{2} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{3} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{4} \right)^{2}}{ 4 E_{y}x}\right) \right][/itex]

    3. The attempt at a solution

    Basically, all I've gotten to simplify is that cpeak cancels such that:

    [itex] 0.01 = \left[ exp \left( -\frac{ u \left( w_{p} \right)^{2}}{ 4 E_{y}x}\right) + exp \left( -\frac{ u \left( w_{p} - y_{1} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{2} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{3} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{4} \right)^{2}}{ 4 E_{y}x}\right) \right][/itex]

    Is there a way of simplifying the exponential terms so that I can take the natural log of the right side of the equation and solve for wp as a function of x?
     
  2. jcsd
  3. Mar 5, 2013 #2

    SammyS

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    WOW!

    Logs aren't of much help in working with sums.

    The only slight help I can see is that all the terms on the right have a common factor of [itex]\displaystyle\ \exp\left(\frac{-u(w_p)^2}{4E_y\,x}\right)\ . [/itex]
     
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