1. Mar 5, 2013

### edge333

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

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

3. The attempt at a solution

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

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

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. Mar 5, 2013

### SammyS

Staff Emeritus
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 $\displaystyle\ \exp\left(\frac{-u(w_p)^2}{4E_y\,x}\right)\ .$