Changes

The propogating wavefront generates spherical wavelets as it propogates. What we want to know is what the wave at the initial point <math> f(x,y,0,t) </math> will look like at <math>f(x,y,D,t)</math>. To do this, we can integrate over the product of this equation times a propogator g.

The propogating wavefront generates spherical wavelets as it propogates. What we want to know is what the wave at the initial point <math> f(x,y,0,t) </math> will look like at <math>f(x,y,D,t)</math>. To do this, we can integrate over the product of this equation times a propogator g.

<math> g( x_i, t_i, x_f, t_f)

<math> g( x_i, t_i, x_f, t_f)\,</math>

Because the propogator is actually in terms  

Because the propogator is actually in terms of the differences between the x and t values, we will write the difference between the x-vectors as <math>\Delta x</math> and the difference between the times as <math>\Delta t</math>.

 

of the differences between these values, we will  

 

write the difference between the x-vectors as  

 

{delta, hereafter d} dx and the difference between  

 

the times as dt.

[integral]dti [integral]g(stuff)f(stuffi)dxi dyi =  

[integral]dti [integral]g(stuff)f(stuffi)dxi dyi =