Equations 1 and 2 can be regarded as describing the evolution of the final boundary of 

 the burned area, R (6) : 



de 



dt 



= (A/R) sin a 



(6) 



— = X cos a 



(7) 



The last four equations can be used to model the final boundary, taking into considera- 

 tion the limitations mentioned above. 



Note that the condition required for eventual completion of the boundary is that the 

 expression under the radical (equations 4 and 5] should be nonnegative, or 



The right hand side of inequality 8 is the rate of advance of the edge of the fire 



in the direction perpendicular to the boundary, so the requirement is intuitively 



obvious. So long as inequality 8 is maintained the crew can make progress in contain- 

 ing the fire. 



To gain further insight into the relative importance of the various factors out- 

 lined above, we introduce simplifying assumptions which permit closed form solutions of 

 equations 6 and 7 and make possible numerical examples. 



First, we assume that the shape of the free-burning fire can be expressed analytic- 

 ally and that the form of the fire is invariant. That is, the boundary of the fire 

 simply expands linearly with time, and, similar to the enlargement of a photograph, main- 

 tains its shape. Tliis assumption implies that tlie fire is fully developed and is burn- 

 ing in continuous, homogeneous fuel on flat terrain or gentle slopes, and that the wind 

 does not change speed or direction. Such idealized conditions never occur exactly, but 

 many fires approximately satisfy these conditions at least for short periods of time. 



Since we assume that the fire boundary grows linearly with time, the shape of the 

 fire, expressed in polar coordinates, is the same as the distribution of radial rates of 



A > 



r/(l + (rVr)2)^ 



(8) 



SIMPLIFYING ASSUMPTIONS 



3 



