Throughout the rest of this paper we will employ dimensionless forms for all 

 parameters and results. Table 1 gives values for the perimeter length and the area of 

 various fire shapes generated using equation 9. In this table the perimeter length is 

 normalized by r(0) and the area by r^(o), where r(0) is the distance from the point 

 of origin to the head (or front) of the fire. These values may be used to compare 

 sensitivities in absolute terms, because later results will be given in terms of the 

 initial fire perimeter length and initial fire area. 



The second simplifying assumption we make is that the fire is to be suppressed 

 (or contained) through the work of two crev;s which divide the effort equally. This 

 assumption not only introduces the simplification of mathematical symmetry, but reflects 

 current practice both in the United States and the Soviet Union. The advantage of this 

 tactic is clear upon a little thought: If the work of suppression proceeds in only one 

 direction from the starting point, then when the crew completes its circuit around the 

 fire edge, it will encounter the fire burning behind the original line of control 

 near the starting point. If the work proceeds in both directions from the starting 

 point, then when the two teams meet on the opposite edge of the fire the containment 

 will be complete. 



The third assumption employed here is that the rate of progress by the suppression 

 crew (A) is constant. This is a good approximation for machine-aided effort, but is 

 clearly not a good approximation for work with handtools or backpack pumps (with the 

 possible exception of the new Soviet technique of backfiring against a line of foam 

 laid down using a backpack cannister) . It would be a little more complex to assume that 

 the rate of suppression is a simple function of the rate of advance of the fire edge 

 perpendicular to the boundary (see inequality 8) which quantity is proportional to the 

 fire intensity as defined by Byram (1959) . For i7istance, one might argue that direct 

 suppression will progress at a rate inversely proportional to the depth of the 

 flaming zone. This depth is, in turn, approximately proportional to the rate of advance 

 of the fire edge (Albini 1976). For the purpose of exploring the sensitivity of burned 

 area and time required for containment to various factors, however, it is sufficient 

 to use a constant work rate. 



Using these simplifying assumptions it is possible to write closed form expressions 

 for the burned area and the time expended. Dividing equation 7 by equation 6 and 

 integrating we obtain the formula for the shape of the final boundary: 



R(e) = R(e ) exp (/'^f(e') de') (lO) 



9 







In this equation the function to be integrated is 



f(e) = r/A+ (rVr) V 1 + (rVr)^ - (r/A)^ 



- (r'/r) (r/A) + V 1 + (rVr)^ - (r/X)2 (II) 



which is, under our assumptions, purely a function of the angle 0, since r (0) is given 

 by equation 9 and 



rVr = (^ r(e))/r(6) (12) 



Note that the value of R(Q) depends upon tb.e choice of the starting point, O^. If tlie 

 effort begins at tlie front edge of the fire on the line of symmetry, then 



6=0; R(e ) r (0) (13) 







6 



