194 MEASUREMENT OF PRESSURES 



to a lower value Sec. By the superposition theorem/^ the observed 

 response R(t) is found to be 



R{t) 



= Voe-^/^ - {So - S^) (^4^) e-'/^ - e-^/^1 



If the decay of the step response is slow, and hence r much greater 

 than d, one finds to a first approximation that 



y(,. ^ Rjt) - (1 - s^/So) [e/{r - e)]R{0) (1 - t/r) 



^^ So- {So- S^^)[d/{T-d)] 



where R{0) is the value of response at time t = 0, and terms of order 

 {I/tY and higher have been neglected. The true time constant 6 can 

 be evaluated from the R{t) curves with sufficient accuracy. If the 

 ratios t/r, 6/t can also be neglected the even simpler formula results 

 that 



y(.. ^ R{t) - (1 - S^/So) {e/T)R{0) 

 ^^ So - {So - S^) {d/r) 



If the area under the V{t) curve to a time t' is required it can be 

 shown that for the same assumptions 



t' V 



r v{i) dt = — — ^ — -- r R{t) dt 



T — U T — U 



If the time t' is sufficiently greater than r that S(t') ~ aS» this equation 

 becomes 



o o 



and the proper calibration value is the limiting value *Soo for long times. 

 Equations of this type have been found useful and practical in correct- 

 ing shock wave records for leakage and dielectric effects in cable cir- 

 cuits. If these effects are not too large, similar correction formulas are 

 readily worked out for other types of applied signal and response. The 

 important conclusion to these arguments is that the presence of such 

 effects may result in serious errors if precautions are not taken to mini- 

 mize and evaluate them. 



^^ Seo, for cxampl(% the hook by Gardner and Barnos (40). 



