370 SECONDARY PRESSURE WAVES 



sumed to be of the same amplitude and duration as the first. This as- 

 sumption, which ignores any acoustic radiation or energy loss in tur- 

 bulence near the minimum, is evidently not a very good one. The 

 duration as defined, however, is useful as a qualitative indication of the 

 extent of times of interest rather than as an accurately measurable 

 quantity of quantitative value, and a more precise estimate is hardly 

 warranted. The calculations so far considered thus show that the peak 

 pressure is fairly sensitive to the migration, but that the duration of 

 positive gauge pressure depends primarily only on the period and is 

 proportional to it. The pressures to be expected at other times are not 

 indicated and must therefore be considered in more detail. 



Explicit calculation of the pressure-tune curve can of course be made 

 by numerical evaluation of the quantity d/dt (a^ da/dt) from computed 

 values of a and t. This tedious and rather inaccurate process would 

 have to be repeated for each case of interest, and does not therefore 

 readily give a general view of what to expect under different conditions. 

 A simpler approach, at the expense of less information, can be made by 

 considering the positive impulse I, or time integral of excess pressure, 

 for the interval of positive pressure. From its definition, this is given 

 by 





'<,_,.,..?fi|-.l(..,|!)'' 



where the limits can be taken to correspond to the values of the inte- 

 grated function when P = Po. If the motion is assumed symmetrical 

 about the minimum, the impulse is twice the value for the limits 

 P = -Pmax, P = Po' For the first limit the bubble radius is a minimum 

 and da^/df" = 0. Hence we have 



J ^ 4.PoL*C'' 



ir ^,dan 



and expressing the bracketed quantity in terms of a* and vertical 

 momentum s gives 



J ^ ^PoL'^C* 





The radius a* corresponding to P = Po was found to be approxi- 

 mately 0.61, and for this large a radius, the term in s^ is negligible and 

 can be set equal to zero. The term k*/a*^y depends on the equation of 

 state for the products, but the impulse will not be seriously in error for 



