the condition that the change of p across the sheet is propor- 

 tional to the curvature and tension in the sheet. However, 

 the latter in turn is dependent upon the distortion of the sheet, 

 which requires the solution of a separate elastic problem for 

 the sheet, with its attendant end conditions regarding rigidity. 



Associated Energy Equation 



In many cases one can bypass some of the mathematical 

 complexities involved in the detailed analysis of the dynamic 

 problem by considering the energy budget alone, or at least 

 as a supplementary relation. The real utility of the energy 

 relation is associated with its inherent appeal from the 

 standpoint of physical understanding of the problem. 



In order to form the energy relation associated with the 

 acoustic wave phenomena, relations (2a), (6), and (14) are 

 convenient as a starting point. Using the definition of X 

 stipulated by (6), the continuity relation is readily rendered 

 in the form 



I 2£ + v .& = o (18) 



A., at 



By forming the scalar product of v on relation (14), multi- 

 plying (18) by p, and then adding the two resulting equations, 

 the following quadratic relation is obtained 



zt 



2 M ° 2X 



■ + v -• (pv) = (2^ 1 +X 1 ) v ■ v 2 v (19) 



where v is the magnitude of the particle velocity. This is 

 the energy relation associated with acoustic disturbances 

 (of sufficiently small energy) in a viscous fluid. 



The terms p v 3 /2 and p 3 /2 X represent, respectively, 



33 



