SECONDARY PRESSURE WAVES 355 



done in Chapter 8 for the problems of interest here. In applying these 

 results to the evaluation of Eq. (9.1), one condition must be respected: 

 that the differentiation d(p/dt be performed for a point fixed in space. 

 The velocity potentials obtained in Chapter 8 are, however, expressed 

 for convenience in coordinates moving with the velocity U of the center 

 of the sphere, and a point fixed in space therefore has a velocity —Um 

 this system. The desired derivative dq)/dt is then 



dip 

 dt 



\dt Jm \dxjt 



where {d(p/dt),n is for fixed positions in the moving coordinates and 

 where x is in the direction of U (positive upward in the cases to be 

 considered). By definition of the velocity potential, u^ = (grad (pY 

 and Eq. (9.1) becomes 



(9 



.2) - = - i(grad ipy + (^) -u(^) - 12 + constant 



Po \dt Jm \dX/t 



9.2. The Pressure Distribution for Gravity Alone 



If boundary surfaces are assumed sufficiently far away from the gas 

 sphere to have a negligible effect, the velocity potential (p for a hollow 

 sphere of variable radius a is given by (Eq. (8.21)) 



r at r^ 



where U is the upward velocity of the center and the coordinates r, 9 

 are measured from this center and the axis x drawn upward as positive. 

 The derivative {d(p/dx)t in Eq, (9.2) is given by 



M ^cos^f^V-^f-^ 

 \dxjt \drj r \ddj 



The potential 12 for gravity is 12 = —gz, where, as in section 8.5, z is the 

 distance below the level of zero pressure. Substitution in Eq. (9.2) 

 then yields Taylor's result (107) : 



(9-3) '~ = a^ + -dtY^) 



Po 



+ 



- — a — - + 5t/ — I cos 

 2r2V dt dt) 



