Figure 14. Sketch illustrating surface disturbance caused by an entering body. 



due to a system of axially symmetric multipole sources located at the origin of the 

 disturbance.* The time-varying strengths of the multipole sources are such as to 

 describe the actual flow velocities outside the immediate vicinity of the disturbance. At 

 some specified stage of the entry, the strength of any such multipole, say of order m, 

 must, from consideration of dimensions, be proportional to UL m+2 . But the sound 

 pressure associated with a multipole source of order m is proportional to the (m + l)th 

 time derivative of the strength of the source. These, with appropriate further dimen- 

 sional considerations, allow the sound pressure to be expressed in the form 



Ps(r,6,t) 



L \ c 



(17) 



The free-surface condition requires that A (corresponding to a simple source) be 

 zero, so that for small values of the Mach number (U/c) the dipole term (/?? = 1) 

 can be expected to predominate. The dependence of the functions A m upon the polar 

 angle 9 can be expressed very simply: In particular, 



Ax{e,r) = Z(t) COS0, (18) 



* The condition for which such a procedure is valid may be stated as follows: (1) 

 all flow velocities involved are very small in comparison with the velocity of sound; and 

 (2) the wavelength of the highest sound frequency of interest is large in comparison with 

 the linear dimensions of the disturbance. Ordinarily this means simply: (C//c)<l; (/L/c)<l. 



260 



