75 89 



surface provided z^ Is such a function of the time that z,. has the value 

 given by [133]. 



The electrostatic analogy can be utilized In all cases to show that 

 MfZ^/2 represents the kinetic energy in the water. This may also be shown 

 from [126] as follows. Let the mass of the diaphragm be negligible, so that 

 M can be set equal to and stress forces can be neglected in *, and let F^ = 

 0. Then Equation [126], multiplied through by z^, can be written, using 



[n3b], 



M^'z/z^ = jj<l>z^f{x,y)dxdy 



Here <i> is now the difference between hydrostatic pressure and the pressure on 

 the back face of the plate, and z^f{x,y) is the velocity; hence the integral 

 represents the rate at which the net pressure is doing work. This must equal 

 the rate at which the kinetic energy of the water is increasing; and the left- 

 hand member of the equation is in fact equal to 



{>:'.') 



dt\2 



Up to this point it has been assumed that the diaphragm is sur- 

 rounded by a fixed plane baffle of infinite extent. If there is no baffle, 

 and the diaphragm forms one side of an air-filled box, the determination of 

 Afj is much more difficult. In order to estimate the order of magnitude of 

 the difference, the value of M, was calculated for a sphere whose surface 

 over one hemisphere moves radially outward while the other hemisphere re- 

 mains at rest. The motion of potential flow is easily written out for this 

 case in terras of spherical harmonics; summation of the resulting series gives 

 M, = 0.8327rpo^ where a is the radius of the sphere and p the density of the 

 surrounding fluid. Had the fluid been confined by a plane baffle continuing 

 the plane of the base of the expanding hemisphere, M, would have been 2jrpo^ 

 Thus removal of the baffle decreases iVf, in the ratio 0.4l6. It is a plaus- 

 ible surmise that the decrease in Af, would be somewhat less for a parabolol- 

 dal diaphragm and somewhat more for a piston. 



THE REDUCTION PRINCIPLE, IN THE CASE OF PROPORTIONAL CONSTRAINT 



Suppose again that only part of the target is movable, the rest 

 constituting an infinite rigid baffle; as before, let the maximum diameter of 

 the movable plate be D. Let M and M, be constants. Then the following state- 

 ments are true: 



1 . Within any time Interval of length D/c, at least once 



d^z, _ (2F. + »V r^,^, 



dt^ M + Mi 



