1^12 SURFACE AND OTHER EFFECTS 



If we consider the pressure on a diaphragm or plate of finite extent, 

 it is evident that the pressure at any point is modified by the pressure 

 at other points on the plate and external to it after the times required 

 for hydrodynamic signals to arrive from these points. The method of 

 support and boundary external to the plate must therefore be con- 

 sidered. If the plate is assumed set in a rigid structure, or baffle, of 

 infinite extent, the velocity Un must be zero at all points external to the 

 plate. The integral over Uon then represents the reflected pressure on 

 an infinite surface and Eq. (10.14) becomes 



(10.15) P(r, = 2P.(r. i) - ^ j j ^^ ^ (^»)^ ..4' 



where the integral is extended over the area ^o of the plate. This 

 equation represents the doubling of pressure which would occur if the 

 surface were perfectly rigid, less the effect of motion of the surface in 

 the direction of the incident wave. 



If no baffle at all is assumed and boundaries behind the plate are 

 ignored, we have Un = Uon external to the area Ao and po duon/dt = 

 — grad Pi, giving 



(10.16) ^(r.O = ^.(M)+|^JJ^^-^(|^)^^A' 



If the incident wave varies slowly enough with distance from the sur- 

 face, the second integral can be neglected. In this limit, the difference 

 between presence and absence of a baffle is that in the latter case the 

 pressure Pi, rather than the doubled pressure 2Pi, is modified by the 

 rarefaction or damping term from motion of the surface. The effect of 

 diffraction from the unbaffled periphery of the surface is thus to reduce 

 and ultimately destroy the doubling of pressure resulting from a rigid 

 surface. The effect of an imperfectly rigid baffle or boundary can be 

 taken into account by suitable assumptions as to the velocity distri- 

 bution on such boundaries, as has been done by Kirkwood (61), and 

 Kennard (57). 



C. The pressure at the center of a circular plate. Only the simple 

 case of a circular plate of radius a in an infinite rigid baffle will be con- 

 sidered further here, and a further simplifying assumption will be made 

 that the plate deflection is always paraboloidal. This assumption im- 

 plicitly requires a suitable constraint by which this profile is artificially 



