the bounding surface. This requires that v n = _or that 

 p = on the surface, a situation which is approached only 

 if a large difference in acoustic impedance {pa) exists 

 across the bounding surface. 



In the case of the viscous fluid, even if p or v n vanishes 

 on the bounding surface of the medium, the energy of free 

 acoustic vibrations will decay at a rate depending upon the 

 square of the time rate of change of p. If the disturbances 

 are nearly simple harmonic (except for a slow decay) then 



2 



(JU 2 p' 



dt 



where uu is the frequency (radians /second) and the bar 

 indicates a time average over an integral number of oscil- 

 lations. Moreover, the energy E can be considered (on the 

 average) as half kinetic and half intrinsic. Accordingly, 

 it follows from (22) that, for total reflection at the bounding 

 surface S, 



^— = -$w 3 E (22b) 



3 t 



E-E n e~^ t (23) 



If energy does radiate through the bounding surface at 

 a rate proportional to the mean energy density (El V) within 

 the medium then the damping modulus Buu 2 in (2 3) must be 

 replaced by 



(^ + y t f) 



36 



