III-79 



Following an approach similar to that used to obtain the equations for 

 the A^, it can be shown that MS (| p| ), the mean square value of the total pres- 

 sure p at the surface, is given by 



Since p = on the surface, it follows that MS(|pI) = 0. The evaluating (III-79) 

 with the values of Aj^ obtained from the finite set of linear equations shows the 

 effect of neglecting the damped plane waves . 



From (III-6), the intensity, I^ , in the z direction of a plane wave 



p = A e^^^"^^"^^^ is given by I^ = ^ ^ ^ I A |^ . The difference between the 



o o 

 energy incident on the surface and that reflected in plane undamped waves is 

 therefore proportional to 



E - Y -£l A 1% (III-80) 



n 



E 



where the summation is over n such that I a + -j^I si. Then 1 - - is the frac- 



K Y 



tion of the incident energy contained in the plane undamped waves and (E/y) is 

 the fraction absorbed. 



Table III-3 gives the results of some calculations done by Heaps (Ref , 



III-13) for a surface of the form S(x) = h cos px with h = 0.15 cm and p = 3.08 



cm"-^ . The coefficients A were obtained from (III-72), where the summation 

 n 



is only over n such that I a + — I ^ 1 . The appearance of non-zero values of 



k 

 MS (I pI ^ ) is a consequence of neglecting terms in (III-72) involving larger values 

 of n. The value of 1 - E/y gives the fraction of the incident energy reflected in 

 plane undamped waves . The figures in parentheses are the corresponding quanti- 

 ties using the Rayleigh values for A . 



From Table III- 3a, it is apparent that at normal incidence (9=0°) the 

 listed values of A^ satisfy the condition MS| p]^ = (and so also the condition 

 of zero surface pressure) much better than the Rayleigh values do and, in fact, 

 account for about 90% of the incident energy, whereas the Rayleigh theory ac- 

 counts for only 68% of the energy. Table III-3b suggests that, at an angle of 

 incidence of 40°, there is little to choose between the two theories . In either 

 case, an appreciable portion (10% or more) of the incident energy must be re- 

 flected in other than plane undamped waves . 



artliur B.%itth3nt. 



S-7001-0307 



