SURFACE REVERBERATION 



263 



and { is the angle of tilt of the transducer axis relative 

 to the horizontal plane. The relation (41) is probably 

 not valid for angles d greater than 30 degrees, at 

 which angle the directivity pattern of any actual 

 transducer is likely to differ appreciably from the 

 ideal. Equation (41) was proved in reference 4 only 

 for rectangular pistons. However, even for the circu- 

 lar pistons used in most Navy gear, the use of equa- 

 tion (41) is probably legitimate as long as the correc- 

 tion factor b{e- ?,0)6'(e - ?,0)(cos 5)-' does not dif- 

 fer too much from unity. For a horizontal trans- 

 ducer (? = 0), typical values of the correction 



b{e,ow(e,o) 



10 log 



^ cose 



are given in Table 1. This table was computed by the 



Table 1 



flJn^»„r»»= in 1^.. b(e,0)b'(e,0) 



use of values of b and b' measured for the EBl-1, 

 which is similar to standard 24-kc Navy gear.^ The 

 use of corrections greater than — 12 db is probably 

 not justified. Some further discussion of the use of 

 this correction is given in Chapter 15. 



Formulas for Js(0) = 10 log Q(0)/27r are given in 

 reference 3. For transducers which are circular or 

 rectangular pistons 



7,(0) = 10 log y - 23.8. (42) 



As was done with the expressions for volume rever- 

 beration, we may define the surface reverberation 

 level R'it) as 



R'(t) = lOlogG(i) - lOlog(F-F') 



= 10 log (^^ + 10 log (^') - 30 log r 



+ Js{B) - 2A. (43) 



Also, we define standard siuface reverberation level 

 R{t) as the level of the average reverberation which 

 would have been received at the time t if the ping 

 length had had some standard value to instead of t. 

 Then, 



Rit) = 101ogG(<j - 10 log (F-F') + lOlogf-V 



(44) 



so that 



R(t) = 10 log (^°) + 10 log (^') - 30 log r 



+ J,{e) - 2A. (45) 



Reverberation strengths can also be defined in a 

 manner similar to that in Section 12.2. 



When using equations (39), (43), and (45) it is 

 necessary to remember that A has been defined as 

 the transmission anomaly along the actual ray path 

 to the surface. This transmission anomaly may differ 

 from A', the value of the transmission anomaly 

 measured in the usual experimental determination of 

 transmission loss. Consider specifically that the pro- 

 jector axis is horizontal and that a ray leaving the 

 projector with the angle of elevation 9 and an azimuth 

 angle (j> of zero reaches the surface after covering the 

 slant range r. Then from the definition of A, 



A = 10 log F+10 log b(,e,0) - 10 log 7-20 log r, 



(46) 



where / is the measured intensity in db at the point 

 where the ray strikes the surface. The measured 

 anomaly A' is usually determined from the equation 



A' = 10 log F - 10 log 7-20 log r' (47) 



where r' is the horizontal range from the projector 

 to the point where the ray strikes the surface. 

 Neglecting the difference between r and r', we have 

 from equations (46) and (47) 



A = A' + 101og6(e,0). (48) 



Further, from equations (40) and (41) we have for a 

 horizontal transducer 



J.(.e] = Js{0) + 10 log b{d,0) -\- 10 log b'idfi) 



- 10 log cos 9. (49) 



Substituting equations (48) and (49) in equation (45) 

 gives 



R(t) = 10 log (^°) -f- 10 log (^) - 30 log r 



+ J.iO) - 2A' - 10 log b{9,0) 



+ 10 log b'i9,0) - 10 log cos 9. (50) 



Under most circumstances cos 6 is sufficiently near 

 unity and the projecting and receiving patterns are 

 sufficiently sjinmetrical, with the result that the last 

 three terms in equation (50) can be neglected. Thus, 

 if the measured transmission anomaly A' defined by 

 equation (47) is used in the analysis of surface 

 reverberation, the correct expression for the standard 

 reverberation level imder most circumstances is 



