Hasselmann and Sahieler 



A useful feature of the relation (17) is that it defines the 

 surface -wave spectrum in absolute energy units independent of 

 electromagnetic calibration factors, which are difficult to establish 

 for long-range ionospheric mode propagation. Using Eqs, (3) and 

 (2) to eliminate the surface- wave spectriam at the Bragg wavenumber, 

 Eq, (17) becomes 



T,(l) (2)8 . . 

 Eg(cod - swg) + Eg(sojg - <0d) = ^ ^^--^^ (s=±) (18) 



where c^'''= / x (wj) d"H is the energy of the first-order Bragg 



'= / X^'^' ("d) «^"d is 

 ratio T<'^ /2T<2» ca 



line. The ratio T\" /2T*'^' can be determined from theory, and 

 X^^ and c^''' may be measured in arbitrary energy units. 



In the relevant limit k*'« k^ T /2T may be deduced 

 from the picture of a short scattering wave ^u = Ab exp {i(k''x - ovt)} 

 riding on a long carrier wave t,Q = Aq exp {i(k° x - Wat)} (wiifch is 

 now, howeyer, assumed to satisfy the wave- wave interaction con- 

 dition Aflkj « i, rather than the wave-facet interaction condition 

 (5) ). For small slopes Agk « 1 , the principal effect of the carrier 

 wave is presumably to alter the phase of the scattered field by raising 

 and lowering the local mean reference surface of the short scattering 

 wavesT, Thus if the first-order backscattered wave in the absence 

 of the carrier wave is of the form 



^^'^ = C^'^ A^A, exp {i(k' + o-^k'')x - i(cOj + o-^w^t + ik^Xj} 



where Aj is the amplitude of the incident field, C is a first- 

 order coupling coefficient, and k' + a^k *« - k , k|« - kj, 

 the modulated scattered wave in'the pr'esence~of tfie carrier wave 

 will be given approximately by 



~ JikiCb (1) _ ,. 4.-,:i,ir \ I') - (D j. (2) nax 



<p - & <p « (1 + 2ik^Cjj)^ = <p -^ <p (19) 



Thus 



^(2) ^ d^^ AoA^A\ exp {- i k'x - i(a)i + Wd)t - IkgXj} (20) 



with C = 2ikl.C . Expressed in terms of a continuous energy 

 spectrum, this is readily found to correspond to a scattering function 

 ratio 



A more detailed investigation indicates that slope effects can be 

 ignored if k° « k^ = k' sin 6. 



384 



