Equations (13) and (14) are one vector equation and one scalar 

 equat ion, respect ively, for the scalar and vector unknowns, (jO and k. 

 However, they are not independent since the curl of equation (13) gives 



8(Vxk) 



hence, equation (14) can be interpreted as just an initial condition for 

 equation (15). This is similar to the irrotational condition, that 

 vorticity is zero, being used for inviscid flow, whereas it may also be 

 considered simply as an initial condition for use with Kelvin's 

 circulation theorem. However, since there are three unknowns, w , k|^, 

 and k2, an extra equation is needed. The waves are locally like a plane 

 wave train so they must satisfy the dispersion equation (5) which may be 

 written: 



w = k • u + cr(k,d) (16) 



It is in this equation that the space and time dependence of the medium 

 enters in u(x,t) and d(x). 



Now (J can be eliminated by using equation (16) in equation (13) to 

 give 



,3 , , . „ X nil da dd . 6 



[■^r^r + (u + C ) • V]k = ^-r -r k„ -5 .- ,x 



3t ^^ ^g ■■ a 9d 3x 6 dx (17) 



after using equation (14) where 3 a/ 8d =CJk/sinh 2kd from equation (4). 

 Equation (17) is in tensor notation for which the repeated subscript 6 

 Indicates summation over g = 1, 2. 



Another useful equation to be obtained from the same three equations 

 is 



[|j+(u + C) .Vl..k.^-.|ff. (18) 



where d has been allowed time variation for full generality. Here and 

 elsewhere 



28 



