Forces and their Relationship to the Structure of the Ocean 327 



In addition, the differential equation J0 = must also be satisfied by i/-. Since the 

 curves ^ — const, are perpendicular to the curves 9 = const, 



\dx 8y '^ 8y dx "' 



They represent stream lines (hence, the name stream function). 



It can easily be shown that every analytical function of the complex variable 

 r = .V + iy satisfies the continuity equation Jcp = 0, i.e. represents a solution for 

 the equations of motion. If this function is given by 



F(z) = F(x + iy), 



then its real part is the velocity potential (p and the imaginary part is the stream func- 

 tion ifj or vice versa. This important consequence allows simpler current systems to be 

 completely worked out kinematically. Use will be made of this later (see Chap. XII, 3). 



In a few important cases the use of the impulse theorems for steady currents in a water 

 mass has considerable advantages. The product of mass and velocity is termed the 

 impulse or momentum; as a vector, like velocity, it has three components. The impulse 

 theorem states that for any arbitrarily limited water mass (the outer boundary sur- 

 faces all together are usually termed "control surface") the change with time of the 

 impulse in it is equal to the sum of the external forces acting on the mass. The internal 

 forces in the system balance each other according to the principle of action and 

 reaction. The change in momentum can be divided into two parts. The first gives the 

 change with time of the impulse in the volume under consideration enclosed by the 

 control surface; for a steady current this term vanishes. The second is the momentum 

 entering or leaving it in unit time through all the boundaries (total control surface). 

 For a steady current the vector sum of all pressures acting on the control surface must 

 be equal to the transport of impulse through it. 



As an example, the following two cases will be considered here. Fig. 136a shows a 

 straight current tube formed by stream lines ; we consider the part between 1 and 2. 

 At the cross-section 1 (surface F^) the current enters with a velocity V^. The water 



Fig. 136a 



mass transported in unit time is pV-^F^, the impulse transport (momentum flux) 

 through Fj into the volume under consideration is p Fj^Fi ; similarly, at cross-section 2 

 (surface F^ an impulse amount pV^Fz leaves the enclosed space; as a "counter action" 

 it has to be taken with a negative sign. The impulse amount remaining in the space is 

 thus p(Ki^Fj — V^F^. In a steady current, in order to secure an equilibrium state, 



