Forces and their Relationship to the Structure of the Ocean 325 



At all inner boundary surfaces, on the other hand, the velocity component perpendicu- 

 lar to the boundary surface must be the same on both sides of the surface. If the values 

 for the quantities on both sides of the boundary are specified by separate indices, 

 then this kinematic boundary condition can be represented as a special case of equation 

 (X. 28) 



/("i - «2) + rn{vi - ^'2) + n{yv\ - w^) = 0. (X.29) 



From the point of view of continuity it is allowed to make a free choice about the 

 velocity component parallel to the inner boundary surface and solid surface, respec- 

 tively. 



If the liquid has Sifree upper surface this will be subject to the condition that all the 

 small fluid elements which belong to it will always remain in the liquid. If/Cv, y, r, /) = 

 is the equation for the free upper surface the foregoing condition requires that 



In addition to the kinematic, there is also a dynamic boundary-surface condition 

 that must be satisfied at inner boundary surfaces as well as at a free surface. This 

 requires that at the discontinuity surface where the individual quantities are subject 

 to sudden changes, the pressure must be the same on both sides of the boundary. If 

 /(x, y, z, /) = is the equation for the discontinuity surface, which may be either 

 moving or stationary, and if/7i and/72 give the pressures in the medium on both sides 

 of the surface as functions of .Vi, y^, z^ and x^, J2, z^, respectively, then the dynamic 

 boundary condition will require that values of x, y, 2 and t, in order to satisfy 

 f{x, y, z, t) = 0, must also satisfy the equation 



PiiXi, >i, Ti, — p^ix^, J2, Z2, /) = 0. (X.31) 



4. Potential Flow, the Bernoulli Equation, Impulse and the Impulse Form of the 

 Hydrodynamic Equations 



In very many cases the velocity components u, v, w can be expressed by a function 

 9, so that 



80 S(D do 



This function then is called the velocity potential, and movement for which a function of 

 ' this type is valid has been termed o. potential flow. By this kind of definition it is shown 

 that if such a potential is present: 



(1) The stream lines will be everywhere perpendicular to the surfaces 9 = const, 

 (equi-potential surfaces of velocity). This follows from (X. 22) when combined 

 with (X. 32). 



(2) The following combinationary relationships : 



du 8v dv 8w 8w 8u 



8y 8x ' 8z 8y ' 8x 8z 



will apply ; these state that the current in the presence of a velocity potential is 

 irrotational (free of vorticity). 



