Theory of Short and Long Waves 17 



In addition to these kinematic boundary conditions there is a dynamic 

 boundary condition to be fulfilled at the free surface, namely that pressure 

 and counter pressure be equal. If we neglect the influence which the atmosphere 

 exerts on the water motion (see p. 83), this condition is simply reduced to 

 the requirements that the pressure at the surface be equal to the uniform 

 air pressure p . If the motion is small, we can neglect in (11.2) the square 

 of the velocity c in a first approximation and. provided the function F(t), 

 and the additive constant /) /o. be supposed merged in the value of dqjfdt 

 we obtain for 



z~0:„— J£. (II.4) 



The equation (II. 1) with the boundary conditions give the possible wave 

 motion in the water mass of the depth h (Airy, 1845; Stokes, 1880, vol. I, 

 p. 197; Lamb. 1932, p. 227). 



If (p is a simple-harmonic function of x 



<F = Pcos(xx-at) . (II. 5) 



According to (II. 1) P must satisfy the differential equation 



cPP 



y}P = , 



dz- 



of which the general solution is P = Ae +xz +Be +xz . The boundary condition at the bottom 



ctpl'dz = for z = — h 

 gives 



A e - yh = Be'*' 1 = hC 



and from (II. 5) results*: 



(f = Ccoshy.iz— h)cos(y.x-at) . (II. 6) 



From the equations (II. 3) and (II. 4) we obtain drjjdt and >/. If we differentiate again with respect 

 to t we get d))\bt. By equating both terms we get the wave frequency in the form: 



a- = gy.\.a.n\\y.h . (II. 7) 



If we write 



t) = Asm(y.x-ot) , (II. 8) 



then we obtain for the velocity potential 



gA coshx(z+h) 



(f = — cos(xx— at). (H.9) 



a cosh y.h 



The relation (II. 8) represents ah infinite train of progressive waves with 

 a harmonic wave profile, travelling in the —a- direction with the velocity 



* Hyperbolic functions are used more often and therefore we give here their relationship with 

 the exponential functions: 



e x —e~ x e Xj re~ x e x —e~ x 



sinh.v = , coshx = , tanh.v = , sech.v = 



; x +e~ x coshx 



