74 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS
dydz'
dzdx*
.(143).
dxdy
Let dS be an element of the bounding surface, I', m'} ri the direction-cosines of the normal drawn outwards, AP, AQ, AJ? the components in the direction of x, y, z of the additional pressure on a plane in the direction of dS. Then by the formulae (9) of my former paper applied to the equations (142), (143) we get
AP = ~2/JZ'^
'+n' ...(144),
dxdy dxdzj v h
with similar expressions for AQ and AJK, and AP, AQ, AJ2 are the components of the pressure which must be applied at the surface, in order to preserve the original motion unaltered by friction.
53. Let us apply this method to the case of oscillatory waves, considered in Art. 51. In this case the bounding surface is nearly horizontal, and its vertical ordinates are very small, and since the squares of small quantities are neglected, we may suppose the surface to coincide with the plane of xz in calculating the system of pressures which must be supplied, in order to keep up the motion. Moreover, since the motion is symmetrical with respect to the plane of xy, there will be no tangential pressure in the direction of z, so that the only pressures we have to calculate are AP2 and ATS. We get from (140), (142), and (143), putting y = 0 after differentiation,
AP == -
c sin (mx — nt),
2c cos (mx — nt). . .(145).
If ult vI be the velocities of the surface, we get from (140), putting y = 0 after differentiation,
Wj = me cos (mx — nt\ v1 = — mo sin (mx — nt).. .(146).
It appears from (145) and (146) that the oblique pressure which must be supplied at the surface in order to keep up the motion is constant in magnitude, and always acts in the direction in which the particles are moving.