208
X
OCEANIC CURRENTS
T the length of a day. Let
water above the mean level, and let
= cos 4π
denote the elevation of the
t
L T
be a bidiurnal tidal wave which would traverse the canal
under the action of a heavenly body, on the equilibrium
theory. Then the tidal wave which is actually produced is
given by the equation
where
4π
t
5 = 5,000 47 (12 — — — <).
-
L T
kAL
tan 4πe
=
2πμh²(gh — А²)'
-
and
=
=
2πgµ³¿ sin 4πе.
KAL
Here k denotes the coefficient of viscosity of water, and A
L
= T
denotes the velocity of propagation of the wave, μ the density
of water, and g the acceleration of gravity. In the calculation
squares and products of small quantities are neglected. For
instance, at the free surface the tangential component of
pressure is taken to be zero for the mean level; whilst in
reality it is zero for the actual level. We find that this error
of the second order may be compensated by supposing a tension
T to act at the surface in the direction of propagation of the
wave, of which the magnitude is the mean of the values of
μX at different times, where X denotes the component of
gravitational attraction along the canal. For the tidal wave
considered above, we have
T=
3 2
4m²μ²g²h³ sin 24πe=
kAL2
Ak
hs
This tension corresponds to a current flowing along the canal
in the direction of the tidal wave, and increasing in velocity
uniformly from the bed of the canal to the velocity
με
at the surface.
4m²h g²μ
2AL2
21.4 2.2
sin247e A
=
h2