III. DYNAMICAL THEORY OF TIDES (cont.)

15. Semi-Diurnal Tide, with Variable Depth

Next let us consider the semi-diurnal tide in the case where q = 1, so that γ = l sin² θ. Then e = E sin² θ, k = 2 , f = 1 ; also υ ₀ = – e = – E sin² θ. Hence by (28) and (26) – 8lυ₀ + 4mlυ₁ = 0, whence υ₁ = 2/mυ₀. Applying the same theorem a second time, υ₂ = (2/m)² υ₀,

If the height of tide is equal to the equilibrium height; but it is inverted, giving low water where the equilibrium theory gives high water. In the case of the earth m = 1/289, and therefore this relation is satisfied if l = a/1156. Hence in a sea 3000 fathoms deep at the equator, and shallowing to the poles, we have inverted semi-diurnal tides of the equilibrium height.






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