Did Isaac Newton derive the expression for the speed of water waves in both shallow water and deep water approximations?

Question

My questions refer to propositions 44-46 of section 8 of book 2 of Isaac Newton's principia, where Newton draws analogy between oscillations of water in a U - shaped tube and water waves in deep water. Newton reasons from this analogy an important early result on water waves - the speed of water waves in the limit of deep water (i.e when the depth of water is much higher than the wavelength) is proportional to the squared root of their wavelength.

Now comes my question. Chandrasekhar's book "Newton's principia for the common reader" states that in the conclusion remarks of these propositions Newton also stated without proof the result for the speed of shallow water waves - that their speed is the squared root of gH. But i was unable to find it in the principia.

So my question is: did Newton derive also the expression for shallow water waves or not?

Answer 1

It seems not.

Newton states in Prop. XLV that the velocity of waves is as the square root of the wavelength, and again in Corol. 2 of Prop. XLVI.

Newton defines wavelength as "the transverse distance either between bottoms of troughs or between tops of crests." Newton notes that he has considered that "the parts of the water go straight up or straight down," but since "this ascent and descent takes place more truly in a circle," he admits that "in this proposition the time has been determined only approximately."

Thus, it seems that there is no ref to the depth of water...