Proposition 51, Theorem 39

Find the velocity of waves.

A pendulum’s length, between the point of suspension and the centre of oscillation, is equal to the width of the waves.

At the time that the pendulum will perform one oscillation, the waves will advance forward nearly a space equal to their width.

The width of the waves is the transverse measure lying between the deepest part of the hollows, or the tops of the ridges.

Let ABCDEF represent the surface of stagnant water ascending and descending in successive waves.

Let A, C, E, etc. be the tops of the waves.

Let B, D, F, etc. be the intermediate hollows.

The motion of the waves is carried on by the successive ascent and descent of the water.

Its parts as A, C, E, etc. which are highest at one time become lowest immediately after.

The motive force that causes the highest parts to fall and the lowest to rise is the weight of the elevated water.

That alternating ascent and descent will:

• be analogous to the reciprocal motion of the water in the canal
• observe the same laws as to the times of its rise and fall.

By Prop. 44, if the distances between the highest places of the waves A, C, E, and the lowest B, D, F are equal to twice the length of any pendulum, the highest parts A, C, E will become the lowest in the time of one oscillation, and in the time of another oscillation will ascend again.

Therefore, between the passage of each wave, the time of two oscillations will intervene; that is, the wave will describe its breadth in the time that pendulum will oscillate twice.

But a pendulum of 4 times that length, and which therefore is equal to the breadth of the waves, will just oscillate once in that time. Q.E.D.

Corollary 1

Therefore waves, whose breadth is equal to 3 1 18 French feet, will advance through a space equal to their breadth in one second of time; and therefore in one minute will go over a space of 183⅓ feet; and in an hour a space of 11000 feet, nearly.

Corollary 2

The speed of greater or less waves will be augmented or diminished in the subduplicate ratio of their breadth.

These things are true upon the supposition that the parts of water ascend or descend in a right line; but, in truth, that ascent and descent is rather performed in a circle;

Therefore I propose the time defined by this Proposition as only near the truth.