FYKOS Year 35 Series 3 Problem 4 — gentle tide · FYKOS 35 #35.3.4

Physics · mechanics

Problem

Close to the shore, the speed of sea waves is influenced by the presence of the sea bed. Assume that the speed of waves

v

is a function of the gravity of Earth

g

and the water depth

h

. We have

v = C g^\alpha h^\beta

. Using dimensional analysis, determine the speed of the waves as a function of the depth. Constant

C

is dimensionless, and cannot be determined using this method. Besides the speed of the waves, swimming Jindra is also interested in the direction of incidence of the waves. Let's define a system of coordinates, where the water surface lies in the

xy

plane. The shoreline follows the equation

y = 0

, the ocean lies in the

y > 0

half-plane. The water depth

h

is given as a function of distance from the shore

h = \gamma y

, where

\gamma = \const

. On the open ocean, where the speed of the waves is constant

c

(not influenced by the depth), plane waves are propagating at incidence angle

\theta_0

to the

x

axis. Find a differential equation

\der{y}{x} = f\!\left(y\right)

describing the shape of the wavefront close to the shore, but do not attempt to solve it, it is far from trivial. Calculate the incidence angle of the wavefront at the shoreline. Bonus Solve the differential equation and find the shape of the wavefront close to the shore.

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