Question

Surface waves on water with wavelengths large compared to depth are described by the equatio

d^2h/dx^2 + d^2h/dy^2 = 1/gd . d^2h/dt^2

where g is the acceleration of gravity, d is the equilibrium depth of the water, and h(x,y,t) is the height of the wave above the surface is equilibrium position. What is the speed of traveling waves described by this equation?

Answer 1

Answer with Explanation:

The general wave equation is given by

`(\partial ^2u)/(\partial x^2)+(\partial ^2u)/(\partial y^2)=(1)/(c^2)(\partial ^2u)/(\partial t^2)`

where

'c' is the velocity of the wave

Comparing with the given equation

`(\partial ^2h)/(\partial x^2)+(\partial ^2h)/(\partial y^2)=(1)/(gd)(\partial ^2h)/(\partial t^2)`

We can see that

`c^2=gd\\\\\therefore c=√(gd)`

Thus the velocity of wave is given by
`v=√(gd)`