Waves in a pool
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A detailed solution is then proposed to you.
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Consider a perfect incompressible fluid (density ) in a basin of width (along Oy).
A wave propagates along (Ox). The velocity field is of the form :
The pressure is hydrostatic in the canal.
Note the height of the water in the canal ( is a constant).
The flow is assumed irrotational.
Applying the Euler equation to a fluid particle, find a differential equation between and assuming small movements.
Show that is indépendent of .
The Euler equation is :
The pressure is hydrostatic, which means it is given by the expression obtained in fluid statics, namely :
Assuming small movements, it keeps only the first order terms (similar approximation to the acoustic approximation for the study of waves in fluids).
The Euler equation becomes :
In projection along the horizontal, the differential equation is obtained between and :
The flow is incompressible, so :
In order of magnitude, if we note the wavelength of the wave (in the direction (Ox)) and a characteristic dimension of the vertical movement (with ) :
We can therefore neglect the vertical component of velocity.
The flow is irrotational :
This leads to, neglecting :
Therefore, does not depend on .
Compute the volume flow rate .
Make a mass balance on a slice of thickness .
Deduce a differential equation between and assuming small movements.
The volume flow rate is :
Mass balance on a slice of thickness , located between and :
is the mass variation in this slice. It can be written in two ways :
For identification :
The first order :
Show that and check each classical wave equation.
Deduce the velocity of propagation of waves in the channel.
It was also the two equations :
By decoupling these two equations, we arrive at the d'Alembert equation :
Where the wave propagation speed is :