# Mechanical waves

# Waves in a pool

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

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.

## Question

Applying the Euler equation to a fluid particle, find a differential equation between and assuming small movements.

Show that is indépendent of .

### Hint

### Solution

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 :

Is :

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 .

## Question

Compute the volume flow rate .

Make a mass balance on a slice of thickness .

Deduce a differential equation between and assuming small movements.

### Hint

### Solution

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 :

Or :

For identification :

The first order :

## Question

Show that and check each classical wave equation.

Deduce the velocity of propagation of waves in the channel.

### Hint

### Solution

It was also the two equations :

And :

By decoupling these two equations, we arrive at the d'Alembert equation :

Where the wave propagation speed is :