Chapter 8

Mechanical waves

Acoustic waves

ComplémentA video (in french) about acoustic waves (with some fun science experiments)

Ondes sonores dans les fluides

FondamentalAcoustic approximation

Experience shows that the propagation of sound waves is generally characterized by a low damping within the fluid in which they propagate.

We therefore neglect the dissipative processes (thermal conductivity and viscosity), which amounts to applying the isentropic character of the propagation of sound waves and thus to assume a perfect fluid.

The only forces considered are the pressure forces (gravity is neglected).

That is , and the characteristics of the fluid at rest (assumed uniform), we note :

  • , the variation of density of the fluid ( )

  • , the variation of fluid pressure, also called acoustic pressure ( )

  • the vector velocity of a fluid particle (no rest)

The acoustic approximation is to consider that the magnitudes , and are infinitesimal of the same order (and their spatial and temporal derivatives also).

In particular, the calculations will be made to order in this infinitely small.

Magnitude of the acoustic pressure :

Overpressure likely to be detected by the ear typically range from (his painfull) to (hearing threshold), covering decades.

FondamentalHypothesis of adiabatic

The propagation of sound in a fluid can be studied by considering that the fluid makes small movements isentropic.

As part of the acoustic approximation, the coefficient of isentropic compressibility gives :

FondamentalLinearization of the conservation equation of mass and the Euler equation

  • The mass conservation equation is :

    Either, with  :

    Whence :

    Linear approximation (or acoustic) : we limit the following order terms.

    Therefore :

  • The equation of motion of the fluid is here Euler's equation (no viscosity) :

    The static volume force (eg, ) is compensated by the gradient of static pressure .

    The influence of variations (for example ) is in practice negligible compared to the pressure gradient .

    The equation of motion becomes, after linearization :

  • More we recall the relationship between acoustic pressure and the change in density :

  • Eliminating variable , one obtains the coupled equations system :

FondamentalEquation propagation

  • For overpressure  :

    Calculating the divergence of the previous left equation :

    Is :

    We recognize the d'Alembert wave equation ; the speed of the sound waves is deduced :

  • For speed  :

    Writing that :

    By taking the gradient while deriving the equation :

    With respect to time :

    We have used the relation : (see the lesson Vector calculus)

    Very often, there will be the velocity field of a form , Therefore . This result can be generalized.

    The equation satisfied by the velocity field becomes :

    We find an identical d'Alembert relation to that obtained for the pressure.

AttentionEquations d'Alembert for the pressure and the velocity field

ExempleOrder of magnitude of the speed of sound in air

For an ideal gas in isentropic evolution, Laplace's law gives :

Or, using the acoustic hypothesis :

The speed of sound is :

Where is the temperature of the gas and its molecular weight.

Numerical applications :

With , and  : .

Note :

  • For a liquid, : the speed of sound is higher in the liquid than in gases.

  • For solids is not within the scope of the study here (see the model of the atom chain coupled by springs).

    It notes, however, that the speed of sound in a solid is even higher than in liquid.

SimulationAcoustics and vibration animations (by Daniel A.Russell)

  • Click HERE :

    An animation to visualize particle motion and pressure for longitudinal sound waves.

ComplémentA video on the Helmholtz resonator

Why blow in a bottle created a sound ? (Reference : "")

Why Blowing in Bottles Makes Sound and Helmholtz Resonance

ComplémentA video about Hot chocolate effect

The hot chocolate effect is a phenomenon of wave mechanics, where the pitch heard from tapping a cup of hot liquid rises after the addition of a soluble powder.

It can be observed in the making of hot chocolate !

Hot chocolate effect

SimulationJava animations

  • Java animations (by Jean-Jacques Rousseau, University of Le Mans) :

    The Helmholtz resonator : click HERE

Dispersion on a chain of atoms
Influence of viscosity on the propagation of sound