Chapter 8

Mechanical waves

To test the understanding of the lesson

Question

  • What is the definition of the phase velocity of a wave phenomenon ?

  • Give the definition of the group velocity of a wave phenomenon.

Hint

Solution

  • Phase velocity :

  • Group velocity :

It was assumed here that was real (no absorption).

Question

Provide part of the acoustic approximation.

Hint

Solution

The acoustic approximation is to consider that the magnitudes , (variation of density around the equilibrium value) andt (change in pressure around the equilibrium value) are infinitesimal of the same order as their derivatives time and space.

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

Magnitude :

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

Question

Defining "acoustic Poynting vector".

Hint

Solution

The "Poynting vector" acoustics :

The conservation equation of energy of a sound wave is written :

Question

Define the acoustic impedance.

Hint

Solution

That is the pressure , the velocity of fluid particles and the volume flow rate .

We can define the acoustic impedance in two different ways : ( is the cross section of the studied acoustic pipe)

We note that :

Question

Define the sound intensity in decibels ( ).

Hint

Solution

We define the sound intensity (or acoustic) in decibels ( ) :

Question

  • What is the expression of the speed of sound in air ?

  • What is its value at under ?

Hint

Solution

  • The speed of sound in air is :

  • At and  :

Question

Why do we say that a wave is propagated even more harm than the nedium is softer and more inert ?

Hint

Solution

  • The sound propagation speed in the solids is :

  • The speed of propagation of a wave in a string is :

We see that is even smaller than the medium is "soft" ( or "weak") and inert ( and "large").

Question

What is called the approximation of continuous medium ?

Hint

Solution

are the vibratory displacement at discrete points , small compared to the wavelength of the wave.

The approximation of continuous medium is to construct a continuous function such that :

Question

Is a beam of rectangular cross section steel, height , width and length, fitted at both ends.

Subjecting this beam to the rise of a temperature of .

The modulus of elasticity of the steel is and its coefficient of expansion. .

What is the compressive stress in the beam ?

Hint

Solution

The constraint (It's a pressure) is given by :

Where is the Young's modulus.

The coefficient of expansion is used to determine the relative change in length :

We deduce :

Previous
Transversal oscillations of a leaded rope