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 :