Mechanical waves

Mechanical waves
These waves need a medium in order to propagate.
Examples
transverse or longitudinal waves on a spring
sound waves in a solid
sound waves in air
waves in water

Speed of Mechanical Waves
Depends on
1) a density factor
2) a stiffness or elasticity factor
eg transverse waves on string/spring
where T is tension
and μ is mass per unit length

In the arrangement shown below an object of mass 5.00 kg hangs from a cord around a light pulley. The length of the cord between point P and the pulley is 2.00 m. When the vibrator is set to a frequency of 150 Hz, a standing wave with six loops is formed. What must be the linear mass density of the cord?
Example 1

Sound wave: how it propagates through a row of particles

Sound is therefore a wave of pressure compressions (more dense regions) and rarefactions (less dense regions) travelling through a medium.
Alternatively it is a wave of longitudinal particle displacements.
There is a phase difference between particle displacements and pressure (of 900)
Our ear drums detect the pressure difference.
Sound travels much faster in a solid than in a gas

Intensity (I) of any wave is the energy per second crossing unit area perpendicular to the wave
(applies to em waves as well as sound waves)
We can show that for any point source of waves the intensity of the source at some position:
where P is the power of the source, and r is the distance of the source from the position

The human ear can detect sounds over a huge range of intensities: about 10-12 Wm-2 to about 102 Wm-2.
The sensation of loudness is approximately logarithmic ie the loudness doubles when intensity increases by a factor of 10.
We define a loudness scale by:
Number of decibels =10 log(I/I0)
where I0 = 10 -12 Wm-2

The intensity of electromagnetic radiation from the sun arriving at the collector panel of a solar water heater is 500 W m-2. How much energy does the heater receive in 1 hour if the area of the collector is 2.5 m2 ?
Example 2

a) A loudspeaker can be considered to be a point source of sound. If the total power of the loudspeaker is 80 mW, what is the intensity, I, at 3 m from the loudspeaker?
b) What is the loudness in decibels at a distance of 0.25 m from the loudspeaker
c) Find the difference in decibels between a point 0.1 m from the loudspeaker and a point 100 m from the speaker.
Example 3

Recall, for SHM, energy = ½mω2A2 so
energy is proportional to A2
All mechanical waves consist of particles undergoing SHM, and so it can be shown that :
For all waves: intensity is proportional to (Amplitude)2