Waves - longitudinal waves

    Longitudinal waves are the disturbances propagating in a medium where the particles of the medium move in the direction of the movement of the wave, that is they vibrate back and forth along the direction of the motion of the wave.At some points the particles come closer than their mean position and the density as well as the pressure of the medium increases are called compressions.The points where the pressure is medium decreases because the particles are separated apart are called rarefactions.Mechanical longitudinal  waves (mechanical waves are those which require a material medium to propagate through) are sometimes also referred as compressional or compression wave.


                             Source:http://scienceprep.org/images/waves.jpg


 Example of a longitudinal wave is sound wave.The displacement of particles as well the alternation in pressure of the medium are examples longitudinal waves.

Mathematically,longitudinal harmonic sound waves can be represented as:



where
y = displacement of a particular point on the propagating sound wave.
x = distance travelled by the particle from the source.
t = time taken to cover the distance. (i.e., x is the distance travelled at time t)
y₀ = amplitude of the oscillation.
c = speed of wave (sound here).
ω = angular frequency of the wave. ( ω = 2πf, f = frequency of the wave).

While in elastic medium having rigidity the harmonic pressure wave can be written as:



y = amplitude of displacement.
x = distance along the axis of the propagation.
k = wave number = 2π/λ.
ω = angular frequency.
φ = phase difference.