Mechanical wave

PROGRESSIVE TRANSVERSE WAVE
Each particle vibrates perpendicular to the direction of propagation with the same amplitude and frequency
Light and all other electromagnetic waves, and water waves are transverse

PROGRESSIVE LONGITUDINAL WAVE
The displacements of the particles cause regions of high density (compressions) and of low density (rarefactions) to be formed along the wave
Each particle vibrates about its mean position with the same amplitude and frequency
Longitudinal wave is a sound wave. This is propagated by alternate compressions and rarefactions of the air

CHARACTERISTICS of MECHANICAL WAVES
f is frequency of a wave
is cyclic frequency of a wave
is period of a wave
is phase velocity of a wave
is group velocity of a wave
is wave vector directed along the normal to the wave surface.
It demonstrates the direction of wave propagation
is wave number. It is modulus of wave vector

is a wavelength
The distance travelled by a wave in the time
in which the particle of the medium completes
one vibration is the wavelength
O is point source
After time t the wave has travelled distance υt
υ is the velocity of the wave
The energy of the wave has reached
the surface of a sphere
of centre O and radius υt
The surface of the sphere is called
the wavefront of the wave at this instant.
Every point on it is vibrating in phase
with every other point
wave front (there are spherical, plane,
cylindrical and other waves
depending on the shape of a wavefront)
S(A) is amplitude of a wave

WAVE EQUATION
Source of wave
is infinitely long plane
is equation of a linear-progressive wave (a plane-progressive wave)
The negative sign indicates that, since the wave moves
from left to right, the vibrations at points such as P
to the right of O will lag that at O
A wave travelling in the opposite direction, arrives at P
before O. Thus, the vibration at P leads that at O.
0
General case:
a) a plane-progressive wave

b) a spherical wave
The amplitude of the wave diminishes as distance r from the source O increases.
This is caused by the decrease of energy density at larger distances from the source
r is the radius of the spherical wavefront at time t
is equation of a spherical wave

is the wave differential equation

If an elastic bar is caused to vibrate
by being strucked, an elastic wave
begins to propagate along the bar
Phase velocity is
PHASE VELOCITY OF WAVE
Δt is impact duration
l is displacement of a wave front
Velocity of displacement of bar points after impact is
Δl is displacement of points of a bar end per Δt
E is Young's modulus of elasticity
is the phase velocity
of a longitudinal elastic wave
is the phase velocity
of a transversal wave
G is shear modulus