
Wave
motion is defined as the mode of transfer of energy through an elastic
medium by repeated oscillations of the particles of the medium about their
mean positions.




Wave
Motion is Doubly Periodic


 When waves are propagated
through a medium, the form of the waves repeats after equal intervals
of time  wave motion is periodic
in time.
 The form of the waves
keeps repeating at equal distance  wave motion is periodic in space.




Terms
Used in the Study of Wave Motion


Amplitude (A): The maximum displacement of a particle of
the medium from its mean position.
SI Unit  metre (m)
Period (T): The time taken for one complete oscillation by any
particle of the medium.
SI Unit  second (s)
Frequency (n): The number of waves passing across any point of the
medium in
one second is called the frequency of the wave.
SI Unit  Hertz (Hz).


Frequency, n =

1


T




Wavelength
(l)


It is the distance between two adjacent particles in the path
of the wave motion which are exactly in the same phase.
It is the distance between two consecutive crests (or troughs or
condensations or rarefactions).
SI Unit  metre (m)




Velocity
(n):


The
distance covered by the wave in one second.
SI Unit  metre/second (m/s).


Relation
between Velocity, Frequency and Wavelength


In a time of one period (T), the
wave travels a distance equal to one wavelength (l).
Hence, velocity, n =

Distance covered

=

l

\ v=

l




time

T

T

But, T =

1

=

, v = nl


n





Simple
Harmonic Progressive Wave


 A wave which travels with
finite velocity without any change in its form is called a progressive
wave.
 If the particles of the
medium perform simple harmonic motion about their mean positions when
the wave travels through the medium, the wave is called a simple
harmonic progressive wave.
 The amplitude and period
of SHM is the same for every particle.


Equation
of a Simple Harmonic Progressive Wave




Consider a simple harmonic progressive wave travelling in the positive direction of the xaxis.

At t = 0, displacement y = 0
Hence, the equation of motion at any instant, for a vibrating particle
such as O, is given by
y = A sin wt,
where A ® amplitude
w ® angular
velocity
For another particle B, at a distance x from O, the equation of the motion is
given by,
y = A sin (wt  f), since it lags
behind in phase.
f is the phase
lag with O.
A path difference of l corresponds to a phase difference of 2p.
\ A path difference of x corresponds to a phase difference of

2px


l


\ f =

2px


l


\ The equation of motion of particle B can be written as,

y = A sin

(

w t 

2px

)


l


Since, this equation is true for any particle,
it is the equation of the wave.

Since, w =

2p


T


y = A sin

(

2pt



2px

)



T

l

\
y = A sin 2p

(

t



x

)



T

l

A simple harmonic progressive wave travelling
along the negative x  axis is represented

by y = A sin 2p

(

t

+

x

)



T

l






Different
Forms


Since n
=

1

,


T

y = A sin 2p

(

nt



x

)


l

\
y = A sin 2pn

(

t



x

)


nl

But v = nl
\
y = A sin 2pn

(

t



x

)


v

But n =

v


l

\
y = A sin

2p

(vt  x)


l



Longitudinal
Waves and Transverse Waves


On the basis of the direction of vibration of particles and direction
of propagation of waves, waves are classified into two categories:
(i) Longitudinal waves
(ii) Transverse waves
 Longitudinal waves are
those in which the particles of the medium vibrate parallel to the
direction of propagation of the waves.
 Transverse waves are
those in which the particles of the medium vibrate perpendicular to
the direction of propagation of the waves.



Longitudinal
Waves

Transverse
Waves


Particles
vibrate parallel to the direction of propagation.

Particles
vibrate perpendicular to the direction of propagation.


Travel
in the form of alternate compressions and rarefactions.

Travel
in the form of crests and troughs.


It
can be propagated through solids, liquids and gases.

It
can be propagated through solids and liquid surfaces.


Longitudinal
waves cannot be polarised.

Transverse
waves can be polarised.


When
longitudinal waves pass through a medium, the pressure and density at any
point vary between maximum and minimum values.

When
transverse waves pass through a medium, there is no change in the
pressure and density of the medium at any point.



Reflection
of Sound Waves and Change of Phase


A) Reflection from a Denser Medium
a) Transverse Wave
 Phase changes by p radians.
 Crest is reflected as a
trough and vice versa.
b) Longitudinal Wave
 Phase changes by p radians.
 Compression is reflected
as a compression and a rarefaction is reflected as a rarefaction.
B) Reflection from a Rarer Medium
a) Transverse Wave
 No change in phase
 A crest is reflected as a
crest and a trough is reflected as a trough.
b) Longitudinal Wave
 No change in phase.
 A compression is
reflected as a rarefaction and vice versa.



