Waves

Wave is a form of disturbance which travels through a material medium due to the repeated periodic
motion of the particles of the medium about their mean positions without any actual transportation of
matter

Waves are mainly of three types: (a) mechanical or elastic waves, (b) electromagnetic waves and (c)
matter waves.

• Mechanical waves

Mechanical waves can be produced or propagated only in a material medium. These waves are
governed by Newton’s laws of motion. For their propagation, the medium must possess inertia and
elasticity. For example, waves on water surface, waves on strings, sound waves etc.

• Electromagnetic Waves

These are the waves which require no material medium for their production and propagation, i.e., they
can pass through vacuum and any other material medium. Common examples of electromagnetic

Waves are visible light; ultra-violet light; radiowaves, microwaves etc.

• Matter waves

These waves are associated with moving particles of matter, like electrons, protons, neutrons etc.

Mechanical waves are of two types:

(i) Transverse wave motion, (ii) Longitudinal wave motion,

• Transverse wave motion

In transverse waves the particles of the medium vibrate at right angles to the direction in which the
wave propagates. Waves on strings and surface of water are transverse waves. These waves can
propagate through those medium which possess shear modulus hence these waves can propagate
through solids or on the surface of liquids only.

• Longitudinal wave motion

In these types of waves, particles of the medium vibrate to and fro about their mean position along the
direction of propagation of energy. These are also called pressure waves. Sound waves are longitudinal
mechanical waves. These waves can propagate through those medium which possess bulk modulus of
elasticity. Hence these waves are possible in all medium…solids, liquids and gases.

Important terms related to waves:

• Wavelength

The distance travelled by the disturbance during the time of one vibration by a medium particle is called
the wavelength (λ). In case of a transverse wave the wavelength may also be defined as the distance
between two successive crests or troughs. In case of a longitudinal wave, the wavelength (λ) is equal to
distance from centre of one compression (or refraction) to another.

• Wave Velocity

Wave velocity is the time rate of propagation of wave motion in the given medium. It is different from
particle velocity. Wave velocity depends upon the nature of medium.

Wave velocity (υ) = frequency (v) x wavelength (λ)

It is also equal to w/k where w is the angular frequency and k is the angular wave number.

• Amplitude

The amplitude of a wave is the maximum displacement of the particles of the medium from their mean
position.

• Frequency

The number of vibrations made by a particle in one second is called Frequency. It is represented by v. Its
unit is hertz (Hz) v =1/T

• Time Period

The time taken by a particle to complete one vibration is called time period.

T = 1/v, it is expressed in seconds.

Angular wave number or propagation constant : It is equal to 2π/ wavelength.

Some important expressions for the speed of waves :

• The velocity of transverse waves in a stretched string is given by v = √ T/m

Where T is the tension in the string and m is the mass per unit length of the string, m is also called
linear mass density of the string. SI unit of m is kg m-1.

• The velocity of the longitudinal wave in an elastic medium is given by: v = √ Y/d where Y is the Young’s
modulus of elasticity of the medium and d is the density of the medium.

• The velocity of longitudinal wave in a fluid is given by : v = √ B/d where B is the Bulk modulus of
elasticity of the medium and d is the density of the medium.

• Newton’s Formula for the velocity of sound in Air

According to Newton, when sound waves travel in air or in a gaseous media, the change is taking place
isothermally and hence, it is found that the speed of sound waves in air is given by :

V = √ P/d where P is the pressure of the air and d is the density of air.

Speed of sound in air at STP conditions, calculated on the basis of Newton’s formula is 280 ms-1.
However, the experimentally determined values is 332 ms-1.
According to Laplace, during propagation of sound waves, the change takes place under adiabatic
conditions. Hence the speed of sound can be calculated as : v = √ yP/d where P is the air pressure, y is
the ratio of molar specific heat at constant pressure to that at constant volume and d is the density of
air.

• Factors Influencing Velocity of Sound

The velocity of sound in any gaseous medium is affected by a large number of factors like density,
pressure, temperature, humidity, wind velocity etc.

(i) The velocity of sound in a gas is inversely proportional to the square root of density of the
gas.
(ii) The velocity of sound is independent of the change in pressure of the gas, provided
temperature remains constant.
(iii) The velocity of sound in a gas is directly proportional to the square root of its absolute
temperature.
(iv) The velocity of sound in moist air is greater than the velocity of sound in dry air.

• General Equation of Progressive Waves

“A progressive wave is one which travels in a given direction with constant amplitude.

As in wave motion, the displacement is a function of space as well as time, hence displacement relation
is expressed as a combined function of position and time as:

Y (x,t) = A sin (kx — ωt + Ф) ……,(1) OR

Y (x,t) = A sin (kx+ ωt + Ф)……….(2)

We may also choose a cosine function instead of sine function. Here A, K, ω and Ф are four constant for
a given wave and are known as amplitude, angular wave number, angular frequency and initial phase
angle of given wave.

• equation 1 represents a progressive wave travelling towards positive X axis and equation 2 represents
a progressive wave travelling towards negative X axis.

• Y represents the displacement of the particles at any instant of time from their mean position and x
represents the position of the disturbance wrt the origin.

• wave velocity can be calculated as :v = dc /dt

And particle velocity can be calculated as : v =dy/dt

• A wave motion can be reflected from a rigid as well as from a free boundary. A travelling wave, at a
rigid boundary or a closed end, is reflected with a phase reversal but the reflection at an open boundary
takes place without any phase change.

• The Principle of Superposition of Wave

i.e., y = y1 + y2 + y3

• Standing waves or Stationary waves

When two progressive waves of the same type (i.e., both longitudinal or both transverse) having the
same amplitude and time period/frequency/ wavelength travelling with same speed along the same
straight line in opposite directions superimpose or overlap , a new set of waves are formed. These are
called stationary waves or standing waves.

At some points the displacement of particles is zero. Such points are called as NODES, while there are
some points at which the displacement of particles is maximum. Such points are called as ANTINODES.

• Mathematical expression for stationary waves :

Let a wave propagates towards positive X axis. The displacement can be represented by the equation:

Y' = A sin (Kx – wt)

Another identical wave travelling in opposite direction i.e. along negative X axis can be represented by
the equation :

Y’' = A sin (Kx +wt)

According to the principle of superposition of waves, the resultant displacement of the particles can be
written as :

Y = Y’ + Y’’

Y = 2A sin( kx ) cos (wt) ……..1

Using the identity, sin C + sin D = 2 sin(C +D/2) * cos (C-D/2)

• In case of a stretched string of length L, the end points of the string are fixed so, Y=0 at x=0 and at x =L

Putting these boundary conditions in equation 1, we get

Sin kL =0 which means that kL = n π where n is a whole number

Similarly in a pipe closed at one end but open at the other end the equation 1 represents the stationary
wave. Boundary conditions in this case are that the displacement of particles at closed end is zero while
at the open end is maximum.

Hence Y =0 at x=0 but Y is maximum at x =L. Putting these conditions in equation 1, we get
• Stationary waves in an open pipe

In case of an open pipe, both ends of the pipe are open like in a whistle.

When air is blown from one end then it creates a pressure change at the other end which makes the
surrounding air to rush into the pipe and a wave is created inside the pipe in opposite direction wrt the
original wave. The two waves differ in phase by π. Hence the two waves can be represented by :

Y’ = A sin ( Kx – wt) and

Y’’ = A sin ( Kx +wt + π)

Due to their superposition, the resultant displacement can be calculated as

Y = Y’ + Y’’ = 2A cos (Kx) sin (wt)

At the end points of the pipe the displacement is maximum. Hence at both x=0 and x=L, Y is maximum.

So we can write that:

The phenomenon of regular rise and fall in the intensity of sound, when two waves of nearly equal
frequencies travelling along the same line and in the same direction superimpose each other is called
beats.

One rise and one fall in the intensity of sound constitutes one beat and the number of beats per second
is called beat frequency. It is given as:

Vb = (v1-v2)

Where v1 and v2 are the frequencies of the two interfering waves; v1 being greater than v2.

•Essential condition for the formation of beats:

For beats to be audible, the difference in the frequency of the two sound waves should be lesser than 10
Hz. If the frequency difference is more than 10 Hz then our ears will not be able to distinguish between
two sounds due to persistence of hearing.