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What is a wave?
If a source particle keeps on vibrating then each particle in turn keeps receiving energy
transferred and keeps transferring energy. Is this energy transfer efficient? In case the mechanical
forces acting on each of the particles is the same, their natural frequency is the same and there is
a situation of resonance. We have a series of particles in the same medium vibrating with the
same frequency causing a chain of efficient energy transfers. This is a wave. A wave is a method
of energy transfer from oscillator to oscillator.
Types of Waves
Waves that comprise vibrating particles of the medium are calledelastic waves. This is
because the restoring force proportional to displacement (that makes the particle undergo simple
harmonic oscillation) arises out of the elastic property of the medium. These are also called
mechanical waves as the simple harmonic oscillators undergo mechanical energy
transformations.
Other than this type of wave, there are also electromagnetic waves, such as visible light
waves, X-rays or radio waves. In this type of wave the energy transformations are from
electrostatic to magnetic and vice versa. There is no need for any medium as there are no
particles undergoing mechanical vibrations.
Waves may also be classified according to whether its particle oscillators undergo
vibrations parallel to the direction of energy propagation or perpendicular to that direction. The
former type are longitudinal waves such as sound waves in any medium or shock waves in an
earthquake. The latter type are calledtransverse waves such as waves in vibrating strings of
musical instruments, or in tuning.
Waves are also classified as progressive waves or stationary waves according to
whether they carry energy forward in an unbounded medium or reflect energy back and forth in a
bounded section of the medium. The differences between these two types is highlighted in a later
section of the chapter.
Wave properties
The distance covered by the energy carrying wave per unit time is called the wave velocity
(v). The SI units of wave velocity are metres/second.
The number of waves crossing any point per second is called the wave frequency (f). This
is the same as the number of oscillations made by the vibrating particles per second. It is also the
same as the frequency of the source particle. The frequency of the wave is thus given by
f = 1/T
where T is the time period of each oscillation.
The units of frequency are hertz (Hz). 1Hz = 1 (oscillation or cycle) per second.
The wavelength ( ) is the distance covered by the wave in one time period of vibration of
the particle oscillators in the wave. In case of some types of waves, a better definition of the
wavelength is : the smallest distance between two oscillations in phase.
The wave velocity is therefore, for all types of waves given by:
v=
/T
or, v = f
Factors affecting Wave velocity
The above equation does not imply that the velocity of a wave in any given medium
depends upon the frequency or the wavelength. Sound waves of all frequencies, (whether shrill
notes or base notes) and all intensities (whether loud sounds or soft sounds) all travel at the same
velocity given physical conditions of the medium remain unchanged. Light waves heat waves
gamma rays and microwaves all travel at the same velocity of 3 x 108 ms-1 in vacuum.
The higher frequencies are waves of smaller wavelengths and vice versa making the wave
velocity independent of the properties of the wave and determined only by the relevant properties
of the medium.
In case of electromagnetic waves, the relevant properties are electrical and magnetic
properties called permittivity ( ) and permeability ( ). For electromagnetic waves, the speed in
vacuum is given by
In case of mechanical waves, the relevant properties are mechanical such as density and
elasticity. Thus the velocity of longitudinal waves in any medium is given by:
where E is the relevant modulus of elasticity and
is the density of the medium.
In the case of sound waves in any solid medium E is the Young’s modulus of that medium
while for fluids E is the bulk modulus of the medium. It should be noted that the greater the
restoring forces (indicated by elasticity) in a medium the faster the wave travels in the medium. On
the other hand, the greater the inertia of the medium (indicated by density) the slower the wave
travels in that medium.
For gases, there is no appreciable heat transfer between the heated compressed layers
and the cooler rarefied layers due to the poor conducting and radiating properties of any gas. E is
thus the adiabatic bulk modulus given by E = P where is the ratio of molar heat capacities of
the gas and P is the ambient pressure. Thus,
In the case of sound waves travelling in air, P is the atmospheric pressure (1.013 x
105 Nm-2), the density of air (1.29 kgm-3), and the mean ratio of molar heat capacities of the
gaseous mixture (= 1.4).
Thus velocity of sound in dry air at 0oC is given by
We have from
This equation indicates that changes in atmospheric pressure cause density of the mixture
to change in the same ratio leaving unaffected the velocity of sound in air at constant temperature.
Unless temperature T changes, or else the composition of the mixture changes affecting M (such
as in change of humidity of air), the velocity of sound remains constant in air.
Transverse mechanical waves require greater intermolecular forces than are available in
fluid media. This type of wave is to be found only on surfaces of liquids (due to surface tension
forces) and in solids as in strings and rods In case of transverse waves in strings, the velocity is
given by
where T is the tension in the string and
is its mass per unit length.
Power and Intensity
The intensity of a wave is defined as the energy per unit time crossing unit area held at
right angles to the direction of propagation of the wave.
Thus,
Units of intensity would be watts/metre2
For waves spreading out radially from a point source, intensity at a distance x from the source is
given by:
where P is the power radiated by the source.
In such a case intensity is inversely proportional to the square of the distance from the source.
If however the source is at infinity, then the radial spread is modified to a parallel spread and the
intensity is thus independent of the distance from the source.
Expression for Intensity of a wave
Consider a length (l) of the medium through which the energy is spreading unidirectionally. That
is, the point source is at an infinite distance. The energy E crossing normally through an area A is
given by:
E = total energy of each oscillator x no. of oscillators
where r is the amplitude and
is the angular frequency of each oscillator.
Time taken for this energy to pass through the given area is
Thus energy crossing the whole area A in time t is given by:
Energy crossing every unit area per unit time is given by:
since nm =
, which is the density of the medium
The intensity of the wave is thus proportional to the square of the amplitude of vibration.
Intensity Level
For sound waves in particular, there is another manner of expressing intensity. The power of the
source is expressed as a ratio with respect to a standard power and this ratio is converted to a
logarithmic scale. The standard of comparison is 1 picowatt (10-12 W) of power which is the
minimum power audible to the human ear at a frequency of 1000 Hz. The power output P to be
measured is expressed as a ratio with this standard power P 0. The inten- sity level is measured in
decibels where,
Numericals
1. The velocity of sound in air at 0oC being 330.0 ms-1, find the change in velocity per oC rise of
temperature.
[0.6 ms-1]
2. If a detonator is exploded on a railway line an observer standing on the rail 2.0 km away hears
two reports. What is the time interval between these reports? (Density of air = 1.4 kg m -3; ratio
of the molar heat capacities of air = 1.40; Atmospheric pressure = 105 Nm-2; Density of steel = 8.0 x 103 kg
m-3; Young’s modulus of steel = 2.0 x 1011 Nm-2)
3. A small piece of cork in a ripple tank oscillates up and down as ripples pass it. If the ripples
travel at 0.20 ms-1, have a wavelength of 15 mm and if the cork moves vertically through a
vertical height of 10 mm as the ripples cross it, find the maximum velocity of the cork. [0.42ms-1]
4. A source of sound of frequency 550 Hz emits waves of wavelength 600 mm in air at 20 oC.
What is the velocity of sound in air at this temperature? What would be the wavelength of the
sound from the source in air at 0oC?
5. A loudspeaker produces a sound intensity level of 8 decibels above a certain reference level at
a point P, 40 m from it. Find (a) the intensity level at a point 30 m from the loudspeaker, (b) the
intensity level at P if the electrical power to the loudspeaker is halved.
[log2 =0.3; log3=0.5]
6. At a point 20 m from a small source of sound the intensity is 0.5 microwatt cm -2. Find a value
for the rate of emission of sound energy from the source and state the assumptions you make
in your calculation.