WEBVTT
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i am m k srivastav the department of physics
iit roorkee this is the first lecture for
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the four lecture series on acoustics
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in these lectures on acoustics we shall begin
with sound generation and propagation take
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up caustic equations consider types of wave
motion and shall concentrate on harmonic waves
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we shall then consider principle of superposition
formation of beats and stationary waves shall
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describe kundt’s tube experiment as an experimental
manifestation of the stationary waves then
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we shall consider the phenomena of reflection
refraction and diffraction of sound waves
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towards the end of the series we shall take
up two more important topics which have wide
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applications these are: ultrasonics their
methods of production which are based on magnetostriction
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and piezoelectric effects and then the applications
of ultrasonics number 2: the acoustics of
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buildings which involves reverberation control
sound quality management and auditorium design
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let us begin the acoustics is a disciplined
extremely broad in scope
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literally covering waves and vibrations in
all media at all frequencies but a wide range
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and at all intensities primarily it is a matter
of communication
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whether it be speech or music signaling and
sonar or ultrasonography we seek to maximize
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our ability to convey permission and at the
same time to minimize the effects of noise
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external or internal
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modern acoustics now encompasses the realm
of ultrasonics and infrasonics in addition
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to the audio range as a result of applications
in material science medicine dentistry oceanology
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marine navigation oceanology okay communications
petroleum and mineral prospecting and music
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and voice synthesis
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let us consider the sound generation and propagation
you see sound is a mechanical disturbance
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sound waves are mechanical waves they travel
through an elastic medium at a speed which
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is a characteristic of that medium it is essentially
a wave phenomena as in the case of a light
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beam
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but acoustics phenomena are mechanical in
nature the wide the particles of the medium
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vibrate while light x-rays gamma rays etcetera
they occur as electromagnetic phenomena acoustics
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signals require a mechanically elastic medium
see this is a very important the medium has
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to be elastic elasticity controls the propagation
so a mechanically last medium is required
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for propagation
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and therefore sound cannot travel through
a vacuum on the other hand the propagation
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of an electromagnetic wave as we know can
occur even in empty space
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consider sound as generated by the vibration
of molecules in a plane surface saying at
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x = 0 it could be a stretched membrane just
a plain sheet placed at x = 0
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the displacement of the surface to the right
as a result of vibrations in the positive
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direction causes the compression of a layer
of air immediately adjacent to the surface
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thereby resulting in an increase in the density
of the air in that layer which is touching
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the surface
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as the pressure of that layer is now greater
than the pressure of the undisturbed atmosphere
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the air molecules in the layer tend to move
in the positive x direction and compress the
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second layer which in time transmits the pressure
impulse to the third layer and so on
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but at the plane surface at x = 0 reverses
its direction of vibration after half a cycle
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of vibration and opposite effect occurs a
reflection of the first layer now occurs and
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this reflection decreases the pressures to
a value below that of the undisturbed atmosphere
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the molecules from the second layer now tend
to move left wards and in the negative x direction
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and a rarefaction impulse now follows the
previously generated compression impulse
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now this succession of outwardly moving rarefractions
and compressions constitutes a wave at a given
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point in the space and alternating increase
in decreasing pressure leading to the corresponding
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increase and decrease in density occurs
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the spatial distance between and one point
on the cycle to the corresponding point on
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the next cycle is the wave length of this
wave motion this is really the distance which
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the wave travels during the time the particle
includes one complete vibration the vibrating
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molecules that transmits the waves do not
on the average change their positions but
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are merely moved back and
forth under the influence of the transmitted
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waves
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the distances these particles move about the
equilibrium positions are called displacement
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amplitudes amplitude with the maximum displacement
which a molecule of atmosphere suffers when
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a wave is passing through the velocity at
which the molecules move back and forth about
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their mean position is termed particle velocity
if it is velocity this is different from the
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speed of sound which is the rate at which
the acoustic waves travel through the medium
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that is the characteristic of the medium the
properties of the medium determine the rate
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at which the sound travels in the medium
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consider this picture it has three parts:
the central portion dealing with the compression
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and rarefactions and then there is a variation
density pressure amplitudes the distance and
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in the bottom where the variation of the particle
velocity is given
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so as i said this figure depicts refraction
and condensation of air molecules subjected
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to the vibrational impact of a plane wall
located at x=0 this plane wall could be ascetic
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sheet it any vibrating sheet the degree of
darkness in the figure is proportional to
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the density of molecules is a measure of the
density of molecules light areas in the figure
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are those of rarefactions
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in the figure as i said the shown mini plots
at the top and the bottom are given as functions
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of x remember the propagation is along the
x axis so as a function of x at a given instant
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of sound propagation at a given instant of
time t
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these are of the local variations of molecular
displacements phi changes and pressure p condensation
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s given by fractional change in density rho
- rho 0 upon rho 0 these three are at the
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top the bottom one corresponds to the variation
of the particle displacement speed just note
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the phase difference between the variations
shown on the top the variations of the particle
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displacement the variation of pressure
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and the variation of condensation on one side
compared to the variation of the particle
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displacement speed u shown at the bottom there
is a phase difference of a quarter vibrations
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pi by 2 you see when the molecular displacement
is at maximum the pressure change is maximum
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correspondingly the density changes maximum
at that instant the particle displacement
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speed is 0
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the wavelength lambda represents the distance
between corresponding points of adjacent cycles
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the variation repeats after every lambda the
lambda barely represents the space periodicity
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the speed of sound in any medium is characteristic
of that medium as i said earlier sound travels
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far more rapidly in solids than it does in
gases a temperature of 20 degrees centigrade
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sound moves at the rate of 344 meters per
second through air at a normal atmospheric
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pressure of 760 millimeters of mercury
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sound velocities are also greater in liquids
than in gases but remain less in order of
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magnitude than those for solids
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for an ideal gas the velocity v of a sound
wave may be obtained from the relation v = the
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square root of gamma times pressure upon the
density which is = the square root of gamma
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times r into capital t where gamma is the
gas constant defined as the thermodynamic
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ratio specific heat cp upon cv p is the gas
pressure rho is the density of the gas r is
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the thermodynamic constant characteristic
of the gas
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and capital t is the absolute temperature
of the gas one thing should be noted: at a
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constant temperature the velocity is not affected
by changes in pressure the reason is pressure
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and the density they change in the same way
actually but pressure and density they are
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proportional to each other at a constant temperature
this is the well-known boyles law
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a simple relation such as the above somehow
does not exist for acoustic velocity in liquids
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but we know that the propagation velocity
does not depend on the temperature of the
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liquid and to a lesser degree on the pressure
some velocity is approximately 1461 meters
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per second in water
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for a solid the propagation speed can be found
from the relation v is = square root of e
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upon rho where e represents young’s modulus
or the modulus of elasticity of the material
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and rho is the material density so this relation
explains why the speed of sound in solids
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is much higher than what it is in gases or
even liquids this is modulus of elasticity
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for solids is very large compared to what
it is for a gas
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let us now obtain acoustic equations consider
an undisturbed fluid at rest having definite
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values of pressure density and temperature
which are uniform and time independent uniform
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means these things do not vary from point
to point time independent means they are steady
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and are not changing with time
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the passage of acoustical signals through
the field results in small perturbations small
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changes in the value of the pressure in the
density and in particle velocity over the
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undisturbed values
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the transmission of sound through the fluid
is sufficiently transient so that there is
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virtually no time for heat transfer to occur
i mean at the point is that these changes
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are taking place so fast frequencies are pretty
high so there is no time for the heat transfer
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to occur and the result is that the ongoing
thermodynamic actions during the sound propagation
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may be taken to be an adiabatic process
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it can be shown that for propagation along
the x axis these perturbations the perturbations
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in pressure perturbations in density perturbations
in particle speed satisfy an equation of the
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form del square psi by del x square is = 1
upon v square times del 2 psi by del t square
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as i said psi could be either pressure or
density or particle speed the constant v is
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same in all these cases it has the dimensions
of l by t distance upon time and is the velocity
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of sound propagation in the fluid
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we shall not derive this equation but i look
like to point out the salient features of
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how it is obtained let us consider the continuity
equation which is an expression of the conservation
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of matter for the fellow of a compressible
fluid similarly consider energy conservation
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equation in a fluid this involves the macroscopic
kinetic energy and the internal energy of
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the fluid these two conservation equations
along with the equation of state for the solute
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they form the basis
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these equations are then solved for small
perturbations first order terms are retained
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we can ignore the second order terms and consider
the process as adiabatic as pointed out earlier
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and that leads to an equation which is given
here
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you see this equation is the second order
equation in time and the variable x and we
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know a partial acoustic original equation
like this shall need two initial conditions
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and two boundary conditions for a well defined
solution but these conditions are really not
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needed if we are interested in ascertaining
the general form of the solution of this equation
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this general solution can be written as psi
of xt remember we are considering propagation
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along the x axis so psi of xt psi could be
as i said earlier it could be particle displacement
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it could be pressure it could be density so
= a function f of x - vt and some function
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g of x + vt these functions f in g are arbitrary
arbitrary means that they can be of any form
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but they have to be functions of this combination
x - vt or x + vt and they should have continuous
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derivatives of the first and second order
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these functions f of x - vt represents waves
moving in the positive x direction and similarly
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the function g of x + vt represents waves
moving in the opposite direction these functions
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represents wave can be understood like this
suppose f represents pressure or density as
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a function of x and t then x and t must increase
simultaneously at the rate given by dx by
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dt = v which is the speed at which the wave
is established
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the simultaneous variation of x and t means
that as t advances as t increases x must also
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increase that is the wave is traveling waves
disturbances is moving towards the positive
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direction of x axis similarly for the function
g of x + vt as t increases s must decrease
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simultaneously again as i said simultaneous
change means as time advances x must change
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in the opposite direction
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and the rate determined by dx by dt = - v
the change in x in the negative direction
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means the wave disturbance is moving towards
the negative direction of x axis all solutions
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of the above differential wave equation must
be of this form: function of x - vt or function
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of x + vt if there is any other form that
will not satisfy the basic equation
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let us now consider types of wave motion several
different types of waves may be generated
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depending upon the motion of a particle in
the medium with respect to the direction of
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propagation transverse waves
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we can define a transverse waves as a wave
motion in which the particles of the medium
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vibrate about their mean position at right
angles to the direction of propagation of
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the wave the propagation is along the x axis
this means the vibrations are in the transverse
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plane in the yz plane within the yz plane
these vibrations can have any direction they
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can also randomly change or change in any
fashion
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the wave is polarized the direction can be
fixed but always be remaining in the transverse
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plane a very common example of transverse
waves is the vibrations of a plug district
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sitar spring for example or the vibrations
of a rod which is clamped at one end or the
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vibrations of a tuning fork all these are
very common examples of transverse waves
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longitudinal waves: you can define a longitudinal
wave as the wave motion in which the particles
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of the medium vibrate about the mean position
again but along the same line as that of the
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direction of propagation of the waves the
sound waves are principally longitudinal with
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the results that the particle motion creates
alternate compression and rarefaction in the
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medium alternate density increase and density
decrease or a pressure increase or a pressure
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decrease the sound passes a given point
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the reason why the sound waves are principally
longitudinal is the most of the time we deal
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with sound waves in air or sometimes in water
what are the proportions you see these substances
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do not have a fixed shape they are not rigid
coefficient of rigidity is 0 or the shear
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coefficient is 0 they cannot support a transverse
wave the only possibility is the longitudinal
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waves and that is why most of the time when
we deal with sound waves we find that they
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are longitudinal and that thing to be noted
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when a sound wave is passing the net fluid
displacement over a vibration cycle is zero
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azimuth movement on one side as on the other
side during a vibration it is the disturbance
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other than the solute that is moving at the
speed of sound the fluid molecules do not
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move far from their original positions those
displacements are very small
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additionally waves may also fall into the
category of being rotational or torsional
25:37.779 --> 25:47.470
waves the particles of the rotational wave
rotate about a common center as the wave advances
25:47.470 --> 25:52.059
the curl of
the ocean wave roaring on a beach and to a
25:52.059 --> 25:55.450
beach is a very common example
25:55.450 --> 26:02.590
the particles of torsional waves move in a
helical fashion which could be considered
26:02.590 --> 26:15.360
a vector combination of longitudinal and transverse
motions such waves will occur in solid substances
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naturally they cannot occur in gases or in
water these are sometimes referred to as shear
26:23.769 --> 26:31.039
waves the solids have the shear coefficient
which is non 0 so the shear waves with all
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solids support
26:33.580 --> 26:41.750
now the most simple form of waves is a harmonic
wave of a single frequency you can call it
26:41.750 --> 26:48.630
even a monochromatic harmonic wave let us
consider a plane progressive simple harmonic
26:48.630 --> 26:58.750
wave originating at the origin o and traveling
in the positive direction of the x axis with
26:58.750 --> 27:01.899
the velocity v as shown in the figure
27:01.899 --> 27:11.020
this figure shows the variation of psi as
a function of x at an instant of time t the
27:11.020 --> 27:18.880
amplitude of the vibration is a of the way
proceeds these successive particles of medium
27:18.880 --> 27:22.149
is set in two simple harmonic vibration
27:22.149 --> 27:29.710
that the time be measured from the instant
when the particle had the origin o is passing
27:29.710 --> 27:36.669
through its equilibrium position this will
set the initial phase of the vibration the
27:36.669 --> 27:45.809
displacement psi the particle at o from its
mean position at any time t is therefore given
27:45.809 --> 27:55.299
by psi = a sine omega t which is equal can
be written as a sine 2 pi by capital t times
27:55.299 --> 28:07.600
t the angular frequency omega is = 2 pi mu
which is = 2 pi by capital t v is the frequency
28:07.600 --> 28:12.680
and capital t is the time period of a vibration
28:12.680 --> 28:20.580
you see this result for psi is = a sine omega
t is very standard result for a particle executing
28:20.580 --> 28:29.539
simple harmonic motion simple harmonic vibratory
motion results whenever the situation is such
28:29.539 --> 28:37.799
that the restoring force due to the elasticity
of the medium is proportional to the displacement
28:37.799 --> 28:47.130
in a medium and the density changes or when
the pressure changes the force on the particle
28:47.130 --> 28:55.600
is again proportional to the change in the
density or change in the pressure the result
28:55.600 --> 29:06.280
is we get a simple harmonically varying particle
displacement or particle velocity or density
29:06.280 --> 29:10.110
changes or pressure changes
29:10.110 --> 29:19.880
if now we consider particle of the medium
at a point a distant x from the origin o the
29:19.880 --> 29:27.450
wave is starting from o would reach this point
a little later in upon x by v seconds that
29:27.450 --> 29:32.070
is the time taken by the wave
29:32.070 --> 29:38.669
it means that this particle the particle at
a really start vibrating x by v seconds later
29:38.669 --> 29:48.169
than the particle at o therefore there is
a phase lag of x by v seconds between this
29:48.169 --> 29:52.370
particle and the particle at the origin particle
at o
29:52.370 --> 30:01.120
consequently the displacement of the particle
at a at times t will be the same as that of
30:01.120 --> 30:14.210
the particle at o at a time x by v seconds
earlier that is at time t - x by v does the
30:14.210 --> 30:26.340
displacement of a particle at a after a time
t can be obtained by substituting t - x by
30:26.340 --> 30:34.460
v in place of t in the earlier equation and
so we get for the vibrations of this particle
30:34.460 --> 30:43.830
whose distance is x from the origin at time
t given by psi is = a times sine of 2 pi by
30:43.830 --> 30:50.070
capital t times t - x upon v
30:50.070 --> 30:57.970
this is the equation of a progressive simple
harmonic wave progressing along the x direction
30:57.970 --> 31:11.940
let us look at its basic properties inherent
characteristics as the wave advances every
31:11.940 --> 31:20.279
particle along the path of the wave executes
identical simple harmonic motion there is
31:20.279 --> 31:29.309
a constant lagging of phase for a particle
at a distance of x this phase lag is 2 pi
31:29.309 --> 31:32.600
by capital t times x upon v
31:32.600 --> 31:45.470
if we write mu for 1 upon t then this phase
lag is 2 pi mu times x upon v or omega times
31:45.470 --> 31:53.500
x upon v another thing is this equation does
not contain any y or z this is the equation
31:53.500 --> 32:04.420
of a plane wave the wave front or the phase
fronts or parallel to yz plane in the yz plane
32:04.420 --> 32:12.540
everything is constant the changes are only
along the x axis so this is the equation of
32:12.540 --> 32:16.110
a plane simple harmonic wave then psi
32:16.110 --> 32:25.750
if it is a particle displacement this particle
displacement could be along the x axis itself
32:25.750 --> 32:33.840
along the direction of propagation if the
wave is longitudinal has happened for the
32:33.840 --> 32:43.049
sound wave within here this will lead to alternate
compressions and rarefactions in the medium
32:43.049 --> 32:50.631
this psi could also be along the transverse
plane if the wave is a transverse plane as
32:50.631 --> 32:58.659
i said in that case i mean in the case of
a transverse wave psi can have any direction
32:58.659 --> 33:00.110
in the first plane
33:00.110 --> 33:07.340
if it is so this psi can also represent the
vibration of a plug of the spring or the vibrations
33:07.340 --> 33:16.009
of a tuning fork and the thing is that the
intensity of the wave is proportional to the
33:16.009 --> 33:20.990
square of the amplitude a there are some other
factors but let us not bother about them at
33:20.990 --> 33:27.309
this stage intensity is proportional to the
square of the amplitude a
33:27.309 --> 33:36.320
this equation can also be written in the following
form psi = a sine 2 pi by lambda times vt
33:36.320 --> 33:46.750
- x just a question of changing the variables
or psi = a sine omega t - kx here t is = lambda
33:46.750 --> 33:56.620
by v and k is 2 pi by lambda it is the propagation
constant the second form psi = a sine omega
33:56.620 --> 34:08.300
t is commonly used so this is all about simple
harmonic plane waves and with this we come
34:08.300 --> 34:10.940
to the end of this lecture thank you