Wave Motion|Physics|What is Wave Motion? Definition|Classification|Characteristics

Wave Motion : A wave is a disturbance which propagates energy and momentum from one place to the other without the transport of matter. Wave motion is a kind of disturbance which travels through a medium due to repeated vibrations of the particles of medium about their mean positions, the disturbance being handed over from one particle to the next.

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(1) Necessary properties of the medium for wave propagation (Wave Motion) :

(i) Elasticity : So that particles can return to their mean position, after having been disturbed.
(ii) Inertia : So that particles can store energy and overshoot their mean position.
(iii) Minimum friction amongst the particles of the medium.
(iv) Uniform density of the medium.

(2) Characteristics of wave motion :

(i) It is a sort of disturbance which travels through a medium.
(ii) Material medium is essential for the propagation of mechanical waves.
(iii) When a wave motion passes through a medium, particles of the medium only vibrate simple harmonically about their mean position. They do leave their position and move with the disturbance.
(iv) There is a continuous phase difference amongst successive particles of the medium i.e., particle 2 starts vibrating slightly later than particle 1 and so on.
(v) The velocity of the particle during their vibration is different at different position.
(vi) The velocity of wave motion through a particular medium is constant. It depends only on the nature of medium not on the frequency, wavelength or intensity.
(vii) Energy is propagated along with the wave motion without any net transport of the medium.

(3) Mechanical waves :

The waves which require medium for their propagation are called mechanical
Example : Waves on string and spring, waves on water surface, sound waves, seismic waves.

(4) Non-mechanical waves :

The waves which do not require medium for their propagation are called
non- mechanical or electromagnetic waves.
Examples : Light, heat (Infrared), radio waves, α – rays, X-rays etc.

(5) Transverse waves :

Particles of the medium execute simple harmonic motion about their mean
position in a direction perpendicular to the direction of propagation of wave motion.
(i) It travels in the form of crests and troughs.
(ii) A crest is a portion of the medium which is raised temporarily above the normal position of rest of the particles of the medium when a transverse wave passes through it.

Transverse waves
Transverse waves

(iii) A trough is a portion of the medium which is depressed temporarily below the normal position of rest
of the particles of the medium, when transverse wave passes through it.
(iv) Examples of transverse wave motion : Movement of string of a sitar or violin, movement of the
membrane of a Tabla or Dholak, movement of kink on a rope, waves set-up on the surface of water.
(v) Transverse waves can be transmitted through solids, they can be setup on the surface of liquids. But
they can not be transmitted into liquids and gases.

Longitudinal waves

(6) Longitudinal waves :

If the particles of a medium vibrate in the direction of wave motion the wave is called longitudinal.

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(i) It travels in the form of compression and rarefaction.
(ii) A compression (C) is a region of the medium in which particles are compressed.
(iii) A rarefaction (R) is a region of the medium in which particles are rarefied.

(iv) Examples sound waves travel through air in the form of longitudinal waves, Vibration of air column in
organ pipes are longitudinal, Vibration of air column above the surface of water in the tube of resonance
apparatus are longitudinal.
(v) These waves can be transmitted through solids, liquids and gases because for these waves propagation,
volume elasticity is necessary.

(7) One dimensional Wave Motion :

Energy is transferred in a single direction only.
Example : Wave propagating in a stretched string.

(8) Two dimensional Wave Motion :

Energy is transferred in a plane in two mutually perpendicular directions.
Example : Wave propagating on the surface of water.

(9) Three dimensional Wave Motion :

Energy in transferred in space in all direction.
Example : Light and sound waves propagating in space.

 Important Terms Regarding Wave Motion

(1) Wavelength :

(i) It is the length of one wave.
(ii) Wavelength is equal to the distance travelled by the wave during the time in which any one particle of the medium completes one vibration about its mean position.

(iii) Wavelength is the distance between any two nearest particles of the medium, vibrating in the same phase.

(iv) Distance travelled by the wave in one time period is known as wavelength.

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(v) In transverse wave motion :
λ = Distance between the centers of two consecutive crests.
λ = Distance between the centers of two consecutive troughs.
λ = Distance in which one trough and one crest are contained.
(vi) In longitudinal wave motion :
λ = Distance between the centers of two consecutive compression.
λ = Distance between the centers of two consecutive rarefaction.
λ = Distance in which one compression and one rarefaction contained.

(2) Frequency : 

(i) Frequency of vibration of a particle is defined as the number of vibrations completed
by particle in one second.

(ii) It is the number of complete wavelengths traversed by the wave in one second.
(iii) Units of frequency are hertz (Hz) and per second.

(3) Time period :

(i) Time period of vibration of particle is defined as the time taken by the particle to complete one vibration about its mean position.
(ii) It is the time taken by the wave to travel a distance equal to one wavelength.

(4) Relation between frequency and time period in Wave Motion :

Time period = 1/Frequency , T = 1/n

(5) Relation between velocity, frequency and wavelength in Wave Motion :

v = n\λ ; Velocity (v) of the wave in a given medium depends on the elastic and inertial property of the medium. Frequency (n) is characterised by the source which produces disturbance. Different sources may produce
vibration of different frequencies. Wavelength (λ)
will differ to keep nλ= v = constant

Sound Wave Motion

The energy to which the human ears are sensitive is known as sound. In general all types of waves are produced in an elastic material medium, Irrespective of whether these are heard or not are known as sound.
According to their frequencies, waves are divided into three categories :

(1) Audible or sound waves :

Range 20 Hz to 20 KHz. These are generated by vibrating bodies such as vocal cords, stretched strings or membrane.

(2) Infrasonic waves :

Frequency lie below 20 Hz.
Example : waves produced during earth quake, ocean waves etc.

(3) Ultrasonic waves :

Frequency greater than 20 KHz. Human ear cannot detect these waves, certain creatures such as mosquito, dog and bat show response to these. As velocity of sound in air is 332 m/sec so the wavelength of ultrasonic λ< 1.66 cm and for infrasonic λ> 16.6 m.

(4) Supersonic speed :

An object moving with a speed greater than the speed of sound is said to move with a supersonic speed.

Mach number : It is the ratio of velocity of source to the velocity of sound.
Mach Number = Velocity of sound/Velocity of source.

 Difference between sound and light waves 

(i) For propagation of sound wave material medium is required but no material medium is required for light waves.
(ii) Sound waves are longitudinal but light waves are transverse.
(iii) Wavelength of sound waves ranges from 1.65 cm to 16.5 meter and for light it ranges from 4000 Å to 2000 Å.

4. Velocity of Sound (Wave motion)

(1) Speed of transverse wave motion :

(i) On a stretched string :

    \[v= \sqrt{\frac{T}{M}}\ , T = Tension\ in\ the\ string; M = Linear\ density\ of\ string\ (mass\ per\ unit\ length)\]

(ii) In a solid body :

    \[v= \sqrt{\frac{\eta }{\rho }}\ , \eta = Modulus\ of\ rigidity , \rho = Density\ of\ the\ material.\]

(2) Speed of longitudinal wave motion :

(i) In a solid medium:

    \[v=\sqrt{\frac{k+\frac{4}{3}\eta }{\rho }} , k = Bulk\ modulus; \eta = Modulus\ of\ rigidity; \rho = Density\ of\ the\ material\]

When the solid is in the form of long bar ;

    \[v= \sqrt{\frac{Y}{\rho}} , Y = Young's\ modulus\ of\ material\ of\ rod\]

(ii) In a liquid medium :

    \[v= \sqrt{\frac{k}{\rho}}\]

(iii) In gases :

    \[v= \sqrt{\frac{k}{\rho}}\]

Velocity of Sound in Elastic Medium

When a sound wave travels through a medium such as air, water or steel, it will set particles of medium into vibration as it passes through it. For this to happen the medium must possess both inertia i.e. mass density (so that kinetic energy may be stored) and elasticity (so that PE may be stored). These two properties of matter determine the velocity of sound.
i.e. velocity of sound is the characteristic of the medium in which wave propagate.

    \[v= \sqrt{\frac{E}{\rho}}; (E = Elasticity\ of\ the\ medium; \rho = Density\ of\ the\ medium)\]

***Important points

(1) As solids are most elastic while gases least i.e. Es>EL>EG. So the velocity of sound is maximum insolids and minimum in gases

vsteel > vwater > vair
5000 m/s > 1500 m/s > 330 m/s
As for sound vwater > vAir while for light vw < vA.
Water is rarer than air for sound and denser for light.
The concept of rarer and denser media for a wave is through the velocity of propagation (and not density). Lesser the velocity, denser is said to be the medium and vice-versa.

(2) Newton’s formula :

He assumed that when sound propagates through air temperature remains constant.(i.e. the process is isothermal)

    \[v_{Air}= \sqrt{\frac{K}{\rho}}=\sqrt{\frac{P}{\rho}}\ As K= E_{\Theta },E_{\Theta }= Isothermal\ Elasticity ; P=Pressure.\]

By calculation vair = 279 m/sec.
However the experimental value of sound in air is 332 m/sec which is greater than that given by Newton’s formula.

(3) Laplace correction :

He modified Newton’s formula assuming that propagation of sound in air as adiabatic process.

    \[v=\sqrt{\frac{k}{\rho }}=\sqrt{\frac{E_{\Phi }}{\rho }} ; (As \ k=E_{\Phi }=\gamma \rho =Adiabatic\ elasticity)\]

    \[v=\sqrt{1.41}\times 279=331.3\ m/s.\ (\gamma _{Air}=1.41)\]

(4) Effect of density :

    \[v=\sqrt{\frac{\gamma P}{\rho }} \Rightarrow v\propto \frac{1}{\sqrt{\rho }}\]

(5) Effect of pressure :

    \[v=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}\]

Velocity of sound is independent of the pressure of gas provided the temperature remains constant.

    \[ (P\propto \rho\ when\ T = constant)\]

(6) Effect of temperature :

    \[v=\sqrt{\frac{\gamma RT}{M}}\Rightarrow v\propto \sqrt{T}\]

When the temperature change is small then vt= v0 (1 +αt)

v0 = velocity of sound at 0°C, vt = velocity of sound at t°C , α = temp-coefficient of velocity of sound. Value of

    \[\alpha =0.608\ \frac{m/s}{o_{C}^{}\textrm{}}= 0.61 (Approx.)\]

Temperature coefficient of velocity of sound is defined as the change in the velocity of sound, when
temperature changes by 1°C.

(7) Effect of humidity :

With increase in humidity, density of air decreases. So with rise in humidity
velocity of sound increases.
This is why sound travels faster in humid air (rainy season) than in dry air (summer) at the same temperature.

(8) Effect of wind velocity :

Because wind drifts the medium (air) along its direction of motion therefore the velocity of sound in a particular direction is the algebraic sum of the velocity of sound and the component of wind velocity in that direction. Resultant velocity of sound along

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    \[SL=v+w\ cos\Theta\]

(9) Sound of any frequency or wavelength travels through a given medium with the same velocity :

(v = constant) For a given medium velocity remains constant. All other factors like phase, loudness pitch,
quality etc. have practically no effect on sound velocity.

(10) Relation between velocity of sound and root mean square velocity

    \[v=\sqrt{\frac{\gamma RT}{M}}\ and\ v=\sqrt{\frac{\gamma RT}{M}}\ so\ \frac{v_{rms}}{v_{sound}}=\sqrt{\frac{3}{\gamma }}\ or\ v_{sound}=[\gamma /3]^{1/2}\ rms\]

(11) There is no atmosphere on moon, therefore propagation of sound is not possible there. To do conversation on moon, the astronaut uses an instrument which can transmit and detect electromagnetic waves.

Reflection and Refraction of Waves

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When sound waves are incident on a boundary between two media, a part of incident waves returns back
into the initial medium (reflection) while the remaining is partly absorbed and partly transmitted into the
second medium (refraction) In case of reflection and refraction of sound
(1) The frequency of the wave remains unchanged that means
ωi= ωr= ωt = ω = constant.
(2) The incident ray, reflected ray, normal and refracted ray all lie in
the same plane.
(3) For reflection angle of incidence (i) = Angle of reflection (r).

(4) For refraction

    \[\frac{sin\ i}{sin\ r}=\frac{v_{i}}{v_{r}}\]

(5) In reflection from a denser medium or rigid support, phase changes by 180° and direction reverses if incident wave is y= Asin(ωt-kx) then reflected wave becomes           y = Ar sin(ωt+kx+π)=- Ar sin(ωt+kx).

(6) In reflection from a rarer medium or free end, phase does not change and direction reverses if incident
wave is y = A1 sin (ωt + kx) then reflected wave becomes y = Ar sin (ωt + kx)

(7) Echo is an example of reflection.
If there is a sound reflector at a distance d from the source then time interval
between original sound and its echo at the site of source will be l=2d/v .

Reflection of Mechanical Waves

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Progressive Wave

(1) These waves propagate in the forward direction of medium with a finite velocity.

(2) Energy and momentum are transmitted in the direction of propagation of waves without actual
transmission of matter.
(3) In progressive waves, equal changes in pressure and density occurs at all points of medium.
(4) Various forms of progressive wave function.

    \[y=A\ sin(\omega t-\Theta )\]

    \[y=A\ sin(\omega t-kx)\]

    \[y=A\ sin(\omega t-\frac{2\pi }{\lambda }x)\]

    \[y=A\ sin\ 2\pi (\frac{t}{T}-\frac{x}{\lambda })\]

    \[y=A\ sin\ \frac{2\pi }{\lambda }(vt-x)\]

Where  y = displacement
A = amplitude
ω = angular frequency
n = frequency
k = propagation constant
T = time period
λ  = wave length
v = wave velocity
t = instantaneous time
x = position of particle from origin

**Important points

(a) If the sign between t and x terms is negative the wave is propagating along positive X-axis and if the
sign is positive then the wave moves in negative X-axis direction.(b) The coefficient of sin or cos functions i.e. Argument of sin or cos function i.e. (ωt – kx) = Phase.

(c) The coefficient of t gives angular frequency, \omega =2\pi n=\frac{2\pi }{T}=vk

(d) The coefficient of x gives propagation constant or wave number, k=\frac{2\pi }{\lambda }=\frac{\omega }{v}

(e) The ratio of coefficient of t to that of x gives wave or phase velocity. i.e. v=\frac{\omega }{v}

(f) When a given wave passes from one medium to another its frequency does not change.

(g) From v=n\lambda \Rightarrow v\propto \lambda \ \because n=constant\Rightarrow \frac{v_{1}}{v_{2}}=\frac{\lambda _{1}}{\lambda _{2}}

(5) Some terms related to progressive waves

(i) Wave number \displaystyle \left (\bar{n} \right )

  : The number of waves present in unit length is defined as the wave number \displaystyle \left (\bar{n} \right )=\frac{1}{\lambda }
Unit = meter-1 ; Dimension = [L–1].

(ii) Propagation constant (k) : \displaystyle k=\frac{\Phi }{x}=\frac{Phase\ difference\ between\ particles}{Distance between them}

k=\frac{\omega }{v}=\frac{Angular velocity}{Wave velocity}\ and\ k=\frac{2\pi }{\lambda }=2\pi \bar{\lambda }

(iii) Wave velocity (v) : The velocity with which the crests and troughs or compression and rarefaction
travel in a medium, is defined as wave velocity, \displaystyle v=\frac{\omega }{k}=n\lambda =\frac{\omega \lambda }{2\pi }=\frac{\lambda }{T}

(iv) Phase and phase difference : Phase of the wave is given by the argument of sine or cosine in the
equation of wave. It is represented by \displaystyle \Theta (x,t)=\frac{2\pi }{\lambda }(vt-x)

(v) At a given position (for fixed value of x) phase changes with time (t).

\displaystyle \frac{d\Phi }{dt}=\frac{2\pi v}{\lambda }=\frac{2\pi }{T}.dt\Rightarrow Phase\ difference=\frac{2\pi }{T}\times Time\ difference.

(vi) At a given time (for fixed value of t) phase changes with position (x).

\displaystyle \frac{d\Phi }{dx}=\frac{2\pi }{\lambda }\Rightarrow d\Phi =\frac{2\pi }{\lambda }.dx\Rightarrow Phase\ difference=\frac{2\pi }{T}\times Path\ difference.

\displaystyle \Rightarrow Time\ difference =\frac{T}{\lambda }\times Path\ difference

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