Waves

Waves

Salient Features of Waves


A wave is a disturbance which propagates energy from one place to the other without the transport of matter.

Waves moving along the spring (or) string are one dimensional waves. Ripples on water are two dimensional waves. Sound waves and light waves which propagate radically from a source are three dimensional waves.

The waves which require medium for their propagation are called mechanical waves. These waves transmit energy and momentum but not matter.

In transverse waves the particles of the medium vibrate perpendicular to the direction of propagation of the wave. Transverse waves are propagated in the form of crests and troughs. Waves on strings are transverse

In longitudinal waves the particles of the medium vibrate parallel to the direction of propagation of the wave longitudinal waves are propagated as compressions and rarefactions.

Longitudinal waves are also known as pressure waves.

In a progressive wave all the particles vibrate with same amplitude, same time period but every particle differs in phase from the neighboring particle.

The distance travelled by the wave in the time the particle of the medium completes one oscillation is known as wave length (l)

The distance travelled by the wave in one second is known as wave velocity (or) Phase velocity

Wave velocity depends only on the nature of medium in which wave propagates and is independent of source producing the waves.
In a given medium



When a given wave propagates from one medium to another, its frequency remains same


If we consider two points at positions on a wave at a given instant, then the phase difference between the two points in between the two points is
Phase difference = (Path difference)

Df = (Dx)
Here is called propagations constant


If two sources emit waves of frequencies n₁and n₂simultaneously, then the phase difference between these two waves after a time t is
= (w₁ ~ w₂) t
= 2 (n₁~ n₂) t


The general equation for plane progressive wave motion is
y = A sin (wt ₀ )

This is the displacement-time equation of any reference particle in the medium.
Here
y = displacement of reference particle from its mean position after t seconds from the commencement of its oscillation.
A = Amplitude of vibration
phase component by virtue of displacement of the particle from its mean position
phase component by virtue of location of the particle relative to source
₀ = initial phase angle (or) epoch
y = A Sin (wt-kx)



This is called differential form of plane progressive wave equation .All solutions of the above equations represent the progressive wave

Energy density = =

Intensity of progressive wave =
=


Velocity of sound
Velocity of sound in a medium is

Here E = Modulus of elasticity of medium
d = density of medium

a)
where y = young's modulus

b) where k = bulk modulus
c) Newton's formula

Here p= pressure of gas
(At compressions and rarefactions the thermodynamic process is isothermal)
d) Newtons - Laplace formula

Where and p = pressure
(At compressions and rarefactions the thermodynamic process is adiabatic)
e)

Factors that influence the velocity of sound in a gas
i) Effect of humidity

If are constants then

ii) Effect of change of pressure: At constant temperature, the velocity of sound in a gas is independent of pressure variations
At constant temperature, = constant
= constant


iii) Effect of temperature
a)
In a given gas,
b) When the change in temperature is small



Note: For small temperature variations at, for every rise of temperature, the increase in velocity of sound is 0.61 ms^-1
c) In two different gases if the velocities of sound are equal then


iv) Effect of relative humidity



v) Amplitude, frequency, phase, loudness pitch, quality have no effect on the velocity of sound in a given medium




Sonometer Experiment
i) The purpose of holes in sonometer experiment in sonometer is to dissipate the energy of air inside box into the surroundings so that resonance does not occur inside the box.
ii) In sonometer box, forced vibrations are desired but not resonance.
iii) Vibrations in platform - transverse
iv) Vibrations in bridge - longitudinal
v) all vibrate with frequency of tuning fork.
vi) Vibrations in string - transverse
vii) Vibrations of tuning fork are communicated to string through platform and bridges.
viii) First law is verified directly, (law of length)
ix) Second, third law is verified by showing (same tuning fork, same wire) (n, m constant).
x) Law of mass is verified by showing (n, T are kept const).


Vibrations of air columns (vibrations of organ pipes) : The wave in a vibrating air column is longitudinal stationary wave with displacement node at closed end and displacement antinode at open end

Closed Pipe

i)

ii)

First overtone is 3rd harmonic
iii)



i.e., second overtone is fifth harmonic
iv) The ratio of frequencies of fundamental and over tones is 1 : 3 : 5 : 7 : .........
v) Only odd harmonic are formed
vi) If end correction is considered then




vii) e = 0.3d where d = diameter of the pipe


Open pipe




ii)

first overtone is second harmonic
iii)

iv) The ratio of frequencies of fundamental and overtones is 1 :2 : 3 : 4: ..........
v) All harmonic are possible
vi) If end correction is considered then


vii) e = 0.6d where d = diameter of the pipe


In a resonating air column experiment, if l₁ and l₂ are the first and second resonating lengths then


i)
ii)
iii)

In resonating air column experiment ,if a graph is drawn taking l on x-axis and 1/n on y-axis then it is a straight line and the intercept on x-axis gives end correction


Beats
(1) Two notes of slightly different frequencies propagating in the same direction superpose
(Amplitudes need not be equal )
(2) Theintensity of resultant wave rises and falls at a point in the medium
(3) Waxing ( rise) and waning (fall) in intensity of sound at a point is called beat .
(4)

Amplitude of resultant wave is
It is a function of time
(5) In case of superposition of two waves of equal frequencies propagating in the same direction.
i) The resultant wave is a harmonic wave of same frequency
ii)
where is the phase difference between the waves
iii)
iv) for where k = 0,1,2,3...

v) For where k = 0,1,2,3....



ECHO :
i) When an observer produces a sound and receives its reflection from an obstacle, the reflected sound is called echo of the original sound.
ii) The Phenomenon involved in echo is "reflection of sound".
iii) Since wavelength of sound is large, large objects alone can produce echo.
iv) If a sound wave is reflected from an obstacle there will be no change in its velocity, wave length & frequency, but its intensity decreases
v) Suppose a person standing between two parallel cliffs fires a bullet. If echoes are heard from the cliffs after t₁and t₂seconds after firing
distance between the cliffs
vi) An observer at a distance d from an obstacle produces a sound and receives its echo after t₁sec. If he walks through a distance x away from the obstacle produces a sound and receives its echo after t₂sec.
then 2d = vt₁
2(d + x) = vt2

If he moves towards the obstacle



Doppler Effect :
(1) The apparent change in the pitch (frequency) due to the relative motion between the source and observer is called Doppler effect
(2) The Doppler effect does not exist in the following conditions
i) Both source and observer are at rest
ii) Both source and observer are moving in the same direction with same speed .
iii) If the source moves perpendicular to the observer
iv) If
(3) Formulae for apparent frequency:
i) When the source and observer are approaching each other.

ii) When the source and observer are receding each other
iii) When the source is following the observer



REVERBERATION :
i) Definition : Reverberation is the persistence of sound in an enclosure, as a result of continuous reflections of sound at the walls even after the source of sound has been turned off.
ii) Threshold of audibility : The minimum intensity of sound upto which it is audible is called threshold of audibility. (value = 10^-12w/m²)
iii) Reverberation time : The interval of time taken by sound to fall its intensity from 10⁶times of threshold audibility to threshold audibility is called reverberation time.
(or)
Reverberation Time : The time required by the sound intensity to decrease to the threshold of audibility from an initial intensity of 10⁶times of this, after the source of sound is turned off is called reverberation time.


SABINE'S FORMULA :
Architectural acoustics was developed by W. Sabine taking the reverberation time as the most important factor. Experimentally it is found that reverberation time is directly proportional to the volume of the enclosure and inversely proportional to the total absorption.