Principle of superposition





Principle of superposition

When two or more waves propagating in a medium arrive at the same point simultaneously, a new wave is produced. This phenomenon is called superposition of According to Young the net displacement at any point of the medium is equal to the algebraic sum of displacements of individual waves arriving at that point simultaneously. This is called the principle of superposition and holds good as long as the amplitude of the waves is not too large. This principle is of extreme importance and can be applied to many types of waves e.g. sound waves, light waves, wave pulses etc. The superposition of harmonic waves gives rise to interference, beats and standing waves.
To recognize the types of waves, the following
equations must be known.

Equation of a straight line

Equation of straight line

y = mx + c

Equation of circle of radius ‘a’ is x2 + y2 = a2

Equation of circle

Equation of ellipse

 

Equation of ellipse


\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

Where ,

a = semi-major axis

b = semi-minor axis

Equation of parabola is

y2 = 4 a x ( fig .a )

Equation of parabola

This is parabola symmetrically about x-axis and

x2 = 4 a y (fig . b)

This is parabola symmetrical about y – axis

Beats:

When two waves of slightly different frequencies travel along the same straight line and along the same direction, they superimpose in such a way that the resultant intensity alternatively increases or decreases. This phenomenon of waxing and waning of sound is called beats. One waxing and one waning forms one beat.
For the sake of simplicity we assume the two waves of slightly different frequencies n1 and n2 (at x =0 ) are represented as

y_1 = a sin 2 \pi n_1 t \text{and} \, y_2 = a \sin 2 \pi n_2 t

From Young’s principle of superposition

y = y_1 + y_2 = a \sin 2 \pi n_1 t + a \sin 2 \pi n_2 t \\ 2a \sin \dfrac{2 \pi ( n_1 + n_2 ) t}{2} \cos \dfrac{2 \pi ( n_1 - n_2 ) t}{2} …..(1)

Let n1 = n and n2 = n + \triangle n such that

\triangle n << n ; then equation (1) gives

y = 2a \sin 2 \pi n t \cos \pi (n_1 - n_2 )t \\ \text{or} \, \, \, y = 2a \cos \pi ( n_1 - n_2 ) t \sin 2 \pi n t \dots (2a) \\ \, = A \sin 2 \pi n t \\ \text{where} A = 2a \cos \pi ( n_1 - n_2 )t is the amplitude of resultant and obviously depends on time.

The amplitude A = 2a \cos \pi ( n_1 - n_2 ) t ….(3)

is maximum for \cos \pi ( n_1 - n_2 ) t = \pm 1 \\ \text{or} \pi (n_1 - n_2 )t = r \pi where r is an integer.

The instant of maximum are given by

t = \dfrac{r}{n_1 - n_2} = 0 , \dfrac{1}{n_1 - n_2} , \dfrac{2}{n_1 - n_2} , \dfrac{3}{n_1 - n_2}, \dots (4)

Obviously time interval between two consecutive maxima

= \dfrac{1}{n_1 - n_2}

Frequency of maxima = ( n1 – n2 ) sec -1

The amplitude is minimum for \cos \pi ( n_1 - n_2 ) t = 0 \\ \text{or} \pi ( n_1 - n_2 ) t = ( 2r + 1 ) \dfrac{\pi}{2} , r = 0,1,2,3, \dots \\ \text{The instant of minima are given by } \\ t = ( r + \dfrac{1}{2} ) ( \dfrac{1}{n_1 - n_2 )} \\ = \dfrac{1}{2 ( n_1 - n_2 )} , \dfrac{3}{2 ( n_1 - n_2 )} , \dfrac{5}{2 ( n_1 - n_2 )} \dots

The time interval between two consecutive minima   = \dfrac{1}{n_1 - n_2} \\ \therefore \text{Frequency of minima} \, = ( n_1 - n_2) \text{sec} ^{-1}

The number of beats produced per second.

= n_1 \to n_2

Remark: When the prongs of a turning fork are loaded, the frequency of fork decreases and when they are filed; the frequency of fork increases.

Stationary Waves:

When two wave trains of same frequency and amplitude travel with the same velocity along the
same straight line in opposite directions, they superimpose to, produce a new type of wave called stationary wave or standing wave.

The name stationary for such type of waves is justified because there is no flow of energy along the
wave. Let the incident wave propagating along Y- axis be

y_1 = a \sin ( \omega t - kx ) …(1)

and the wave reflected from the boundary traveling along negative X-axis is

y_2 = \pm a \sin ( \omega t - kx )….(2)

The positive and negative signs are used: if the boundary is free or rigid respectively.

 Case (i). If boundary is free: then equation (2) is,

y_2 = a \sin ( \omega t + kx ). ….(3)

The resultant displacement due to these incident and reflected waves is

y = y_1 + y_2 \\ = a \sin ( \omega t - kx ) + a \sin ( \omega t + kx ) \\ = 2a \sin \dfrac{\omega t - kx + \omega t + kx}{2} \cos \dfrac{\omega t - kx - \omega t - kx }{2} \\ = 2a \sin \omega t \cos kx = 2a \cos kx sin \omega t \dots (4) \\ = A \sin \omega t ….(5)

Where A = 2 a \cos kx = 2a \cos \dfrac{2 \pi x}{\lambda} \dots (6) \\ \, ( \text{since} k = \dfrac{\omega }{v} = \dfrac{2 \pi}{\lambda} )

is the amplitude of resultant wave.

At positions where \cos \dfrac{2 \pi x}{\lambda} = \pm 1 , the displacement is maximum. Such points are called antinodes and are given by

\dfrac{2 \pi x}{\lambda} = 0 , \pi , 2 \pi , 3 \pi , \dots = r \pi ( r = 0,1,2,\dots ) \\ \therefore \, \, \, \, \, x = \dfrac{r \lambda}{2} = 0 , \dfrac{\lambda}{2} , \lambda , \dfrac{3 \lambda}{2} , 2 \lambda \dots ( 7 )

The separation between two consecutive antinodes is \dfrac{\lambda}{2}
At positions where \cos \dfrac{2 \pi x}{\lambda} = 0, the displacement is always zero. Such points are called nodes and are given by

\dfrac{2 \pi x}{\lambda} = \dfrac{\pi}{2} , \dfrac{3 \pi}{2} , \dfrac{5 \pi}{2} = ( 2r + 1 ) \dfrac{\pi}{2} ( r = 0,1,2,\dots ) \\ \therefore \, \, x = ( 2 r + 1 ) \dfrac{\lambda}{4} = \dfrac{\lambda}{4} , \dfrac{3 \lambda}{4} , \dfrac{5 \lambda}{4} \, \, \, \, \dots (8)

The separation between two consecutive nodes is  \dfrac{\lambda}{2}
From (7) and (8) it is obvious that at free boundary always an antinodes is farmed. Midway between the antinodes, there are nodes.
The separation between a node and neighboring antinodes is \dfrac{\lambda}{2}

Case (ii)  If the boundary is rigid , then

y_2 = - a \sin ( \omega t + k x ) \, \, \, \, \, \dots ( 9 ) \\ \therefore\text{The resultant displacement} \\ y = y_1 + y_2 = a \sin ( \omega t - k x ) - a \sin ( \omega t + k x ) \\ = 2a \cos \dfrac{\omega t - k x + \omega t + k x}{2} \\ \, \, \, \, \, \, \, \sin ( \dfrac{\omega t - k x - \omega t - k x}{2} \\ = - 2a \sin kx \cos \omega t \, \, \, \, \, \dots (10) \\ = A \cos \omega t \, \, \, \, \, \, \dots (11) \therefore

Amplitude of resultant disturbance,

A = - 2 a \sin k x = - 2 a \sin \dfrac{2 \pi}{x} \, \, \, \dots (12)

The positions of maximum displacement or antinodes are

\sin \dfrac{2 \pi x}{\lambda} = \pm 1 \\ or \, \, \, \dfrac{2 \pi x}{\lambda} = ( 2r + 1 ) \dfrac{\pi}{2} ( r = 0 , 1 , 2 , 3 , \dots ) \\ \therefore \, \, \, x = ( 2r + 1 ) \dfrac{\lambda}{4} = \dfrac{\lambda}{4} , \dfrac{3 \lambda}{4} , \dfrac{5 \lambda}{4} \dots
The positions of zero displacement or nodes are

\sin \dfrac{2 \pi x}{\lambda} = 0 \text{or} \dfrac{2 \pi x}{\lambda} = r \pi ( r = 0 , 1 , 2 , 3 , \dots ) \\ \therefore \, \, \, x = \dfrac{r \lambda}{2} = 0 , \dfrac{\lambda}{2} , \dfrac{3 \lambda}{2} , \dfrac{5 \lambda}{2} \dots

Thus a node is always formed at rigid boundary lowest.

Fundamental tone, harmonics and overtones:

The sound of frequency produced by a musical instrument is called the fundamental tone. The sounds of other frequencies produced by the musical instrument are called overtones. The overtones whose frequencies are integral multiplies of the fundamental frequency are called the harmonics. The fundamental tone is also called the first harmonic. If first harmonic is n, then the tones of frequencies 2n, 3n, 4n … are called the second, the third and the fourth harmonic respectively. If frequencies of sound emitted by an instrument are n, 1.5n, 2n, 2.5n, 3n etc, then the notes of frequencies 1.5n, 2n, 2.5n, 3n are overtones, while those of frequencies 2n, 3n are second and third harmonics respectively.

Stationary Waves in Strings Fixed at Both Ends:

For transverse vibrations in string, we have

Speed of waves v = n \lambda = \sqrt{\dfrac{T}{M}}

where T is tension in string and m is mass per unit length of string

\therefore \text{Frequency} n = \dfrac{1}{\lambda} \sqrt{\dfrac{T}{m}} \, \, \, (1)

When the string is plucked in the middle, it vibrates in one loop with nodes at fixed ends and antinodes in the middle; so that length of string

tones

l = \dfrac{\lambda _1}{2} or \, \lambda _1 = 2l \\ \, \therefore \, \, \, n = n_1 = \dfrac{1}{\lambda} \sqrt{\dfrac{T}{M}} = \dfrac{1}{2l} \sqrt{\dfrac{T}{M}}

This tone is emitted is called the fundamental or first harmonic.

If the wire is plucked at one fourth of its length, the string vibrates in two loops, so that

l = \dfrac{\lambda _2}{2} + \dfrac{\lambda _2}{2} = \lambda _2 \\ \therefore \text{frequency} \\ n_2 = \dfrac{1}{\lambda _2} \sqrt{\dfrac{T}{m}} = 2 . \dfrac{1}{2l} \sqrt{\dfrac{T}{m}} = 2n

This tone is called the second harmonic or first overtone.
In the string is plucked at one sixth of its length, the string vibrates in three loops, so that

l = \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} = \dfrac{3 \lambda _3}{2} \\ \therefore \text{frequency} \\ n_3 = \dfrac{1}{\lambda _3} \sqrt{\dfrac{T}{m}} = 3 . \dfrac{1}{2l} \sqrt{T}{m} = 3n

This tone is called third harmonic or second overtone.
In general when the string vibrates in p-loops, the frequency

n_p = \dfrac{1}{\lambda _p} \sqrt{\dfrac{T}{m}} = \dfrac{p}{2 l} \sqrt{\dfrac{T}{m}} = pn

This tone is called the pth. Thus in the case of string fixed at both ends, all harmonics even and odd are present.

Melde’s Law:

If N is frequency of tuning fork for a given length ‘l’ of a string vibrating in p-loops under 2; tension T, then Melde’s law states
T.p2 = constant or p \sqrt{T} = constant

Vibrations of Air columns in Organ Pipes

The minimum frequency produced in a pipe is called fundamental and other notes are called overtones.

Open organ pipe :

An antinodes is always formed at the open end. Accordingly different notes produced in open pipe are shown in fig. In fundamental mode if \lambda _1 is wavelength, then

l = \dfrac{\lambda _1}{2} or \, \, \, \, \lambda_1 = 2l \\ \therefore \text{fundamental frequency} \\ n = \dfrac{v}{\lambda_1} = \dfrac{v}{2l} \, \, \dots (1)

For first overtone if \lambda _2 is the wavelength , then

l = \dfrac{\lambda_2}{2} + \dfrac{\lambda_2}{2} = \lambda_2 \dots (2) \\ \text{frequency of first overtone} , \\ n_2 = \dfrac{v}{\lambda _2} = \dfrac{v}{l} = 2 \dfrac{v}{2l} = 2n \dots (3)

This frequency is double of fundamental frequency and is therefore called second harmonic.

For second overtone, if \lambda _3 is the wavelength, then

l = \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} = \dfrac{3 \lambda _3}{2} \text{or} \lambda = \dfrac{2l}{3} \\ \therefore \text{frequency of second overtone} \\ n_3 = \dfrac{v}{\lambda _3} = 3 . \dfrac{v}{2l} = 3n \dots (4)

This is the third harmonic Frequencies

Frequencies n1 : n2 : n3 : … = 1 : 2: 3:…  i.e. in open organ pipe all harmonics even or odd are present.

Stationary Waves and Harmonics in Closed Organ Pipe:

A node is always formed at closed end and antinodes at open end. Accordingly different harmonies produced in closed organ pipe are shown in fig.

In fundamental mode, if\lambda _1 is the wavelength of vibrations, then

l = \dfrac{\lambda _1}{4} \text{or} \lambda_1 = 4l \\ \therefore \text{fundamental frequency} n_1 = \dfrac{v}{\lambda _1} = \dfrac{v}{4l} \dots (5)

 

Frequency of first overtone, if \lambda _2 is the wavelength , then

l = \dfrac{\lambda_2}{2} + \dfrac{\lambda _2}{4} = \dfrac{3 \lambda _2}{4} or \lambda _2 = \dfrac{4l}{3} \\ \therefore{frequency of first overtone} , \\ n_2 = \dfrac{v}{\lambda_2} = 3. \dfrac{v}{4l} = 3n

This frequency is three times of fundamental. Therefore in closed pipe the first overtone is third
harmonic.
For second overtone; if \lambda _3 is the wavelength, then

l = \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{2} + \dfrac{\lambda _3}{4} = \dfrac{5 \lambda _3}{4} \text{or} \lambda _3 = 4l \\ \text{frequency of second overtone ,} \\ n_2 = \dfrac{v}{\lambda _3} = 5 . \dfrac{v}{4l} = 5n

Thus in closed pipe

n1 : n2 : n3 : … = 1 : 3 : 5 : …

Hence in closed organ pipe only odd harmonics are present.

End Correction :
So far we have considered that the antinodes is formed exactly at the open end of the pipe; but actually due to finite momentum of the particles the reflection takes place a little above the open end; that is why the antinodes is formed a little above the open end. For this a correction is applied being known as end correction.This is denoted by ‘c’ and its value to 0.6r. r being radius of the pipe. If lo is the length of pipe, then for closed pipe ‘l’ is replaced by lo + e while for open pipe ‘I’ is replaced by 1o + 2e .



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