Standing Waves

Pardeep Kumar

PGT- PHYSICS

JNV RAIGAD

Standing (Stationary) Waves

• A stationary wave is a wave which is not moving ,i.e. it is at rest.
• When two waves with the same frequency, wavelength and amplitude travelling in opposite directions will interfere they produce a standing wave. 
• Conditions to have a standing wave:- Two travelling waves can produce a standing wave, if the waves are moving in opposite directions and they have the same amplitude and frequency.
• At certain instances when the peaks of both the waves will overlap. Then both the peaks will add up to form the resultant wave.
• At certain instances when the peak of the one wave combine with the negative of the second wave .Then the net amplitude will become 0.
• As a result a standing wave is produced. In case of stationary wave the waveform does not move.

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Explanation:-

• Consider I^st wave in the figure and suppose we have a rigid wall which does not move. When an incident wave hits the rigid wall it reflects back with a phase difference of π.
• Consider II^nd wave in the figure, when the reflected wave travels towards the left there is another incident wave which is coming towards right.
• The incident wave is continuously coming come from left to right and the reflected wave will keep continuing from right to left.
• At some instant of time there will be two waves one going towards right and one going towards left as a result these two waves will overlap and form a standing wave.

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Mathematically:�

• Wave travelling towards left y_l(x,t) =a sin(kx– ωt) and towards right y_r(x,t) =a sin (kx + ωt)
• The principle of superposition gives, for the combined wave
• y (x, t) = y_l(x, t) + y_r(x, t)= a sin (kx – ωt) + a sin (kx + ωt)
• y(x,t)= (2a sin kx) cos ωt (By calculating and simplifying)
• The above equation represents the standing wave expression.
• Amplitude = 2a sin kx.
• The amplitude is dependent on the position of the particle.
• The cos ωt represents the time dependent variation or the phase of the standing wave.

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Difference between the travelling wave and stationary wave

Nodes and Antinodes of Standing Wave�

• The amplitude of a standing wave doesn’t remain the same throughout the wave.
• It keeps on changing as it is a function of x.
• At certain positions the value of amplitude is maximum and at certain positions the value of amplitude is 0.
• Nodes: - Nodes represent the positions of zero amplitude.
• Antinodes: - Antinodes represent the positions of maximum amplitude.

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Nodes:-�

• At nodes, amplitude is 0.
• In case of the standing wave amplitude is given as :- 2asinkx
• => 2asinkx = 0 , =>sinkx = 0, =>sinkx =sin n π => kx=n π
• The value of x represents position of nodes where amplitude is 0.
• x=(nπ)/k … equation(i)
• From the definition

of k=(2π)/λ ... equation(ii)

• The position of nodes is represented by: - x=(n λ)/2 from(i) and (ii),where n=1, 2,3…

Note: -Half a wavelength (λ/2) separates two consecutive nodes.

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Antinodes:-�

• At antinodes, amplitude is maximum.
• In case of the standing wave amplitude is given as :- 2asinkx
• => 2asinkx = maximum.
• This value is maximum only when sinkx=1.
• => sinkx = sin(n+(1/2))π
• =>kx=(n+(1/2))π
• =>((2 π)/ λ) x= (n+(1/2)) π
• The position of nodes is represented by:- x= (n+(1/2))( λ/2) ; n=0,1,2,3,4 …

Note: -

• Half a wavelength separates two consecutive nodes.
• Antinodes are located half way between pairs of nodes.

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The formation of a standing wave in a stretched string.

• Two sinusoidal waves of same amplitude travel along the string in opposite directions. The set of pictures represent the state of displacements at four different times. The displacement at positions marked as N is zero at all times. These positions are called nodes.

Nodes and Antinodes: system closed at both ends�

• System closed at both ends means both the ends are rigid boundaries.
• Whenever there is rigid body there is no displacement at the boundary. This implies at boundary amplitude is always 0. Nodes are formed at boundary.
• Standing waves on a string of length L fixed at both ends have restricted wavelength.
• This means wave will vibrate for certain specific values of wavelength.
• At both ends, nodes will be formed.=>Amplitude=0.
• Expression for node x =(nλ)/2.This value is true when x is 0 and L.
• When x=L:- L=(nλ)/2 =>λ=(2L)/n ; n=1,2,3,4,…..
• λ cannot take any value but it can take values which satisfy λ=(2L)/n this expression.
• That is why we can say that the standing wave on a string which is tied on both ends has the restricted wavelength.
• As wavelength is restricted therefore wave number is also restricted.
• ν =v/λ (relation between wavelength and frequency)
• Corresponding frequencies which a standing wave can have is given as: -ν= (vn)/2Lwhere v= speed of the travelling wave.
• These frequencies are known as natural frequency or modes of oscillations.

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Modes of Oscillations:-�

• ν= (vn)/2L where v=speed of the travelling wave, L=length of the string,  n=any natural number.
• First Harmonic:-
• For n=1, mode of oscillation is known as Fundamental mode.
• Therefore ν₁=v/(2L).This is the lowest possible value of frequency.
• Therefore ν₁is the lowest possible mode of the frequency.
• 2 nodes at the ends and 1 antinode.
• Second Harmonic:-
• For n=2, ν₂=(2v)/ (2L) =v/L
• This is second harmonic mode of oscillation.
• 3 nodes at the ends and 2 antinodes.
• Third Harmonic:-
• For n=3,ν₃ = (3v)/ (2L).
• This is third harmonic mode of oscillation.
• 4 nodes and 3 antinodes.

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Problem:-

•  Find the frequency of note emitted (fundamental note) by a string 1m long and stretched by a load of 20 kg, if this string weighs 4.9 g. Given, g = 980 cm s^–2?

Answer:-

L = 100 cm T = 20 kg = 20 × 1000 × 980 dyne

m= 4.9/100 = 0.049 g cm^-1

• Now the frequency of fundamental note produced,

ν= (1/2L) √ (T/m)

ν = 1/ (2x100) √(20x1000x980)/(0.049)

=100Hz

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