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    Explain the Formation of Stationary Waves by Analytical Method. Show that Nodes and Antinodes Are Equally Spaced in Stationary Waves.

    Explain the Formation of Stationary Waves by Analytical Method. Show that Nodes and Antinodes Are Equally Spaced in Stationary Waves.

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    Explain the formation of stationary waves by analytical method. Show that nodes and antinodes are equally spaced in stationary waves.

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    SOLUTION

    Analytical treatment of Stationary Waves : Consider two simple harmonic progressive waves having same amplitude (a), frequency (n) and period (T) travelling along same medium in opposite direction. The wave along positive direction of X-axis is given by :

    λ Y1=αsin2π(nt-xλ)

    ............................(1)

    The wave along negative direction of x-axis is given by:

    λ Y2=αsin2π(nt+xλ)

    ............................(2)

    By principle of superposition, the resultant displacement of stationary wave is given by:

    Y=Y1+Y2 λ λ

    ∴Y=αsin2π(nt-xλ)+αsin2π(nt+xλ)

    λ λ λ λ

    ∴Y=2αsin 2π2 (nt-xλ+nt+xλ).cos 2π2(nt-xλ-ntxλ)

    Using sin C+sin D = 2 sin

    (C+D2)cos(C-D2) and cos θθ (-θ)=cosθ) λ

    ∴Y=2asin2πnt.cos2π(-xλ)

    λ

    ∴Y=2a[sin(2πnt).cos (2πxλ)]

    λ

    ∴Y=[2acos(2πxλ)]sin2π nt

    ....(3) ∴Y=Asin(2π nt) where A =  2a cos λ 2πxλ

    where A is the amplitude of the resutant stationary wave.

    The above expression shows that resultant wave is a simple harmonic motion having same period but new amplitude. The absence of the term (x) in the sine function shows that the resutant waves do not move forward or backward. Such waves are called stationary waves.

    Conditions for antinodes:

    The points of medium which vibrate with maximum amplitude are called antinodes. Antinode is formed when A is maximum.

    A=±2a λ ∴2acos 2π xλ=±2a λ ∴ cos 2πxλ=±1 λ ∴2πxλ=0,π,2π λ ∴2πxλ=pπ λ ∴x=λP2

    Where P = 0,1,2.......

    When λ 2πxλ=0, then  x  = 0 . When λ 2πxλ=π then  x = λ / 2. When λ 2πxλ=2π then  x = λ. λ λ ∴x=0,λ2,λ.....

    The particles at these points vibrate with minimum amplitude. Such points are called antinodes. The distance between two successive antinodes is λ/2.

    Conditions for nodes:

    The points of medium which vibrate with minimum amplitudes are called nodes. Amplitude at nodes is zero.

    A = 0 ∴2acos(2πxλ)=0 ∴cos 2πxλ=0 ∴2πxλ=π2,3π2,5π2 x=(2P-14)

    where p = 1,2,.........

    When 2πxλ=π2, then x = λ/4. When 2πxλ=3π2, then x = 3λ/4. When 2πxλ=5π2, then x = 5λ/4.

    ∴x=λ4,3λ4,5λ4,...........

    The particles at these points vibrate with minimum or zero amplitude. The distance between any two successive nodes is λ/2.

    Thus the distance between two successive antinodes and nodes is equal to λ/2. Therefore antinodes and nodes are equally spaced in a stationary wave.

    Concept: Formation of Stationary Waves on String

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    Explain the formation of stationary waves by analytical method. Show the formation of the stationary wave diagramatically.

    Click here👆to get an answer to your question ✍️ Explain the formation of stationary waves by analytical method. Show the formation of the stationary wave diagramatically.

    Question

    Explain the formation of stationary waves by analytical method. Show the formation of the stationary wave diagramatically.

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    Updated on : 2022-09-05

    When two progressive waves of same amplitude and wavelength travelling along a straight line in opposite directions, they superimpose on each other which results in formation of stationary waves.

    Solution Verified by Toppr

    Consider a progressive wave of amplitude a and wavelength λ travelling in the x-axis direction.

    y 1 ​ =a sin2π( T t ​ − λ x ​ )

    This wave is reflected from a free end and it travels in the negative x-axis direction. It will have same characteristics expect x changes to −x

    y 2 ​ =a sin2π( T t ​ + λ x ​ )

    Now, according to principle of superposition, the resultant displacement will be:

    y=y 1 ​ +y 2 ​ y=a sin2π( T t ​ − λ x ​ )+a sin2π( T t ​ + λ x ​ )

    Using sinA+sinB=2sin[(A+B)/2]cos[(A−B)/2],

    y=a[2sin( T 2πt ​ )cos( λ 2πx ​ )] y=Asin( T 2πt ​ ) where A=2a cos( λ 2πx ​ )

    The above equation of y is an equation of stationary wave and it's amplitude is A=2a cos(

    λ 2πx ​

    ). This represents that at some values of x the resultant amplitude is maximum known as antinodes and for some values of x it will be minimum (zero) known as nodes.

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    Explain the formation of stationary waves by analytical method. Show the formation of the stationary wave diagrammatically.

    Explain the formation of stationary waves by analytical method. Show the formation of the stationary wave diagrammatically.. Ans: Hint: In the analytical method of studying the formation of stationary waves, we consider two progressive waves inciden...

    Explain the formation of stationary waves by analytical method. Show the formation of the stationary wave diagrammatically.

    Last updated date: 15th Jan 2023

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    Hint: In the analytical method of studying the formation of stationary waves, we consider two progressive waves incident on each other from opposite directions. The resultant of this superposition is calculated by using the superposition principle.Complete step-by-step solution

    Consider two waves having the same amplitude and frequency traveling towards each other as shown in the diagram.

    When they reach each other, they undergo interference where they superimpose on each other. The resultant of this type of superimposition is the stationary waves. As the name implies, they are not traveling but stay stationary.

    Consider a traveling wave of the following form.

    y 1 =Asin2π( t T − x λ ) y1=Asin⁡2π(tT−xλ)

    Here A is the amplitude of this wave while

    λ λ

    represents the wavelength of this wave. It is traveling in the positive x-direction.

    Now consider this same wave getting reflected from an obstruction as a result of which it starts travelling back in the negative x direction. It can be written as

    y 2 =Asin2π( t T + x λ ) y2=Asin⁡2π(tT+xλ)

    Now these two waves will superimpose on each other and their resultant is given by the principle of superposition in the following way.

    y= y 1 + y 2 y=y1+y2

    Now inserting the known expressions and solving for the resultant displacement, we get

    y=Asin2π( t T − x λ )+Asin2π( t T + x λ )

    y=Asin⁡2π(tT−xλ)+Asin⁡2π(tT+xλ)

    Using the identity, sinA+sinB=2sin{ A+B 2 }cos{ A+B 2 }

    sin⁡A+sin⁡B=2sin⁡{A+B2}cos⁡{A+B2}

    , we get y=A[2sin( 2πt T )cos( 2πx λ )] y= A ′ sin( 2πt T ) ...(i)

    y=A[2sin⁡(2πtT)cos⁡(2πxλ)]y=A′sin⁡(2πtT) ...(i)

    Here A ′ =2Acos( 2πx λ ) A′=2Acos⁡(2πxλ)

    The equation (i) is known as the equation of stationary waves having amplitude A’. From this equation, we understand that we will get some points of maximum and minimum displacement in the stationary wave called antinodes and nodes respectively.

    These diagrams show the resultant stationary waves formed when the incident and reflected wave superimpose at different phase angles.

    Note: The waves are stationary because they are formed by two waves having equal motion but in opposite directions. The equal and opposite nature of two waves results in the stationary nature of the stationary waves. This phenomenon is observed when the incident and reflected waves superimpose on each other.

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