State And Explain Stationary Wave - Long Question | Physics Class 12

Physics | Long Question [4 marks]

State and Explain Stationary Wave ? 
[HSEB 2063] | Class : 12

Solution :

If two or more wave are travelling simultaneously through a medium, the sesultant displacement of a particle of the medium is equal to the vector cal sum of the individual displacement given to it by two waves. Let us consider two harmonic wave of equal amplitude and frequency travelling in the same medium in opposite to each other are y1 = A sin(kx – ωt) and y= A sin(kx + ωt)

By the principle of superposition;

y = y1 + y2

Where y is resultant displacement

y = A sin(kx – ωt ) + A sin (kx + ωt)

y = A[Sin(kx- ωt) + Sin (kx – ωt)]

                        kx – ωt + kx + ωt               kx – ωt + kx + ωt

y = A2Sin ------------------------- . Cos -----------------------

                                2                                        2

y = A 2 Sin kx Cos (-ωt)

y = A 2 Sin kx Cos ωt  [ *.* Cos (-θ) = Cos θ]

y = 2 A Cos ωt Sin Kx

y = a Cos ωt

Where a = 2A Sin kx is amplitude of the resultant wave. The point at which the amplitude is zero, is called node and the point where the amplitude is maximum, is called antinodes.

At Nodes,

Sin kx = 0[as 2A = 0 ]

      kx = 0, π,  2π, 3π …………

       2π

Or, ---- x = 0, π, 2π ……….

       λ 

                 λ           3 λ     4 λ

.*. x =  0, ---  , λ,  -----, ------  ………….

                 2            2       2

Similarly for antinodes,

kx = ± 1

               π       π      π

or, kx =  ----   ----   ----  …….

               2       2      2

                λ        π       π

or, x =   ------,  -----, -----  ………

               4         4      4

Hence in general nodes occure at nλ and antinodes occure at
(2n + 1) λ , where n = 0, 1, 2, 3 ………