Question

Derive the equation of stationary wave and deduce the conditionfor nodes and antinodes.

Answer 1

A progressive wave with a uniform velocity v, angular frequency ω, wavelength λ and wave number k that is travelling in the positive x direction is given by:

      y₁(x, t) =  displacement of particle at x at time t = A Sin (k x - ω t)
                   k = 2π/λ

Let another progressive wave of equal magnitude, frequency, wave number also travel in the same medium, but in the negative x direction.
 
     y₂ (x, t) = A Sin (k x + ω t)

When both waves interfere, the resultant displacement of particles is:
     y(x, t) = A [ Sin (k x+ ω t) + Sin (k x - ω t) ]
               = (2 A  Sin k x ) Cos ωt

The effect of two equal waves in opposite directions is that the amplitude at each value of x is fixed in time.  Amplitude of the vibrating particles in the wave = 0  when k x = n π, where n is an integer.  Amplitude is maximum when  k x = (n + 1/2) π.

Since all the elements in the medium oscillate with same phase, move all synchronously and have fixed positions in space, they are called standing or stationary waves.

nodes = points where amplitude of oscillation = 0,   ie., k x = n π
           k  = 2 π/λ          x = n λ / 2          here n = 0,1,2,3,4,5...

antinodes = points where amplitude =  maximum :    ie., x = (n +1/2) π
                         x= (n + 1/2) λ/2            here n = 0,1,2,3,4
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Suppose the second wave is a wave reflected from an obstruction that the progressive wave encounters on the medium.  It has a phase difference of π.

      y₁ (x, t) = A Sin (k x + ω t)
      y₂ (x, t) = A Sin (k x - ω t - π)  = A Sin (ω t - k x)
  
Superposition gives,   y(x, t) = A Sin ωt  *  Cos kx

Thus standing waves of same nature will result.  Nodes and antinodes will be interchanges as compared to the above.
   Nodes       x = (n+1/2) λ/2
   anti nodes    at  x = n λ/2 

Answer 2

y1=a SIn (wt-kx)

y2=a SIn (wt+kx+fi)

fi=π

y2=asin (wt+kx+π)

y2= -a sin (wt+kx)

Y=y1+y2

after solving we get,

y=-2asin kx cos wt

at x=0

y=-2a sin k(0)cos wt

y=0

at x=L. L=lembda/2

k=2π/lembda

y =-2a sin kx cos wt

after putting possible values

y=-2asin 2π/lembda ×lembda/2 cos wt

y=0