Stationary Wave

The resultant wave formed due to the superposition of two progressive waves of same (or nearly same) amplitude and frequency travelling in opposite direction is known as stationary wave
In stationary wave, there is no propagation of energy in any direction. Only amplitude and velocity of vibration change periodically with time.
​An electronic orbit around the nucleus of an atom is an example of stationary wave. When a progressive wave and its reflected wave are superimposed, stationary wave is formed.

Stationary Wave

Characteristics of a Stationary Wave

  • It does not move forward or backward. The energy is not transferred across any plane. It is confined in the region where it is formed. So, it is known as a stationary wave.
  • In this wave, nodes and antinodes are formed alternatively. Separation between any two consecutive nodes or antinodes is $\frac{λ}{2}$.
  • The amplitude is minimum at nodes and maximum at antinodes.
  • The pressure is maximum at nodes and minimum at antinodes.
  • The distance between two alternative nodes or antinodes is the wavelength of the wave.
  • Particles of the medium vibrate in different manner.
  • The length of this wave is fixed. It is a disturbance between two fixed points and cannot move beyond these points.
  • The direction of motion of particles is reversed after half vibration or half time period.
  • Except at nodes, each particle vibrates about their mean position: their motion is simple harmonic with time period equal to the time period of the wave.

Equation of a Stationary Wave

​Consider two progressive waves having same amplitude $a$ and wavelength $λ$ travelling in opposite direction with the velocity $v$ simultaneously. Then, the equations of these waves are as follows; \[y_1=a\sin(ωt-kx)\] \[y_2=a\sin(ωt+kx)\] According to the principle of superposition of waves, \[y=y_1+y_2\] \[y=a\sin(ωt-kx)+a\sin(ωt+kx)\] \[y=a[\sin(ωt-kx)+\sin(ωt+kx)]\] \[y=2a\sin(ωt)\cos(kx)\] \[y=A\sin ωt\] This is the equation for stationary wave. This equation represents a simple harmonic wave whose amplitude is $A$. \[A=2a\cos kx\] The amplitude will have different values for different values of $x$.

Condition for maximum amplitude (Formation of Antinode)

​The amplitude $A=2a\cos kx$ will be maximum if, \[\cos kx=±1\] \[kx=nπ\] Where, $n = 0, 1, 2, 3, …$ \[\frac{2π}{λ}x=nπ \;\;\; [k=\frac{2π}{λ}]\] \[x=\frac{nλ}{2}\] This is the condition for the formation of antinode. For $n = 0, 1, 2, 3, …$ \[x=0,\frac{λ}{2},λ,\frac{3λ}{2},…,\frac{nλ}{2}\] These are the points where antinodes are formed.

Distance between any two consecutive antinodes \[=\frac{nλ}{2}-\frac{(n-1)λ}{2}=\frac{λ}{2}\]

Condition for minimum amplitude (Formation of Node)

​The amplitude $A=2a\cos kx$ will be minimum if, \[\cos kx=0\] \[kx=(2n+1)\frac{π}{2}\] Where, $n = 0, 1, 2, 3, …$ \[\frac{2π}{λ}x=(2n+1)\frac{π}{2}\;\;\;[k=\frac{2π}{λ}]\] \[x=(2n+1)\frac{λ}{4}\] This is the condition for the formation of node. For $n = 0, 1, 2, 3, …$ \[x=\frac{λ}{4},\frac{3λ}{4},\frac{5λ}{4},\frac{7λ}{4},…,(2n+1)\frac{λ}{4} \] These are the points where nodes are formed.

Distance between any two consecutive nodes \[=(2n+1)\frac{λ}{4}-(2n-1)\frac{λ}{4}=\frac{λ}{2}\]

Distance between any consecutive node and antinode \[=(2n+1)\frac{λ}{4}-\frac{nλ}{2}=\frac{λ}{4}\]

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