Standing waves

Hello dear students we know that, two or more waves travelling along the same path can undergo overlapping. This overlapping result some important phenomena like interference, which can be studied in detail with the help of principle of superposition of wave, stated as,

When two or more waves travelling through a medium arrive at a point of the medium simultaneously each wave produced its own displacement at the point independently of the other and the resultant displacement at the point is equal to the vector sum of the displacement due to all the waves.

Then if y1, y2, ……. yN be the displacements of waves then

y= y1+ y2+ ……. +yN

Where the displacement of simple harmonic progressive wave is given as,

Let’s discuss the concept of stationary wave……………!

Consider two progressive waves of same amplitude a period T, wavelength y & speed v are travelling through the same medium, along but opposite in direction.

A progressive wave travelling along positive direction of x-axis is represented by,

From equation (5) and (6) we can conclude that when, two simple harmonic progressive waves overlap, the resultant wave is also simple harmonic wave. Here it is known as stationary wave or standing wave. Following diagram shows the stationary wave produced on stretched string.

In the figure above, the point in medium which is vibrating with maximum amplitude is known as antinode (A) and the point in medium which vibrates with minimum amplitude is known as nodes (N).

Condition for nodes and antinodes are discussed as below,

Antinode:-

The points of medium at which the amplitude of vibration of particle maximum are called antinodes. For antinodes amplitude should be maximum, and it will possible if the resultant amplitude which ωt is cosine function will reduce to 1.

Node: –

The point of medium at which the amplitude of vibrations of particles is zero are called nodes.

For nodes,     A = 0

As 2a ≠0          ∴ cos 2πx’/λ = 0

∴2πx’/λ = π/2, 3π/2, 5π/2,  …..

∴ x’ = λ/4, 3λ/4, 5λ/4,  …..

∴x^’ =(2P-1) λ/4        where P = 0, 1, 2,…

The distance between two nodes = λ/2

i.e. the distance between adjacent node and antinode is x – x’= λ/2 – λ/4 = λ/4.

Thus in stationary waves nodes and antinodes are alternatively placed in medium.

Following are some important properties of stationary wave…….!

  1. When two identical progressive waves travelling through the medium along the same line in opposite direction, the stationary waves are formed.
  2. The point where amplitude is maximum, called antinodes.
  3. The points where amplitude is zero are called nodes.
  4. The distance between two successive nodes or antinodes is λ/2.
  5. The distance between node & adjacent antinodes is λ/2.

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