When two progressive waves having equal amplitude and frequency moving in opposite directions along the same line in a bounded medium and are superimposed then the wave produced as a result of this superposition is not seen moving in any direction. These type of waves are called standing waves. Similar to progressive waves, standing waves are also of two types: transverse and longitudinal. Transverse standing waves are generated in stretched strings and longitudinal standing waves are generated in air columns. The music from musical instruments is produced due to standing waves. Sitar, Violin. Piano, Guitar. Iktara, etc. produce transverse standing waves and Flute, Whistle, Tabla, etc. produce longitudinal standing wave.
The medium in which the wave moves along a constant line is called linear medium and if the length of linear medium is not infinity, meaning it is finite, then the medium is called bounded (fixed) medium. In linear bounded medium at the end a wave is incident and the second wave is obtained by the reflection of the first wave. Superposition of those two waves leads to the formation of standing waves. To understand the formation of such waves we will study figure.
Here two waves of same frequency and amplitude propagation in opposite directions are shown. One wave is shown from left side to right side and second wave is shown from right side to left side by constant line and dotted line respectively. Initially (at t = 0) both waves are superimposed
in opposite phases due to this all the medium particles are in the mean position. After t = T/4 (where T is the time period of both the waves) time both the waves move ahead by λ/4 in opposite directions. Additional path difference or π phase difference is generated and therefore both the waves meet in the same phase. It is clear from the resultant wave diagram that in this position displacement of some particles [A1, A2, A3, A4, A5] is maximum and some particles (N1, N2, N3, N4) are still in the mean position. After t = T/2 time again the waves are superimposed in opposite phases; now the resultant displacement of all the particles will be zero.
After time t = 3T/4 waves are in the same phase. At this time the displacement of the medium particles A1, A2, A3, A4 and A5 is again maximum, but their phase is opposite. It is clear from the diagram, after t = T again the initial position is reached.
From the above discussion we can get the conclusion that:
1. Some medium particles (A1, A2, A3, A4 and A5) always vibrate with the maximum amplitude on both the sides of the mean position. They are called antinode. Velocity (dy/dt) of particles at antinodes is maximum and pressure or change in density is zero.
\([\because\ \frac{dy}{dx} = 0]\)
2. Particles (N1, N2, N3, N4) between the consecutive antinodes are always stationary. These are called node. These are positioned at same distance. Velocity of particles [dy/dt] at nodes is zero and the pressure or density change is maximum.
3. Distance between two successive antinodes or two successive nodes is λ/2 and the distance between a node and the consecutive nearest antinode is λ/4; where λ is the wavelength.
4. Antinode’s energy is not transferred to nodes meaning these waves do not transfer energy and the waves do not propagate in the medium, but remain in their position and diverge and compress in the boundaries of the medium. Due to this property of the waves they are called standing waves.
5. In standing waves the time period of all the particles is same but the amplitude of all the particles is different. Similarly: the particles between two successive nodes vibrate in the same phase but their amplitudes are different.
6. Particles on both the sides of a node vibrate in opposite phase and particles on both the sides of an antinode vibrate in the same phase.
7. Two times in a time period all the particles of the medium come together in their equilibrium position. Meaning in one time period the displacement of all the particles is zero twice.
