Stationary Waves

Stationary or standing waves are formed in a medium when two waves having equal amplitude and frequency moving in opposite directions along the same line, interfere in a confined space. Generally, such waves are formed by the superposition of a forward wave and the reflected wave. Both longitudinal and transverse types of waves can form a stationary wave.

When reflection occurs at a free end, there is no reversal of phase. i.e., a crest returns as a crest and a trough as a trough.

Example,

1) A rope held vertically in the hand with the lower end hanging free, made to vibrate briskly at the upper end

2) An open end of an organ pipe into which air is blown

When reflection occurs at a fixed end, there is a reversal of phase but there is no change in amplitude, frequency and velocity. i.e., a crest returns as a trough and a trough as a crest.

Example,

1) A guitar string plucked in the middle.

2) A closed end of an organ pipe, into which air is blown.

Figure(a) shows two sinusoidal waves A and B having the same amplitude and frequency, traveling in opposite directions. At an instant of time t = 0, the resultant displacement graph is a straight line. All the particles of the medium affected by the two waves are at their equilibrium positions.

 

 

Figure shows the situation after a time , where T is the period of oscillation of the particles of the medium. It is seen in the figure that the wave A has advanced through a distance l/4 towards the right and the wave B has advanced through the same distance towards the left.

 

 

The two crests and the two troughs will add up, giving rise to a bigger wave pattern as shown in the figure. The particles 1, 3, 5 and 7 are at their extreme positions while the particles 2, 4 and 6 are at their equilibrium positions.

 

 

Figure (c) shows the situation at the end of . The wave A has advanced through a distance l/4 towards the right and the wave B through the same distance towards the left. All the particles are now in their equilibrium positions and the resultant wave pattern is a straight line.

 

 

Figure (d) shows the situation at the end of . The wave A has advanced through a distance towards the right and the wave B through the same distance towards the left. The resultant wave pattern is bigger than either wave as the amplitudes add up. It may be noted that the particles 1, 3, 5 and 7 are again at their equilibrium positions.

 

 

Figure (e) shows the dynamic condition of the particles at the end of t = T. The wave A has advanced through a distance l, towards the right and the wave B through the same distance, towards the left. The resultant displacement pattern is a straight line. All the particles are in their equilibrium position.

From the above discussion, it follows that some particles like 2,4 and 6 always remain at rest while particles like 1, 3, 5 and 7 vibrate (simple harmonic) about their mean positions. Such particles have the maximum amplitude, equal to twice that of the individual waves.

The resultant displacements at times are shown in figure (f). The positions of particles like 2, 4 and 6, which always remain at their mean positions are called 'nodes'. The positions of particles like 1, 3, 5 and 7 at which the resultant amplitude is maximum, are called 'antinodes'. The distance between any two consecutive nodes or antinodes is equal to l/2. Between a node and an antinode, the amplitude of the particles varies from zero to 2 A.