Standing waves (or) Stationary wave pattern

STANDING WAVES (OR) STATIONARY WAVE PATTERN

Definition

When two waves executing simple harmonic motion travel in opposite directions in a straight line, with same amplitude, same frequency and same time period, then the resultant wave obtained is called a standing wave (or) stationary wave pattern.

Explanation

Let us consider two wave trains 'A' and 'B' of same amptitude, same frequency and wavelength, as shown in Fig. 4.18. & Fig. 4.19.

If both the wave travel in opposite direction, then at time t = 0 sec, the resultant wave (or) displacement curve is a straight line as shown in Fig.4.20, from which, we can see that all the particles are at their mean positions.

SPECIAL CASES

Case (i): At time t = T/ 4

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At time t = T/4, the wave 'A' advances through a distance λ/4 towards right as shown in Fig.4.21, and the wave 'B' advances through a distance λ/4 towards left as shown in Fig. 4.22.

The resultant displacement pattern is as shown in Fig.4.23, from which we can observe that the particles at 1,3,5,7,9 are at their extreme positions (PQRST), whereas the particles at 2,4,6,8 are at their mean positions.

Case (ii) At time t = T/2

At time t = T/2, the wave A advances through a distance λ/2 towards right as shown in Fig. 4.24, and the wave 'B' advances through a distance λ/2 towards left as shown in Fig. 4.25.

The resultant wave (or) displacement pattern is again a straight line as shown in Fig. 4.26, i.e., all the particles are at their mean positions.

Case (iii): At time t: 3T/4

At time t = 3T / 4, the wave 'A' advances through a distance of 3λ/4 towards right as shown in Fig. 4.27 and the wave 'B' advances through a 3λ/4 distance towards left as shown in Fig. 4.28.

The resultant displacement pattern is as shown in Fig. 4.29, from which we can see that the particles 1,3,5,7 9 are at extreme positions and the particles at 2,4,6,8 are at mean positions.

Case (iv): At time t = T

At time t = T the wave 'A' advances through a distance of ' λ' towards right as shown in Fig. 4.30 and the wave 'B' advances through a distance of ' λ' towards left as shown in Fig.4.31.

The resultant displacement is again a straight line as shown in Fig. 4.32, wherein which all the particles are at their mean positions.

Conclusions

Thus, from the above cases we can conclude, the following points, viz.,

1) The particles of the medium at 2,4,6,8 etc always remain at their mean positions and are called NODES as shown in Fig.4.33.

Thus, node is a position of zero displacement with maximum strain.

2) The particles of the medium at 1,3,5,7,9 etc continue to vibrate with simple harmonic motion about their mean positions with DOUBLE the amplitude of each wave and are called ANTINODES as shown in Fig.4.33.

Thus, Antinode is the maximum displacement with minimum strain.

3) Though the wave pattern appears to be stationary, the resultant displacement pattern at various intervals of time (t=0, T/4, T/2, 3T/4, T) is as shown in Fig.4.33.