7.4 Stationary Waves & their Formation

7.4 Stationary Waves & their Formation

Stationary waves, also known as standing waves, are the waves that oscillate in a fixed position, without moving or propagating. They are formed by the superposition of two waves with the same frequency and amplitude, travelling in opposite directions. The resulting wave pattern remains stationary, with nodes (points of zero amplitude) and antinodes (points of maximum amplitude) at fixed positions. Examples include waves on a string, and sound waves in a pipe. The term "standing wave" describes that the wave pattern remains fixed in space, oscillating between positive and negative values, without moving forward or backward.

Let us consider the superposition of two waves moving along a string in opposite directions. Figures 7.6 (a) and (b) show the profile of two such waves at instants $t = 0$, $T/4$, $T/2$, $3T/4$ and $T$, where $T$ is the time period of the wave. We are interested in finding out the displacements of the points $1$, $2$, $3$, $4$, $5$, $6$ and $7$ at these instants as the waves superpose. It is obvious that the points $1$, $2$, $\ldots$, $7$ are distant $\lambda/4$ apart, being the wavelength of the waves. We can determine the resultant displacement of these points by applying the principle of superposition.

Figure 7.6 (c) shows the resultant displacement of the points $1$, $3$, $5$ and $7$ at the instants $t = 0$, $T/4$, $T/2$, $3T/4$ and $T$. It can be seen that the resultant displacement of these points is always zero. These points of the medium are known as nodes. Here, the distance between two consecutive nodes is $\lambda/2$.

Figure 7.6 (d) shows the resultant displacement of the points $2$, $4$ and $6$ at the instants $t = 0$, $T/4$, $T/2$, $3T/4$ and $T$. The figure shows that these points are moving with an amplitude which is the sum of the amplitudes of the component waves represented by arrows. These points are known as antinodes. They are situated midway between the nodes and are $\lambda/2$ apart. The distance between a node and the next antinode is $\lambda/4$. Such a pattern of nodes and antinodes is known as a stationary or standing wave.

Energy in a wave transfers because of the motion of the particles of the medium. The nodes always remain at rest, so energy cannot flow past these points. Hence, energy remains "standing" in the medium between nodes, although it alternates between potential and kinetic forms at the antinodes. When the antinodes are all at their extreme displacements, the energy stored is wholly potential and when they are simultaneously passing through their equilibrium positions, the energy is wholly kinetic.