Question

The distance between the two successive nodes is:
$\begin{align}
  & \text{A}\text{. }\dfrac{\lambda }{4} \\
 & \text{B}\text{. }\dfrac{\lambda }{2} \\
 & \text{C}\text{. }\lambda \\
 & \text{D}\text{. }2\lambda \\
\end{align}$

Answer 1

Hint: A standing wave pattern always consists of an alternating pattern of nodes and antinodes. Nodes and antinodes are the points known to form stationary waves. If we assume a stationary wave traveling in a medium, the approximate distance between a node and the immediate next antinode is actually one-fourth of the given wavelength. We can determine the distance between the two successive nodes by considering the positions where a node and an antinode is formed in a wave within a medium.

Complete step by step answer:
A standing wave pattern is described as an interference phenomenon. It is formed as a result of the perfectly timed occurrence of interference of two waves passing through the same medium. A standing wave pattern is basically not a wave, but it is the pattern which results from the presence of two different waves of the same frequency with different directions of propagation within the same medium.

One main characteristic of a standing wave pattern is that there are some points along the medium that appear to be standing still while the wave propagates. These points are described as the points of no displacement and are referred to as Nodes. There are other points along the medium that perform vibrations between a large positive and large negative displacement. These are the points that actually undergo the maximum displacement during each vibrational cycle of the standing wave. These points are described as the opposite of nodes, and are referred to as Antinodes.

A standing wave pattern always consisted of an alternating pattern of nodes and antinodes. The nodes are produced at the specific locations where destructive interference occurs. For instance, nodes form at locations where the crest of one wave meets the trough of another wave while the antinodes are produced at the specific locations where constructive interference occurs. For instance, if the crest of one wave meets the crest of another wave, a point of large positive displacement results, while if the trough of one wave meets the trough of another wave, a point of large negative displacement results. Antinodes are the points which always vibrate back and forth between these points of large positive and large negative displacement.
Nodes and antinodes are known to generate the stationary waves. In a given stationary wave, the distance between any given two successive nodes or any two successive antinodes is always half of the wavelength.
The distance between the two successive nodes is $\dfrac{\lambda }{2}$.
Hence, the correct option is B.

Note:
Nodes and antinodes points should not be confused with crests and troughs in a wave. When we discuss the motion of a traveling wave, it is obvious to refer to a point of large maximum displacement as a crest and a point of large negative displacement as a trough. These points represent the position of the disturbance that travels from one location to another through the medium. A node on the other hand is a point on the medium that is staying in the same location while the wave is travelling, while an antinode vibrates back and forth between the maximum upward and the maximum downward displacement. The nodes and antinodes are not actually part of a wave. Since a standing wave is not technically a wave, but an interference pattern, an antinode is not technically a point on the wave. The nodes and antinodes are merely unique points located inside the medium that make up the wave pattern.