two ends of a string in her hands and then moves her hands up and down so that a wave travels from one end of the string to the other end. In which case will she have a standing wave along the string?
To determine in which case a standing wave will occur along the string when it is moved up and down, we need to understand the concept of standing waves and the conditions required for their formation.
A standing wave is a pattern produced when two waves of the same frequency traveling in opposite directions interfere with each other. This interference causes certain parts of the wave to cancel out, creating stationary points called nodes and regions of maximum displacement called antinodes.
For a standing wave to occur on a string, the following conditions must be met:
1. The string must be fixed at both ends.
2. The frequency of the waves traveling in opposite directions must be the same.
3. The waves must have proper phase relation to each other.
Based on these conditions, let's consider the two cases:
Case 1: When the girl moves her hands up and down at the same frequency and the string is fixed at both ends.
In this case, if the girl moves her hands in such a way that waves are generated with the same frequency and travel in opposite directions along the string, the conditions for a standing wave are met. The waves will interfere with each other and produce a standing wave pattern along the string.
This is because the fixed ends of the string act as nodes, while the areas where the string moves up and down with maximum displacement act as antinodes. The wave motion will combine in a way that creates stable nodes and antinodes.
Case 2: When the girl moves her hands up and down at different frequencies or the string is not fixed at both ends.
In this case, the conditions for a standing wave are not met. If the frequencies of the waves traveling in opposite directions are different, or if the ends of the string are not fixed, the interference pattern will not result in a stable standing wave.
Therefore, to have a standing wave along the string, the girl needs to move her hands up and down at the same frequency, and the string should be fixed at both ends.