Question

Select correct statement regarding waves on a string (all symbols have their usual meanings).
This question has multiple correct options
A. Power transfer through any point in a standing wave is $\mu v_{p}^{2}$.
B. Energy is conserved between consecutive nodes and antinodes.
C. Two travelling waves of the same frequency, which are moving in opposite directions, must form standing waves.
D. Speed of particles is maximum where slope is maximum.

Answer 1

When two waves of same amplitude and same frequency travelling on the string in opposite directions superimpose, the resultant wave formed is called a standing wave. A standing wave is a wave that does not propagate and oscillates at one place.

Complete answer:
The given four options speak about a stand wave on a string.When two waves of same amplitude and same frequency travelling on the string in opposite directions superimpose, the resultant wave formed is called a standing wave. A standing wave is a wave that does not propagate and oscillates at one place. A standing wave has fixed points where the amplitude is minimum. These points are called nodes. The fixed points with maximum amplitude are called antinodes.

Since the wave does not propagate, the energy is confined in one place. The power transferred at a node and an antinode is zero. However, at other points the power transferred is non zero. This means that power is transferred between a node and an antinode.

Since the energy is confined, the energy is conserved at all the points on the string. Where the kinetic energy of the particle is maximum, the elastic potential energy is minimum (at antinode) and where the kinetic energy is minimum, the elastic potential is maximum (at node).

This means that energy is conserved between consecutive nodes and antinodes.The speed of a particle on a string is given as ${{v}_{p}}=\dfrac{\omega }{k}\times slope$, where k is the wave number and $\omega$ is the angular frequency. Therefore, the speed of the particle is maximum when the slope is maximum.

Hence, with the help of the above discussed data, the correct options are B and D.

Note: The option C is incorrect because for the standing wave to be formed, the two waves must be of same frequency and same amplitude. Only with the same frequency a standing wave is not formed. The option A is incorrect because the dimensional formula of the expression given for the energy is not the same as the dimensional formula for energy.