A standing wave is setup in a string fixed at both ends. Then

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The total energy of a point midway between a node and an adjacent antinode is constant

The sum of total energy per unit length at nodes and antinodes is constant

If the string has a standing wave and a component of travelling wave, then the kinetic energy of the points of minimum amplitude is non-zero at some instant of time.

If the string has a standing wave and a component of travelling wave, then the potential energy of the points of maximum amplitude is non zero at some instant of time.

answer is A, C, D.

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Detailed Solution

$\begin{array}{l} y = 2 Asin (kx - ωt) \\ {(\frac{\partial K}{\partial x})}_{antinodes} = 2 {μA}^{2} ω^{2} \cos^{2} ωt \\ {(\frac{\partial U}{\partial x})}_{nodes} = 2 {μA}^{2} ω^{2} \cos^{2} ωt \\ {(\frac{\partial U}{\partial x})}_{antinodes} + {(\frac{\partial U}{\partial x})}_{nodes} = 2 {μA}^{2} ω^{2} Cons \tan t \end{array}$
For a point midway between a node and an adjacent antinode
$\begin{array}{l} \frac{\partial K}{\partial x} = {μA}^{2} ω^{2} \sin^{2} ωt \\ \frac{\partial U}{\partial x} = {μA}^{2} ω^{2} \cos^{2} ωt \\ Let y = Asin (kx - ωt) + 2 Asin kxcos ωt \\ \frac{\partial K}{\partial x} = \frac{1}{2} μ {(\frac{\partial y}{\partial t})}^{2} = \frac{1}{2} μ [- 2 Aωsin kxsin ωt - Aωcos (kx - ωt)]^{2} \\ \frac{\partial K}{\partial x} = \frac{1}{2} T {(\frac{\partial y}{\partial t})}^{2} = \frac{1}{2} V^{2} μ [2 Aωcos kxcos ωt + Akcos (kx - ωt)]^{2} \end{array}$

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