A standing wave $y = A \sin k x . \cos ω t$ is established in a string fixed at its ends.

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The value of instantaneous power transfer at a cross section of the string when the string is passing through its mean position zero.

The value of instantaneous power transfer at a cross section of the string when the string is at its extreme position is zero.

The frequency of the power transmitted through a cross section is changing with time .

The value of instantaneous power transfer at a cross section of the string when the string is passing through its mean position maximum.

answer is A, B.

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Detailed Solution

(A) Velocity at a point is perpendicular to the tension. Hence power is zero
(b) Velocity at all points is zero. Hence power is zero.
(c) $y = a \sin k x \cos w; p = - (T \frac{\partial y}{\partial x}) (\frac{\partial y}{\partial t})$
Where the first component is transverse component of tension and the term in second bracket is the velocity at a point.
P is positive when energy is transmitted in the direction of positive $x$ . When energy is transferred in other direction $ρ$ is negative
The $(-)$ sign in the above expression is to ensure this sign convention.
$∴ P = - T (K A \sin ω t \cos k x) (ω A \cos ω t \sin k x) = - \frac{T}{4} A^{2} ω k (\sin 2 k x)$ $(\cos 2 ω t)$
Hence frequency of oscillation of P at a given point is twice that of the wave
Wave frequency, $f = \frac{ω}{2 π}$
Power frequency $= \frac{ω}{π}$
Let amplitude of wave on are $m$ and $ρ$ be a due to individual sources