Consider the waves propagating on a stretched membrane of negligible thickness, having surface mass density $σ$ and coefficient of surface tension T. A transverse wave $S (x, y, t)$ propagates along the membrane. AAssume $S (x, y, t) < < λ$ . Consider an element in the membrane with natural dimensions $(d x \times d y) .$ Then which of the following options are correct.

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acceleration of centre of mass of the membrane element is $\frac{T}{σ} (\frac{\partial^{2} s}{\partial x^{2}} + \frac{\partial^{2} s}{\partial y^{2}})$

speed of propagation of surface waves is $\sqrt{\frac{2 T}{σ}}$

speed of propagation of surface waves is $\sqrt{\frac{T}{σ}}$

acceleration of centre of mass of the membrane element is $\frac{2 T}{σ} (\frac{\partial^{2} s}{\partial x^{2}} + \frac{\partial^{2} s}{\partial y^{2}})$

answer is C, D.

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Detailed Solution

For a small element (dx, dy) the force acting on it in $T d y \sin θ \approx T d y \tan θ$
$∴ F = T d y d x \frac{δ^{2} s}{δ y^{2}}$
same can be said in x direction
$∴ F_{n e t} = m a$
$σ \frac{\partial^{2} s}{\partial t^{2}} = T d x d y (\frac{\partial^{2} s}{\partial λ^{2}} + \frac{\partial^{2} s}{d y^{2}})$
$ν = \sqrt{\frac{T}{σ}}$