The particle displacement of a travelling lon | vedclass

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Question

(D) In a longitudinal wave,the displacement $\xi$ is given by $\xi = A \sin(kx - \omega t)$. The strain is given by $\partial \xi / \partial x = Ak \c

Vedclass · Accepted Answer

All of the above

Vedclass · Answer

(D) In a longitudinal wave,the displacement $\xi$ is given by $\xi = A \sin(kx - \omega t)$. The strain is given by $\partial \xi / \partial x = Ak \cos(kx - \omega t)$.At a compression zone $(C)$,the displacement gradient $\partial \xi / \partial x$ is at its negative maximum,and at a rarefaction zone $(R)$,it is at its positive maximum. Thus,their magnitudes are equal: $|\partial \xi / \partial x|_C = |\partial \xi / \partial x|_R$.At the midpoint between $C$ and $R$,the displacement gradient is zero,which corresponds to the equilibrium position where particle velocity is maximum,but the displacement is zero. However,the pressure variation is given by $\Delta P = -B (\partial \xi / \partial x)$,where $B$ is the Bulk modulus.The pressure difference between $C$ and $R$ is $\Delta P_C - \Delta P_R = -B (\partial \xi / \partial x)_C - (-B (\partial \xi / \partial x)_R)$. Since $(\partial \xi / \partial x)_C = -|\partial \xi / \partial x|$ and $(\partial \xi / \partial x)_R = +|\partial \xi / \partial x|$,the difference is $2B |\partial \xi / \partial x|$.Therefore,all statements are correct.

The particle displacement of a travelling longitudinal wave is represented by $\xi = \xi(x, t)$. The midpoints of a compression zone and an adjacent rarefaction zone are represented by the letters $C$ and $R$. Which of the following is true?

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