Question

Figure shows a wave motion in which displacements of particles at an instant is represented as shown. Choose the correct options.

• A. If the wave represents a transverse wave in a stretched string, then an element in the region $B$ has max. kinetic energy
• B. If it represents a transverse wave in a stretched string, then an element in the region $B$ has maximum potential energy
• C. If it represents sound wave travelling in air, point $D$ represents rarefaction
• D. If it represents sound wave travelling in air, point $A$ has normal atmospheric pressure.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A If the wave represents a transverse wave in a stretched string, then an element in the region $B$ has max. kinetic energy
B If it represents a transverse wave in a stretched string, then an element in the region $B$ has maximum potential energy
C If it represents sound wave travelling in air, point $D$ represents rarefaction
D If it represents sound wave travelling in air, point $A$ has normal atmospheric pressure.

For a string segment (Transverse wave),
Potential energy $U\propto (\frac{dy}{dx}{)}^2$
Kinetic energy $K\propto (\frac{dy}{dt}{)}^2$
$PE$ is maximum, when slope is maximum.
$KE$ is maximum when speed of particle is maximum.
Hence, at the mean position $\left(B\right)$, both $PE$ and $KE$ are maximum.

For a longitudinal wave,

Displacement of the particles to the left of $D$ is towards left while that to the right is towards right. Hence it will be a rarefaction.
Variation of pressure $\Delta P=-B\frac{dy}{dx}$
At $A$, Slope is zero. Variation of pressure is zero. Hence the pressure at point $A$ is atmospheric.