Q.During propagation of a plane progressive mechanical wave (Note: more than one of the given options may be correct.)

Concept: Properties of Plane Progressive Mechanical Waves

A plane progressive mechanical wave is characterized by the transfer of energy and momentum through a medium without net transport of matter.

1. In a progressive wave, the phase of vibration varies with position. Particles at different locations generally vibrate with different phases, unless they are separated by an integer multiple of wavelengths. Therefore, option (A) is incorrect.
2. For a plane progressive wave propagating in a uniform medium, the amplitude of oscillation is constant for all particles of the medium. Therefore, option (B) is correct. …

A plane progressive mechanical wave is a disturbance that travels through a medium, transferring energy without transferring matter. The term "plane" implies that the wavefronts are planar, meaning the wave characteristics are uniform across any plane perpendicular to the direction of propagation. "Progressive" means the wave moves forward, carrying energy. "Mechanical" means it requires a material medium for its propagation, unlike electromagnetic waves. Understanding these fundamental characteristics is key to evaluating the given options.

Let's analyze each option:

1. Option (A): all the particles are vibrating in the same phase.

   In a progressive wave, the disturbance propagates through the medium. This means that different particles in the medium reach their maximum displacement, zero displacement, etc., at different times. If all particles were vibrating in the same phase, they would all move up and down simultaneously, which would mean the wave is not progressing. Instead, the phase of vibration changes continuously as you move along the direction of wave propagation. For example, in a sinusoidal wave described by $y(x, t) = A \sin(kx - \omega t)$, the phase is $(kx - \omega t)$. For a fixed time $t$, as $x$ changes, the phase changes. Only particles separated by an integer multiple of the wavelength ($\lambda$) will be in the same phase. Therefore, this statement is incorrect.

2. Option (B): amplitude of all the particles is equal.

   For an ideal plane progressive wave propagating through a uniform, homogeneous, and isotropic medium, there is no loss of energy due to damping or spreading. Consequently, the energy carried by the wave remains constant, and thus, the maximum displacement (amplitude) of every oscillating particle in the medium is the same. If the medium were non-uniform or if there were significant damping, the amplitude might decrease with distance. However, for a standard "plane progressive mechanical wave," equal amplitude is an inherent characteristic. Therefore, this statement is correct.

3. Option (C): particles of the medium executes S.H.M. …

Step 1: For a progressive wave $y = A\sin(kx \mp \omega t + \phi)$, the phase $(kx \mp \omega t)$ depends on $x$, so particles at different positions are generally out of phase (only those separated by a whole number of wavelengths share phase). Rules out (a).

Step 2: In an ideal, undamped, uniform medium, every particle oscillates with the same maximum displacement $A$ — the wave's amplitude. Confirms (b). …