From the equation = 𝐴 sin 2𝜋𝜆 (𝑣𝑡 − 𝑥) , establish the relation between particlevelocity and wave velocity.

The given equation represents a sinusoidal wave propagating in the positive x-direction. The equation can be written as:

y = A sin 2πλ (vt - x)

where:

• y is the displacement of the particle
• A is the amplitude of the wave
• λ is the wavelength
• v is the wave velocity
• t is the time
• x is the position of the particle

The particle velocity (vp) is the rate of change of displacement with respect to time. So, we differentiate y with respect to t to get the particle velocity.

vp = dy/dt = A * 2πλ * v * cos(2πλ (vt - x))

The wave velocity (v) is the speed at which the wave propagates in space. It is given by the ratio of the wavelength to the period (T) of the wave.

v = λ / T

Therefore, the particle velocity is a function of the wave velocity, the amplitude of the wave, the wavelength, and the position and time of the particle. The particle velocity and wave velocity are related, but they are not the same thing. The particle velocity varies with time and position, while the wave velocity is a constant for a given medium.