Let a disturbance y be propagated as a plane wave along the...

Let a disturbance $y$ be propagated as a plane wave along the $x$-axis. The wave profiles at the instants $t=t_{1}$ and $t=t_{2}$ are represented, respectively, as $y_{1}=f\left(x_{1}-v t_{1}\right)$ and $y_{2}=f\left(x_{2}-v t_{2}\right) .$ The wave is propagating without change of shape. (A) The velocity of the wave is $2 v$. (B) The velocity of the wave is $v=\frac{x_{2}-x_{1}}{t_{2}}$. (C) The particle velocity is $v_{p}=v$. (D) None of these.

Let a disturbance $y$ be propagated as a plane wave along the $x$-axis. The wave profiles at the instants $t=t_{1}$ and $t=t_{2}$ are represented, respectively, as $y_{1}=f\left(x_{1}-v t_{1}\right)$ and $y_{2}=f\left(x_{2}-v t_{2}\right) .$ The wave is
propagating without change of shape.
(A) The velocity of the wave is $2 v$.
(B) The velocity of the wave is $v=\frac{x_{2}-x_{1}}{t_{2}}$.
(C) The particle velocity is $v_{p}=v$.
(D) None of these.

Key Concepts

Plane Waves

A plane wave is a wave whose wavefronts (surfaces of constant phase) are infinite, parallel planes. In this context, the displacement function y = f(x - vt) represents a plane wave moving in the x-direction. This form indicates that every point of the wave profile is shifted uniformly by an amount vt over time, keeping the shape of the waveform unaltered.

Phase Velocity

The phase velocity is the speed at which a particular phase point (or any feature of constant phase) on the wave moves. For a wave described by y = f(x - vt), the constant phase condition implies x - vt = constant, leading directly to the phase velocity being v. This concept is key when analyzing how the wave translates through space over time.

Particle Velocity

Particle velocity refers to the speed at which individual particles of the medium move as the wave passes through, rather than the speed at which the wave form travels. In the wave function y = f(x - vt), the time derivative of y gives the particle velocity, v_p = -v f?(x - vt), which generally varies with position and does not equal the constant phase speed v. This distinction is important in understanding motion within the medium versus the propagation of the wave itself.

Wave Shape Invariance (Non-dispersive Propagation)

When a wave propagates without a change in shape, it is said to be non-dispersive. This means that each part of the wave travels at the same speed, and the overall profile f of the wave remains unchanged as it moves through the medium. The functional form f(x - vt) demonstrates this invariance, with the parameter v representing the unaltered phase speed.

Transcript

Please give Ace some feedback