Physics · Waves · NEET
No. Wave velocity v = w/k is a single constant value for a given wave in a given medium; it is the speed of the moving pattern. Particle velocity v_p = dy/dt is the speed of one medium particle doing SHM, so it changes every instant and is different for different particles. They are never the same number by definition.
In a mechanical wave, energy moves forward but matter does not. Each particle only oscillates about its own fixed rest point (up-down for transverse, back-forth for longitudinal). The wave pattern advances because neighbouring particles start their oscillation slightly later, so the shape appears to move even though no particle leaves its spot.
For y = a sin(wt - kx), particle velocity is v_p = dy/dt = a*w*cos(wt - kx). This oscillates between +a*w and -a*w. The maximum particle speed (particle velocity amplitude) is v_p(max) = a*w = a*omega.
Wave velocity is constant for a given medium because it depends only on the medium's properties (for a string v = sqrt(T/mu), for sound it depends on the gas). It does not depend on amplitude or on time. Only the particle velocity changes with time and position.
Particle velocity = -(wave velocity) * (slope of the wave), i.e. v_p = -v * (dy/dx). This comes from dy/dt = -(w/k)*(dy/dx). So the particle velocity at any point equals the negative of the wave speed multiplied by the slope of the y-x graph at that point.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The maximum particle velocity (particle velocity amplitude) is v_p(max) = a*w = a*omega, where a is amplitude and w is angular frequency. A particle reaches this maximum speed when it passes through its mean (rest) position.
Particle velocity is zero at the extreme positions, that is at a crest and at a trough, because the particle momentarily stops before reversing direction, just like a body in SHM at its turning points.
Yes. Since v_p(max) = a*w and wave velocity v = w/k, the ratio is v_p(max)/v = a*k. If a*k > 1 the maximum particle speed exceeds the wave speed. For most real waves a*k is small, but it is not forbidden.
Differentiate the displacement y with respect to time t, keeping x constant. For y = a sin(wt - kx), v_p = dy/dt = a*w*cos(wt - kx). Substitute the given x and t to get the value at that instant.
It checks whether you can tell apart the pattern speed (w/k, f*lambda) from the medium-particle speed (a*w). Many questions give a wave equation and ask for particle velocity or its maximum, so mixing up the two formulas loses easy marks.
