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In a longitudinal wave, both particle velocity and wave velocity occur in the same direction. It might seem intuitive to assume that the particle velocity should be identical to the wave velocity, given that both are consequences of particle motion.

To explore this, consider differentiating the equation $A\sin\left(\frac{2\pi}{\lambda}(x - vt)\right)$ with respect to time, aiming to derive the velocity. However, velocity ($v$) is already a parameter within this equation, suggesting an intrinsic complexity in equating particle velocity directly with wave velocity.

So, is there a difference between particle velocity and wave velocity in a longitudinal wave? If a difference exists, how can it be described?

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It might seem intuitive to assume that the particle velocity should be identical to the wave velocity

No, particle velocity and wave velocity are different things.

Look at the longitudinal wave in this animation (taken from Physics Lens - Longitudinal and Transverse Waves). It travels to the right with speed $v$,

animation of a longitudinal and a transverse wave

Focus on a single particle (like the one marked in red) which has equilibrium position $x_m$ and oscillates with amplitude $A$ around this equilibrium. The position of this particle is $$x(t) = x_m + A\sin\left(\frac{2\pi}{\lambda}(x_m-vt) \right)$$ Notice that we needed to distinguish between equilibrium position $x_m$ and position $x(t)$ at time $t$.

Then you get the velocity of this particle by differentiating the $x(t)$ from above with respect to time. You get $$\frac{dx(t)}{dt} = -A \frac{2\pi}{\lambda}v\cos\left(\frac{2\pi}{\lambda}(x_m-vt) \right)$$ You see the particle's velocity oscillates around $0$ between $+A\frac{2\pi}{\lambda}v$ and $-A\frac{2\pi}{\lambda}v$.

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    $\begingroup$ So the wave goes steadily marching on at speed $v$ while the particle's speed varies between 0 and $2\pi A v/\lambda$. $\endgroup$ Mar 2 at 12:10
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Yes, there is a difference between particle velocity ($v_p$) and wave velocity ($v$) in a longitudinal wave.

The wave velocity is the speed at which wavefronts propagate, calculated as $v = \lambda f$, where $\lambda$ is the wavelength and $f$ is the frequency of the wave. This velocity remains constant for a wave traveling through a medium. On the other hand, particle velocity refers to the oscillation speed of the medium's particles. For a wave described by $A\sin(2\pi\frac{x-vt}{\lambda})$, where $A$, $\lambda$, $x$, $v$, and $t$ represent amplitude, wavelength, displacement, wave velocity, and time respectively, the particle velocity is derived by differentiating with respect to time:

$$ v_p = \frac{\partial}{\partial t}A\sin\left(2\pi\frac{x-vt}{\lambda}\right) = -A\frac{2\pi v}{\lambda}\cos\left(2\pi\frac{x-vt}{\lambda}\right). $$

Thus $v_p$ varies with respect to time which makes sense since particles do undergo simple harmonic motion, unlike $v$, which remains constant.

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This wave is a disturbance of a medium that travels steadily onward.

A given time, some particle is being disturbed. It moves back and forth. This is particle velocity.

Wave velocity is about how long it takes for the disturbance to reach a nearby particle.

If the two velocities were the same thing, the particles would travel steadily onward with the wave.

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