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Wize University Physics Textbook (Master) > Waves: Mechanical

Particle Velocity and Acceleration

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Particle Velocity and Acceleration in Mechanical Waves


In Sinusoidal mechanical waves, each particle of the medium undergoes a simple harmonic motion. The collective motion of all particles creates the wave.


Imagine the wave function describing the propagation of the wave to be:

y(x,t)=Acos⁡(kx+ωt+φ)y\left(x,t\right)=A\cos\left(kx+\omega t+\varphi\right)


Particle velocity (also known as transverse velocity) is the velocity of each particle in the medium in its SHM:


vy(x,t)=dy(x,t)dt=−Aωsin⁡(kx+ωt+φ)\boxed{v_y\left(x,t\right)=\frac{dy\left(x,t\right)}{dt}=-A\omega\sin\left(kx+\omega t+\varphi\right)}





Similarly, one can find the acceleration of each particle in the medium as:


ay(x,t)=dv(x,t)dt=−Aω2cos⁡(kx+ωt+φ)\boxed{a_y\left(x,t\right)=\frac{dv\left(x,t\right)}{dt}=-A\omega^2\cos\left(kx+\omega t+\varphi\right)}






Watch Out!
Note that above equations for velocity and acceleration of particles in the wave medium are derived based on the wave function showed on top. If we use a different wave function (for example the one with sine function or the one for a wave moving toward right), then these equations might be slightly different accordingly.









Exam Tip
It is important to distinguish wave speed and particles' velocity. Particles' velocity is the velocity of each particle in the wave medium while wave speed is the speed at which the whole wave pattern travels in space.



Example: Displacement, Velocity, Acceleration

The velocity of a particle on a traveling wave is given by the equation v(x,t)=8cos⁡(3x+4t+π6)v(x,t)=8\cos\bigg(3x+4t+\dfrac{\pi}{6}\bigg). Give the equation of its position y(x,t)y(x,t) and acceleration a(x,t)a(x,t). Find the wave speed.

We're given the velocity equation:

v(x,t)=Aωcos⁡(kx+ωt+φ)v(x,t)=A\omega\cos(kx+\omega t+\varphi)

v(x,t)=8cos⁡(3x+4t+π6)v(x,t)=8\cos\bigg(3x+4t+\dfrac{\pi}{6}\bigg)

Therefore Aω=8A\omega=8 , and since w=4w=4 we get:

A=8ω=84=2A=\dfrac{8}{\omega}=\dfrac{8}{4}=2

So the position function is (integrate velocity):

y(x,t)=−Asin⁡(kx+ωt+φ)y(x,t)=-A\sin(kx+\omega t+\varphi)

y(x,t)=−2sin⁡(3x+4t+π6)y(x,t)=-2\sin\bigg(3x+4t+\dfrac{\pi}{6}\bigg)

And the acceleration is (differentiate velocity):

a(x,t)=Aω2sin⁡(kx+ωt+φ)a(x,t)=A\omega^2\sin(kx+\omega t+\varphi)

a(x,t)=2⋅42⋅sin⁡(3x+4t+π6)a(x,t)=2\cdot4^2\cdot\sin\bigg(3x+4t+\dfrac{\pi}{6}\bigg)

=32sin⁡(3x+4t+π6)=32\sin\bigg(3x+4t+\dfrac{\pi}{6}\bigg)

Wave speed:

This is NOT the v(x,t)v(x,t) function above, which is the particle velocity. We need to find v=λfv=\lambda f.

We have k=3=2πλk=3=\dfrac{2\pi}{\lambda} so λ=2π3\lambda = \dfrac{2\pi}{3}

Also, ω=4=2πf\omega=4=2\pi f so f=2πf=\dfrac{2}{\pi}

Wave speed is: v=λf=2π3⋅2π=43v=\lambda f=\dfrac{2\pi}{3} \cdot\dfrac{2}{\pi}=\dfrac{4}{3} (m/s)


Exam Tip
You can use the shortcut v=ωkv=\dfrac{\omega}{k} to find the wave speed.

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Practice: Displacement, Velocity, Acceleration


A 250250 Hz transverse wave in a rope is traveling with a speed of 1010 m/s toward left.

a) Write the wave equation (displacement) if the amplitude of the wave is 33 cm, and at t=0t=0 and x=0x=0 the displacement is a minimum.

b) Write the velocity and acceleration equations.

c) Draw a snapshot of the wave at t=2t=2 ms.