Question

A wave travels along the position $x-$direction with a speed of $20m{s}^{-1}$. The amplitude of the wave is $0.20cm$ and the wavelength $2.0cm$

$\left(a\right)$ Write a suitable wave equation that describes this wave.

$\left(b\right)$ What is the displacement and velocity of the particle at$x=2.0cm$ at time$t=0$according to the wave equation written$?$ Can you get different values of this quantity if the wave equation is written in a different fashion$?$

Answer 1

Step 1. Given data.

Wavelength is $\lambda =2cm$

Speed is $V=2.0m/s$

Amplitude is $A=0.20cm$

Step 2. Equation of wave along the x-axis.

We know that the wave equation is given by $y=A\sin (kx–wt)$

Now, $k$ is given by $k=\frac{2\pi }{\lambda }=\pi c{m}^{-1}$ also

Where $\lambda$ is the wavelength

$T$ is given as $T=\frac{\lambda }{V}=\frac{2}{2000}={10}^{-3}sec$

Where $\lambda$ is the wavelength

$V$ is the speed

So, the wave equation is,$y=(0.2)\sin \pi x–\left(2\pi {10}^3\right)t$

Step 3. Calculating displacement and velocity

As we obtain the wave equation as $y=(0.2)\sin \pi x–\left(2\pi {10}^3\right)t$

Now, by applying the given boundary conditions $x=2cm$ and $t=0$

We get, $y=(0.2)\sin (\pi /2)=0$

The particle velocity is given by

$$\begin{gathered} v=AV\cos \left(\pi x\right) \\ =0.2x2000\pi x\cos 2\pi \\ =400\pi \\ =4\pi m/s \end{gathered}$$

Therefore, the displacement is $0$and particle velocity $v=4\pi m/s$

Yes, we can get different values if the amplitude and angular frequency are changed.