Question

A transverse wave of amplitude $0.01m$ and frequency $500Hz$ is travelling along a stretched string with a speed of $200\dfrac{m}{s}$. Find the displacement of a particle at a distance of $0.7m$ from the origin after $0.01s$. Also find the phase difference between the points where the wave reaches from the origin.

Answer 1

Hint: A transverse wave has a speed of propagation $v$, which is given by the formula, $v=n\lambda $. We can apply the equation motion of the transverse wave to find the displacement of the wave after a fixed interval of time.

Formulae used:
$v=n\lambda $
$y=a\sin \left[ 2\pi \left( \dfrac{t}{T}-\dfrac{x}{\lambda } \right) \right]$

Complete step by step answer:
A transverse wave is a moving wave in which particles oscillate in a direction perpendicular to the direction of the wave or the path of propagation of the wave. Transverse waves are characterised by peaks and valleys, which are called crests and troughs. An example of a transverse wave can be the waves produced on a horizontal string by anchoring one of its ends and moving the other end up and down.
Equation of transverse wave is given by:
$y=a\sin (\omega t-\phi )$
$\omega =2\pi f$and $\phi =\dfrac{2\pi x}{\lambda }$
We are given,
Amplitude of the transverse wave $a=0.01m$
Frequency of transverse wave $n=500Hz$
Velocity of transverse wave $v=200m{{s}^{-1}}$
From the formula,$v=n\lambda $, where $\lambda $ is the wavelength of transverse wave
 $\lambda =\dfrac{v}{n}=\dfrac{200}{500}=0.4$
Equation of transverse wave,
$y=a\sin \left[ 2\pi \left( \dfrac{t}{T}-\dfrac{x}{\lambda } \right) \right]$
Putting values of given quantities in the above equation, we get,
$\begin{align}
  & y=0.01\sin \left[ 2\pi \left( 0.01\times 500-\dfrac{0.7}{0.4} \right) \right] \\
 & =0.01\sin \left[ 2\pi (5-1.75) \right] \\
 & =0.01\sin {{1170}^{\circ }} \\
 & =0.01\sin (3\times {{360}^{\circ }}+{{90}^{\circ }})=0.01\sin ({{90}^{\circ }})
\end{align}$
$y=0.01m$
Displacement of particle is $0.01m$
Now,
$\begin{align}
  & \Delta \phi =\dfrac{2\pi }{T}\times \Delta t \\
 & =2\pi \times 500\times 0.01=10\pi
\end{align}$
$\Delta \phi =10\pi $
Required phase difference is $10\pi $

Additional information:
Longitudinal waves are the moving waves in which displacement of the medium is in the direction same as the direction of propagation of the wave. These waves are also known as compression waves because they produce compression and rarefaction when travelling through a medium and known as pressure waves because they produce increase and decrease in the pressure while propagating.
Equation of Longitudinal wave:
\[y(x,t)={{y}_{o}}\cos \left[ \omega \left( t-\dfrac{x}{c} \right) \right]\]
Where, $y$ is the displacement of the wave, $x$ is the amount of distance that the point has travelled from the source of wave, $t$ is the time elapsed, ${{y}_{o}}$ is the amplitude of wave oscillations, and $\omega $ is the angular frequency of the wave

Note: Transverse waves always oscillate in the Z - Y plane, but they travel along the x-axis. Also, keep in mind that in the wave equation, we always consider the speed of the wave itself, not the speed of particles of the wave.