Question

A wave is described by the equation
$y=\left(1·0\mathrm{mm}\right)\sin \pi \left(\frac{x}{2·0\mathrm{cm}}-\frac{t}{0·01\mathrm{s}}\right).$
(a) Find the time period and the wavelength? (b) Write the equation for the velocity of the particles. Find the speed of the particle at x = 1⋅0 cm at time t = 0⋅01 s. (c) What are the speeds of the particles at x = 3⋅0 cm, 5⋅0 cm and 7⋅0 cm at t = 0⋅01 s?
(d) What are the speeds of the particles at x = 1⋅0 cm at t = 0⋅011, 0⋅012, and 0⋅013 s?

Answer 1

The wave equation is represented by
$y=\left(1·0\mathrm{mm}\right)\sin \mathrm{\pi }\left(\frac{x}{2·0\mathrm{cm}}-\frac{t}{0·01\mathrm{s}}\right)$
Let:
Time period = T
Wavelength = λ

$$\begin{gathered} \left(a\right)T=2\times 0.01=0.02\mathrm{s}=20\mathrm{ms} \\ \lambda =2\times 2=4\mathrm{cm} \end{gathered}$$

(b) Equation for the velocity of the particle:
$\nu =\frac{dy}{dt}=\frac{\mathrm{d}}{dt}\left[\sin 2\mathrm{\pi }\left(\frac{\mathrm{x}}{4}-\frac{t}{0.02}\right)\right]$
$$\begin{gathered} =-0.50\cos 2\pi \left\{\left(\frac{x}{4}\right)-\left(\frac{t}{0.02}\right)\right\}\times \frac{1}{0.02} \\ \Rightarrow \nu =-0.50\cos 2\pi \left\{\left(\frac{x}{4}\right)-\left(\frac{t}{0.02}\right)\right\} \\ \mathrm{At}x=1\mathrm{and}t=0.01\mathrm{s},\nu =-0.50\cos 2\pi \left\{\frac{1}{4}-\frac{1}{2}\right\}=0. \end{gathered}$$

(c) (i) Speed of the particle:
$\mathrm{At}x=3\mathrm{cm}\mathrm{and}t=0.01\mathrm{s},$ $\nu =-0.50\cos 2\pi \left\{\frac{3}{4}-\frac{1}{2}\right\}=0$.
(ii) $\mathrm{At}x=5\mathrm{cm}\mathrm{and}t=0.01\mathrm{s},$
$$\begin{gathered} \nu =0 \\ \left(\mathrm{iii}\right)\mathrm{At}x=7\mathrm{cm}\mathrm{and}t=0.1\mathrm{s},\nu =0. \\ \left(\mathrm{iv}\right)\mathrm{At}x=1\mathrm{cm}\mathrm{and}t=0.011\mathrm{s}, \\ \nu =50\cos 2\pi \left\{\left(\frac{1}{4}\right)-\left(\frac{0.011}{0.02}\right)\right\} \\ =-50\cos \left(\frac{3\pi }{5}\right)=-9.7\mathrm{cm}/\mathrm{s} \end{gathered}$$
(By changing the value of t, the other two can be calculated.)