Question

Exercise:

[ A transverse harmonic wave on a string is described by

$y(x,t)=3.0sin(36t+0.018x+\pi /4)$
where x and y are in cm and t in s. The positive direction of x is from left to right. ]

For the wave described in the above Exercise, plot the displacement (y) versus (t) graphs for x=0, 2 and 4 cm. What are the shapes of these of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?

Answer 1

All the waves have different phases.
The given transverse harmonic wave is:

$y(x,t)=3.0sin(36t+0.018x+\frac{\pi }{4})$ ...(i)

For $x=0$, the equation reduces to:

$y(0,t)=3.0sin(36t+\frac{\pi }{4})$

Also,

$\omega =2\pi /t=36rad/s^{-1}$

$\therefore t=\pi /18s$

Now, plotting y vs. t graphs using the different values of t, as listed in the given table

t (s)	0 & T/8	2T/8 & 3T/8	4T/8 & 5T/8	6T/8 & 7T/8
y (cm)	$3\sqrt{2}\&3$	$3\sqrt{2}\&0$	$-3\sqrt{2}\&-3$	$-3\sqrt{2}\&0$

For $x=0,x=2$, and $x=4$, the phases of the three waves will get changed. This is because amplitude and frequency are invariant for any change in $x$. The y-t plots of the three waves are shown in the given figure.