Question

For the wave described in Exercise 15.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes 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)= 3.0 sin (36t + 0.018x + pi/4)` ...(i)

For x = 0, the equation reduces to:

`y (0,t) = 3.0 sin (36t + pi/4)`

Also, `omega = (2pi)/T = 36 " rad/s"^(-1)`

`:. T = pi/8 s`

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)	`(3sqrt2)/2`	3	`(3sqrt2)/2`	0	`(-3sqrt2)/2`	-3	`(-3sqrt2)/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.

Answer 2

The transverse harmonic wave is

`y(x,t) = 3.0 sin (36t + 0.018x + pi/4)`

for x = 0

`y(0,t) = 3 sin(36t + 0 + pi/4) = 3 sin (36t + pi/4)`  ...1

Here `omega = (2pi)/T = 36 => T =(2pi)/36`

To plot a(y) versus (t) graph, different values of y corresponding to the values of t may be tabulated as under (by making use of equation 1)

 

t(s)	0	T/8	2T/8	3T/8	4T/8	5T/8	6T/8	7T/8	T
y(cm)	`(3sqrt2)/2`	3	`(3sqrt2)/2`	0	`(-3sqrt2)/2`	-3	`(-3sqrt2)/2`	0	`3/sqrt2`

Using the values of t and y (as in the table), a graph is plotted as under The graph obtained is sinusoidal.
Similar graphs are obtained for y x = 2 cm and x = 4 cm. The (incm) oscillatory motion in the travelling wave only differs in respect of phase. Amplitude and frequency of oscillatory motion remains the same in all the cases.