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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? - Physics

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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?

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Solution 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.

Solution 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.

Concept: The Speed of a Travelling Wave
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NCERT Class 11 Physics
Chapter 15 Waves
Q 9 | Page 387
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