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?

#### 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.