For the wave, $y(x, t)=3 \sin (36 t+0.018 x+\pi / 4)$ plot the displacement ( $y$ ) versus (t) graphs for $x=$ 0,2 and $4 \mathrm{~cm}$.
(i) Give the shapes of these plots.
(ii) With respect to which aspects (amplitude, frequency or phase) does the oscillatory motion in a travelling wave differ from one point to another?

Given wave is,

$y(x, t)=3 \sin (36 t+0.018 x+\pi / 4)$…..(1)

Putting $x=0$, the equation becomes:

$y(0, t)=3 \sin (36 t+0+\pi / 4)$…..(2)

Also,

$\omega=2 \pi / \mathrm{t}=36 \mathrm{rad} / \mathrm{s}$

$\Rightarrow t=\pi / 18$ secs

The displacement ( $y$ ) vs. (t) graphs using different values of t is,

$\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline \mathrm{t} & 0 & \mathrm{~T} / 8 & 2 \mathrm{~T} / 8 & 3 \mathrm{~T} / 8 & 4 \mathrm{~T} / 8 & 5 \mathrm{~T} / 8 & 6 \mathrm{~T} / 8 & 7 \mathrm{~T} / 8 & \mathrm{~T} \\
\hline \mathrm{y} & \frac{3}{\sqrt{2}} & 3 & \frac{3}{\sqrt{2}} & 0 & \frac{-3}{\sqrt{2}} & -3 & \frac{-3}{\sqrt{2}} & 0 & \frac{3}{\sqrt{2}} \\
\hline
\end{array}$

Similarly, graphs are obtained for x=0, x=2, and x=4. The oscillatory motion in the travelling wave is different from each other only in terms of phase. Amplitude and frequency are invariant for any change in x. The y-t plots of the three waves are shown in the given figure: