A harmonic oscillator vibrates with amplitude of $4 \mathrm{cm}$ and performs $150$ oscillations in one minute. If the initial phase is $45°$ and it starts moving away from the origin, then the equation of motion is:

The displacement of a wave disturbance propagating in the positive $x$ direction is given by, $y=\frac{1}{1+x^2}$ at, $t=0$ and $y=\frac{1}{1+(x-1{)}^2}$ at, $t=2 \mathrm{s}$, where, $x$ and $y$ are in meter. The shape of the wave disturbance does not change during the propagation. What is the velocity of the wave?

A traveling wave pulse is given by, $y=\frac{10}{5+(x+2t{)}^2}$.

Here, $x$ and $y$ are in meter and $t$ is in second. In which direction and with what velocity is the pulse propagating? What is the amplitude of the pulse?

If at $t=0$, a travelling wave pulse on a string is described by the function, $y=\frac{10}{\left(x^2+2\right)}$ where $x$ and $y$ are in meters and $t$ in seconds, what will be the wave function representing the pulse at time $t$ if the pulse is propagating along the positive $x$-axis with speed $2 \mathrm{m} {\mathrm{s}}^{-1}$?

Consider a sinusoidal travelling wave shown in the figure. The wave velocity is $+40 \mathrm{cm} {\mathrm{s}}^{-1}$.

Find,

(a) the frequency
(b) the phase difference between points which are $2.5 \mathrm{cm}$ apart.
(c) how long it takes for the phase at a given position to change by $60°$.
(d) the velocity of a particle at point $P$ at the instant shown.

The equation of a travelling wave is, $y(x,t)=0.02\sin \left(\frac{x}{0.05}+\frac{t}{0.01}\right)\mathrm{m}$.
Find,
(a) the wave velocity and
(b) the particle velocity at $x=0.2 \mathrm{m}$ and $t=0.3 \mathrm{s}$.
Given, $\cos \left(\mathrm{\theta }\right)=-0.85$ where, $\mathrm{\theta }=34 \mathrm{rad}$.

Transverse waves on a string have wave speed $12.0 \mathrm{m} {\mathrm{s}}^{-1}$, amplitude $0.05 \mathrm{m}$ and wavelength $0.4 \mathrm{m}$. The waves travel in the $X-$direction and at $t=0$, the $x=0$ end of the string has zero displacement and is moving upwards.

(a) Write a wave function describing the wave.
(b) Find the transverse displacement of a point at $x=0.25 \mathrm{m}$ at time, $t=0.15 \mathrm{s}$.
(c) How much time must elapse from the instant in part (b) until the point at $x=0.25$ $\mathrm{m}$ has zero displacement?