Displacement Relation in a Progressive Wave

Write an expression of progressive harmonic wave and explain various parameters used in the expression.

What are the parameters used to describe a progressive harmonic wave?

A travelling harmonic wave on a string is described by $y(x,t)=7.5\hspace{0.17em}sin\hspace{0.17em}\left(0.0050x+12t+\frac{\pi }{4}\right)$. Locate the points of the string which have the same transverse displacements and velocity as the $x=1cm$ point at $t=2s,5s$ and $11s$.

Explain why (or how) the shape of a pulse gets distorted during propagation in a dispersive medium.

A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $45Hz$ The mass of the wire is $3.5\times {10}^{-2}k\mathrm{g }$and its linear mass density is $4.0\times {10}^{-2}kg{m}^{-1}$. What is the tension in the string?

For the travelling harmonic wave $y\left(x,t\right)=2.0 cos2\mathrm{\pi }(10t-0.0080x+0.35)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\mathrm{s}$. Calculate the phase difference between oscillatory motion of two points separated by a distance of $\frac{3\lambda }{4}$.

For the travelling harmonic wave $y\left(x,t\right)=2.0 cos2\mathrm{\pi }(10t-0.0080x+0.35)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\mathrm{s}$. Calculate the phase difference between oscillatory motion of two points separated by a distance of $\frac{\lambda }{2}$.

For the travelling harmonic wave $y\left(x,t\right)=2.0 cos2\mathrm{\pi }(10t-0.0080x+0.35)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\mathrm{s}$. Calculate the phase difference between oscillatory motion of two points separated by a distance of $0.5 \mathrm{m}$.

A transverse harmonic wave on a string is described by
$y\left(x,t\right)=3.0sin(36t+0.018x+\frac{\pi }{4})$
where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\mathrm{s}$. The positive direction of $x$ is from left to right.

What is the least distance between two successive crests in the wave?

A travelling harmonic wave on a string is described by $y(x,t)=7.5\sin \left(0.0050x+12t+\frac{\pi }{4}\right)$. What are the displacement and velocity of oscillation of a point at $x=1 \mathrm{cm}$, and $t=1 \mathrm{s}$? Is this velocity equal to the velocity of wave propagation? ($x$ is in $\mathrm{cm}$, $t$ is in $\mathrm{s}$, $\sin (12.79 \mathrm{rad})=0.2217$ and $\cos (12.79 \mathrm{rad})=0.975$) 

Given below is the function of $x$ and $t$ to represent the displacement (transverse or longitudinal) of an elastic wave. State if it represents (i) a travelling wave, (ii) a stationary wave or (iii) none at all: $y=2cos\left(3x\right)sin\left(10t\right)$.
 

For the travelling harmonic wave $y\left(x,t\right)=2.0 cos2\mathrm{\pi }(10t-0.0080x+0.35)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\mathrm{s}$. Calculate the phase difference between oscillatory motion of two points separated by a distance of $4\mathrm{m}$.

A transverse harmonic wave on a string is described by
$y\left(x,t\right)=3.0\sin (36t+0.018x+\frac{\mathrm{\pi }}{4})$
where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\mathrm{s}$. The positive direction of $x$ is from left to right.

Plot the displacement $\left(y\right)$ versus $\left(t\right)$ graphs for $x=0,2$ and $4\mathrm{cm}$. 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?