91Ó°ÊÓ

1. Displacement (y) vs t graph at x=0
2. The wave equation is,$y=y_{m}sin(kx-\mathrm{Ó\neg }t+\mathrm{Ï•})$

We can find the equation of the wave at x = 0 given in the graph. From the slope of the graph, we can predict the approximate value of the phase constant. Then, from the equation of the wave at t = 0, we can easily get the value of the phase constant.

Formula:

The transverse velocity of the wave (or slope),

$v=\frac{dy}{dt}\left(i\right)$

The equation of the wave is given as

$y=y_{m}sin(kx-\mathrm{Ó\neg }t+\mathrm{Ï•})$

At x = 0, it becomes,

$y=y_{m}sin(-\mathrm{Ó\neg }t+\mathrm{Ï•})$

This is the equation for the given graph.

The slope of the graph using equation (i) gives the velocity, which is given as:

$$\begin{gathered} \frac{dy}{dt}=\frac{d}{dt}\left(y_m\sin \left(-\mathrm{Ó\neg }t-\mathrm{Ï•}\right)\right) \\ =-\mathrm{Ó\neg }\left(y_m\cos \left(-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)\right) \end{gathered}$$

From the given graph, we can conclude that the slope of the graph at t = 0 is positive.

Hence,

$$\begin{gathered} -\mathrm{Ó\neg }\left(y_m\cos \left(-\mathrm{Ó\neg }\left(0\right)+\mathrm{Ï•}\right)\right)>0 \\ -\mathrm{Ó\neg }\left(y_m\cos \left(\mathrm{Ï•}\right)\right)>0 \\ -\cos \left(\mathrm{Ï•}\right)>0 \end{gathered}$$

This implies that$\mathrm{ϕ}$is in between$\frac{\mathrm{π}}{2}\mathrm{and}\mathrm{π} \mathrm{or} \mathrm{π} \mathrm{and}\frac{3\mathrm{π}}{2}$.

The equation of the wave at t = 0 is,

$$\begin{gathered} y=y_m\sin \left(-\mathrm{Ó\neg }\left(0\right)+\mathrm{Ï•}\right) \\ y=y_m\sin \left(\mathrm{Ï•}\right) \\ \mathrm{Ï•}={\sin }^{-1}\frac{y}{y_m} \end{gathered}$$

From the given graph, we can figure out that ${\mathrm{y}}_{\mathrm{m}}=6 \mathrm{mm}\mathrm{and} \mathrm{y}(\mathrm{t}=0)=2 \mathrm{mm}$. Hence,

role="math" localid="1660981403661" $$\begin{gathered} \mathrm{Ï•}={\sin }^{-1}\frac{2}{6} \\ =2.8rad \end{gathered}$$

Since$\mathrm{²õ¾\pm ²ÔÏ•}=\sin (\mathrm{Ï€}-\mathrm{Ï•}),\mathrm{and} \mathrm{Ï•}=2.8 \mathrm{rad}$

Also,

$$\begin{gathered} \mathrm{Ï•}=2.8-2\mathrm{Ï€} \\ =-3.48 \\ ~-3.5\mathrm{rad} \end{gathered}$$

Therefore, the value of $\mathrm{ϕ} \mathrm{is}2.8\mathrm{rad}$or -3.5 rad.