91Ó°ÊÓ

Angular frequency is$\mathrm{Ó\neg }=440rad/s$

Displacement$y=+4.5mm$

Transverse velocity is$u=-0.75m/s$

Wave equation is $y\left(x,t\right)=y_m\sin \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)$

We use the concept of wave motion. We can differentiate the given equation for velocity and take the ratio of displacement by velocity. At initial values of x and t, we get the equation for, plugging the given values, we get the value for phase constant.

Formulae:

The velocity of the body in motion, $v=\frac{dv}{dt}........\left(1\right)$

We know that velocity of the wave using equation (1) is given as:
$$\begin{gathered} u\left(x,t\right)=\frac{dy}{dt} \\ =\frac{d{y}_m\sin \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)}{dt} \\ =-y_m\mathrm{Ó\neg }\cos \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right) \end{gathered}$$

We know the equation of wave is given as:

$y\left(x,t\right)=y_m\sin \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)$

We can take the ratio ofand

$$\begin{gathered} \frac{y\left(x,t\right)}{u\left(x,t\right)}=\frac{\left(y_m\sin \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)\right)}{-y_m\mathrm{Ó\neg }\cos \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)} \\ \frac{y\left(x,t\right)}{u\left(x,t\right)}=\frac{\left(\sin \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)\right)}{-\mathrm{Ó\neg }\cos \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)} \\ =\frac{\tan \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)}{\mathrm{Ó\neg }} \end{gathered}$$

Plugging the given values in the above equation, we get

$$\begin{gathered} \frac{4.5}{-0.75}=\frac{\tan \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)}{\mathrm{Ó\neg }} \\ \mathrm{an}\left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)=\frac{4.5×{10}^{-3}×440}{0.75} \\ \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)={\tan }^{-1}\left(2.640\right) \\ \left(kx-\mathrm{Ó\neg }t+\mathrm{Ï•}\right)=69.25° \end{gathered}$$

For x = 0 then t = 0 , we get the value of phase constant as:

$$\begin{gathered} \mathrm{ϕ}=69.25° \\ =69.25°×\frac{\mathrm{π}\mathrm{rad}}{180°} \\ =1.2rad \end{gathered}$$

Hence, the value of phase constant is $1.2rad$.