91Ó°ÊÓ

The given $FIGUREEX15.5$carries the information of amplitude, frequency and phase constant.

The amplitude is the length of the peak from the time axis. From figure , it is clear that the amplitude is $a=20cm$

The frequency is the number of waves per unit time. From $FIGUREEX15.5$, in $8s$the number of waves completed is localid="1650197099580" $1.$

Therefore, frequency is,

localid="1650197155998" $f=\frac{numberofwaves}{time}=\frac{1}{8s}=0.125Hz$

Suppose the equation of the wave is,

$x=a\sin \left(\mathrm{Ó\neg }t+\mathrm{θ}\right)$

Here, $\mathrm{Ó\neg }$is the angular frequency of the wave, and $\mathrm{θ}$is initial phase constant.

The time period of the wave is,

localid="1650197198931" $T=8s$

Therefore, the angular frequency is,

localid="1650197228672" $\mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{T}=\frac{2\mathrm{Ï€}}{8}=\frac{\mathrm{Ï€}}{4}rad/s$

For initial phaselocalid="1650197283691" $t=0s$and$x=10cm$

The equation takes the form,

localid="1650285529821" $$\begin{gathered} 10cm=(20cm)\sin \left(\frac{\mathrm{π}}{4}\left(0s\right)+\mathrm{θ}\right) \\ \sin \left(\mathrm{θ}\right)=\frac{1}{2} \\ \sin \mathrm{θ}=\frac{1}{2}=\sin {150}^0 \\ \mathrm{θ}={150}^0 \end{gathered}$$

Therefore, the phase constant or the initial phase islocalid="1650285650108" ${150}^0$