Question

Two waves, each having a frequency of $100Hz$ and a wavelength of $2.0cm,$ are travelling in the same direction on a string. What is the phase difference between the waves $\left(a\right)$ if the second wave was produced $0.015s$ later than the first one at the same place, $\left(b\right)$ if the two waves were produced at a same instant but the first one was produced a distance $4.0cm,$ behind the second one ? $\left(c\right)$ If each of the waves has an amplitude of $2.0mm,$ what would be the amplitudes of the resultant waves in part $\left(a\right)$ and $\left(b\right)$ ?

Answer 1

$f=100Hz,\lambda =2cm=2\times {10}^{-2}m$

$\therefore$ wave speed $v=f\lambda =2m/s$

(a) in $0.015\sec 1^{st}$ wave has traveled

$x=0.015\times 2=0.03m=$ path different

$\therefore$ Corresponding phase difference, $\varphi =2\pi x/\lambda =\left\{2\pi /\right(2\times {10}^{-2}\left)\right\}\times 0.03=3\pi$

(b) Path different $x=4cm=0.04m$

$\Rightarrow \varphi =\left(2\pi /\lambda \right)x=\left\{\right(2\pi /2\times {10}^{-2})\times 0.04\}=4\pi$.

(c) The waves have same frequency, same wavelength and same amplitude.

Let, $y_1=r\sin wt,y_2=r\sin (wt+\varphi )$

$\Rightarrow y=y_1+y_2=r[\sin wt+(wt+\varphi \left)\right]$

$=2r\sin (wt+\varphi /2)\cos \left(\varphi /2\right)$

$\therefore$ resultant amplitude $=2r\cos \varphi /2$

So, when $\varphi =3\pi ,r=2\times {10}^{-3}m$

$R_{res}=2\times (2\times {10}^{-3})\cos \left(3\pi /2\right)=0$

Again, when $\varphi =4\pi ,R_{res}=2\times (2\times {10}^{-3})\cos \left(4\pi /2\right)=4mm$