Superposition of Waves

Problems from IIT JEE

Problem (IIT JEE 2015): Four harmonic waves of equal frequencies and equal intensities $I_0$ have phase angles $0$, $\pi/3$, $2\pi/3$ and $\pi$. When they are superposed, the intensity of the resulting wave is $nI_0$. The value of $n$ is,

Solution: The intensity of a wave is proportional to the square of its amplitude i.e., $I_0=cA^2$, where $c$ is a constant. The amplitudes of four harmonic waves are equal as their intensities are equal. Let these waves be travelling along the $x$ direction with wave vector $k$ and angular frequency $\omega$. The resultant displacement of these waves is given by, \begin{align} y=& y_1+y_2+y_3+y_4\nonumber\\ =& A\sin(\omega t-kx+0)+A\sin(\omega t-kx+\pi/3)\nonumber\\ & +A\sin(\omega t-kx+2\pi/3)+A\sin(\omega t-kx+\pi) \nonumber\\ =& A\sin(\omega t-kx)-A\sin(\omega t-kx)\nonumber\\ & +A\sin(\omega t-kx+\pi/3)+A\sin(\omega t-kx+2\pi/3) \nonumber\\ =&2A\sin(\omega t-kx+\pi/2)\cos(\pi/6)=\sqrt{3}A \cos(\omega t-kx). \end{align} The amplitude of the resultant wave is $A_r=\sqrt{3}A$ and its intensity is $I_r=cA_r^2=3cA^2=3I_0$.

Note that $y_1$ and $y_4$ are out of phase and interfere destructively. The displacement $y_2$ and $y_3$ have a phase difference of $\delta=\pi/3$. Thus, we can arrive at the resultant intensity by using the formula, \begin{align} I_r=I_0+I_0+2\sqrt{I_0}\sqrt{I_0}\cos\delta=2I_0+2I_0\cos (\pi/3)=3I_0. \end{align}

Problem (IIT JEE 1992): The displacement $y$ of a particle executing periodic motion is given by $y=4\cos^2\left(t/2\right)\sin(1000t)$. This expression may be considered to be a result of the superposition of \lldots\ independent harmonic motions.

  1. two
  2. three
  3. four
  4. five

Solution: Given displacement can be written as, \begin{align} y&=4\cos^2(t/2)\sin(1000t)=2(1+\cos t)\sin(1000t)\nonumber\\ &=2\sin(1000t)+2\sin(1000t)\cos(t)\nonumber\\ &=2\sin(1000t)+\sin(999t)+\sin(1001t).\nonumber \end{align} Thus, $y$ is a superposition of three independent harmonic motions of angular frequencies ${999}\;\mathrm{rad/s}$, ${1000}\;\mathrm{rad/s}$,, and ${1001}\;\mathrm{rad/s}$,.