Question

Two waves having the same frequency, wavelength, and amplitude are traveling in the same direction. If they differ in phase by π/2 and each has an amplitude of 0.060 m, what is the amplitude of the resultant wave?

cm

Answer 1

Let the waves be
   $x_1=a\sin\theta ,~~x_2=a\sin(\theta +\phi)$
where, $\phi$ is the phase difference.
So, the resultant wave is
   $x=x_1+x_2=a\sin\theta+a\sin(\theta +\phi)$
$\Rightarrow x=a\big[\sin\theta+\sin(\theta +\phi)\big]$
  $\Rightarrow x=2a\cos(\phi/2)\sin(\theta +\phi/2)$
So, the amplitude of the resultant wave is given by
   $a'=2a\cos(\phi/2)$
So, given the values,
   $a=0.060~m=6.0~cm,~~\phi=\pi/2$
we get the resultant wave amplitude as
   $a'=2\times 6.0\times \cos(\pi/4)~cm$
  $\Rightarrow a'=8.485~cm$