Question

Five sinusoidal waves have the same frequency 500hz but their amplitudes are in the ratio $2:\dfrac{1}{2}:\dfrac{1}{2}:1:1$ and their phase angles $0,\dfrac{\pi }{6},\dfrac{\pi }{3},\dfrac{\pi }{2},\pi $ respectively. The phase angle of resultant wave obtained by the superposition of these five waves is:
$\begin{align}
  & a){{30}^{0}} \\
 & b){{45}^{0}} \\
 & c){{60}^{0}} \\
 & d){{90}^{0}} \\
\end{align}$

Answer 1

Hint: Find the net amplitude of the given five waves as the five waves are superimposed. Phase angle also plays an important role in finding the effective amplitude. Next, the amplitude is equal to summation of five net amplitudes. Thus, find the phase angle of the resultant amplitude.
Formulas used:
$A=\sum\limits_{i=1}^{5}{{{x}_{i}}}$

Complete answer:
Let us first find the phasors of the given five waves,
$\begin{align}
  & {{x}_{1}}=2\hat{i} \\
 & {{x}_{2}}=\dfrac{\sqrt{3}}{4}i+\dfrac{1}{4}j \\
 & {{x}_{3}}=\dfrac{1}{4}i+\dfrac{\sqrt{3}}{4}j \\
 & {{x}_{4}}=j \\
 & {{x}_{5}}=-i \\
\end{align}$
When waves superimpose, the amplitude of the resultant wave will be equal to the sum of the waves superimposed.
Therefore,
$\begin{align}
  & {{x}_{i}}=\sum\limits_{i=1}^{5}{{{x}_{i}}} \\
 & \Rightarrow x=\dfrac{5+\sqrt{3}}{4}i+\dfrac{5+\sqrt{3}}{4}j \\
\end{align}$
The phase angle of the resultant wave will be,
$\begin{align}
  & \theta ={{\tan }^{-1}}(\dfrac{\dfrac{5+\sqrt{3}}{4}}{\dfrac{5+\sqrt{3}}{4}}) \\
 & \Rightarrow \theta ={{45}^{0}} \\
\end{align}$

Therefore, the correct option is option b.

Additional information:
When a medium is disturbed simultaneously by a large number of waves, the resultant displacement at any point in the medium is equal to the sum of the displacement of individual waves. This is known as the principle of superposition. If 2 waves of the same frequency are moving in the same direction superimpose, then the phenomena are called interference. If the superimposing waves are coherent waves, we can observe the phenomenon of interference. The application of superposition principle are stationary waves, beats interference. Constructive interference is observed when 2 waves of the same phase are superimposed. The resultant amplitude will be equal to the sum of the amplitudes of the individual waves. The intensity is also maximum.

Note:
The basic assumption in superposition principle is that the medium must be non-dispersive since the gaussian wave pulses do not change their shape as they propagate. If the medium was dispersive, the waves would change their shape.