A) \[\frac{\pi }{3}\]
B) \[\frac{2\pi }{3}\]
C) \[\frac{\pi }{6}\]
D) \[\frac{\pi }{4}\]
Correct Answer: B
Solution :
The given progressive waves are |
\[{{y}_{1}}=a\sin \,(\omega t+{{\phi }_{1}})\] |
\[{{y}_{2}}=a\sin \,(\omega t+{{\phi }_{2}})\] |
The resultant of two waves is |
\[y={{y}_{1}}+{{y}_{2}}\] |
\[=a\,[\sin \,(\omega t+{{\phi }_{1}})+\sin \,(\omega t+{{\phi }_{2}})]\] |
If A is the amplitude of resultant wave, then |
\[A=a\] (given) |
\[\therefore \] \[{{A}^{2}}={{a}^{2}}+{{a}^{2}}+2{{a}^{2}}\cos \phi \] |
or \[{{a}^{2}}={{a}^{2}}+{{a}^{2}}+2{{a}^{2}}\cos \phi \] |
or \[\cos \phi =-\frac{1}{2}=\cos {{120}^{o}}\] |
\[\therefore \] \[\phi ={{120}^{o}}=\frac{2\pi }{3}\] |
Thus, \[{{\phi }_{1}}-{{\phi }_{2}}=\frac{2\pi }{3}\] |
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