A) \[\frac{a}{\sqrt{2}},\frac{f}{2}\]
B) \[\frac{a}{\sqrt{2}},f\]
C) \[2a,\frac{f}{2}\]
D) \[\sqrt{2}a,f\]
Correct Answer: D
Solution :
The two given waves are \[{{y}_{1}}=a\sin 2\pi ft\] \[{{y}_{2}}=a\sin \left( 2\pi ft+\frac{\pi }{2} \right)\] \[\therefore \] Resultant displacement \[y={{y}_{1}}+{{y}_{2}}\] \[=a\sin 2\pi ft+a\,\sin \left( 2\pi \,ft\,+\frac{\pi }{2} \right)\] or \[y=2a\,\sin \left( 2\pi ft+\frac{\pi }{4} \right)\cos \frac{\pi }{4}\] \[=\frac{2a}{\sqrt{2}}\sin \left( 2\pi ft+\frac{\pi }{4} \right)\] \[=\sqrt{2}a\sin \left( 2\pi ft+\frac{\pi }{4} \right)\] Thus, the resultant wave has amplitude \[\sqrt{2}a\] and frequency\[f\].
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