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JEE Main & Advanced Physics Wave Mechanics Question Bank Mock Test - Waves and Acoustics

  • question_answer
    The equations of a travelling and stationary waves are \[{{y}_{1}}=a\sin (\omega t-kx)\] and\[{{y}_{2}}=a\sin \,kx\,cos\,\omega t\]. The phase differences between two points \[{{x}_{1}}=\frac{\pi }{4k\,}\,and\,{{x}_{2}}=\frac{4\pi }{3k}\] are \[{{\phi }_{1}}\] and \[{{\phi }_{2}}\] respectively for two waves, where k is the wave number. The ratio of \[{{\phi }_{1}}/{{\phi }_{2}}\] is

    A) 6/7                   

    B) 16/3

    C) 12/13   

    D) 13/12

    Correct Answer: C

    Solution :

    [c] \[\Delta x={{x}_{2}}-{{x}_{1}}=\left( \frac{4}{3}-\frac{1}{4} \right)\frac{\pi }{k}=\frac{13}{12}\frac{\pi }{k}\] \[\sin k{{x}_{1}}=\sin k\left( \frac{\pi }{4k} \right)=\sin \frac{\pi }{4}\ne 0\] \[\sin k{{x}_{2}}=\sin k\left( \frac{4\pi }{3k} \right)=\sin \left( \pi +\frac{\pi }{3} \right)\ne 0\] \[{{x}_{1}}\] and \[{{x}_{2}}\]are not the nodes \[\frac{2\pi }{k}>\Delta x>\frac{\pi }{k}\Rightarrow \lambda >\Delta x>\frac{\lambda }{2}\] For \[{{\phi }_{1}}=\pi ,{{\phi }_{2}}=k(\Delta x)=k\left( \frac{13\pi }{12k} \right)=\frac{13\pi }{12}\] \[\frac{{{\phi }_{1}}}{{{\phi }_{2}}}=\frac{\pi }{(13\pi /12)}=\frac{12}{13}\]


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