A) \[c=\frac{1}{\pi a}\]
B) \[c=\pi a\]
C) \[b=ac\]
D) \[b=\frac{1}{ac}\]
Correct Answer: A
Solution :
[a] Equation of the harmonic progressive wave given \[y=a\text{ }sin\,2\pi \left( bt-cx \right)\] Here \[2\pi v=\omega =2\pi b\Rightarrow v=b\] \[K=\frac{2\pi }{\lambda }=2\pi c\Rightarrow \frac{1}{\lambda }=c\] (here c is the symbol given for \[\frac{1}{\lambda }\] and not the velocity) \[\therefore \] Velocity of the wave \[=v\lambda =b\frac{1}{c}=\frac{b}{c}\] \[\frac{dy}{dt}=a2\pi b\text{ }cos2\pi \left( bt-cx \right)=a\omega cos(\omega t-kx)\] Maximum particle velocity \[=a\omega =a2\pi b=2\pi ab\] given this is \[2\times \frac{b}{c}\] i.e., \[2\pi a=\frac{2}{c}or\text{ c}=\frac{1}{\pi a}\]
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