The equation of a simple harmonic progressive | vedclass

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Question

(B) The general equation of a simple harmonic progressive wave is $y=A \sin 2 \pi \left( \frac{t}{T} - \frac{x}{\lambda} \right)$.Given equation: $y=a

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$\frac{1}{4 \pi a}$

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(B) The general equation of a simple harmonic progressive wave is $y=A \sin 2 \pi \left( \frac{t}{T} - \frac{x}{\lambda} \right)$.Given equation: $y=a \sin 2 \pi(b t-c x)$.Comparing the two,we get amplitude $A=a$,frequency $n = \frac{1}{T} = b$,and wave number $\frac{1}{\lambda} = c$.The maximum particle velocity is given by $(v_p)_{\max} = A \omega = A(2 \pi n) = a(2 \pi b) = 2 \pi ab$.The wave velocity is given by $v = n \lambda = \frac{n}{1/\lambda} = \frac{b}{c}$.According to the problem,$(v_p)_{\max} = \frac{1}{2} v$.Substituting the values: $2 \pi ab = \frac{1}{2} \left( \frac{b}{c} \right)$.Canceling $b$ from both sides: $2 \pi a = \frac{1}{2c}$.Solving for $c$: $c = \frac{1}{4 \pi a}$.

The equation of a simple harmonic progressive wave is given by $y=a \sin 2 \pi(b t-c x)$. The maximum particle velocity will be half the wave velocity,if $c=$

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