When a wave travels in a medium, the particle displacement is given by the equation Y = a sin ⁡ 2 π b t - c x where, a , b and c are constant. The maximum particle velocity will be twice the wave velocity, if

Step-by-step solution

Given, Y = a sin ⁡ 2 π b t - c x ...(i) and standard form Y = A sin ⁡ ω t - k x Thus, we have ω = 2 π b and K = 2 π c or 2 π λ = 2 π c ⇒ c = 1 λ ∴ Speed of wave, v = ω K = 2 π b 2 π c = b c Differentiating Equation (i) w.r.t.t, we get d Y d t = Speed of particle, v p = a cos ⁡ 2 π b t - c x ⋅ 2 π b v p = 2 π a b cos ⁡ 2 π b t - c x For v p m a x = 2 π a b According to the question's condition, v p m a x = 2 v ⇒ 2 π a b = 2 ⋅ b c (Putting values) ⇒ π a = 1 c or a c = 1 π