A transverse wave is represented by : y = 10πsin(2πTt - 2πλx). For what value of the wavelength the wave velocity is twice the maximum particle velocity?

Answer is A

We have, y = Asin(2πx/λ - 2πt/T)
Differentiating above, we get particle velocity as
Vp = Acos(2πx/λ - 2πt/T) * (2π/T)
(i)
For velocity to be maximum, cos(2πx/λ - 2πt/T) = 1
Therefore, from (i) we have
Vp(max) = 10π * (2π/T)
Vp(max) = 20πf
Or, Vp(max) = 20πfλ/λ
Vp(max) = 20π/λ * (λf)
Since, λf = Vw (wave velocity)
Vp(max) = 20Vw/λ
Since, 2Vp(max) = Vw
2 * (20Vw/λ) = Vw
40Vw/λ = Vw
λ = 40cm