The wave equation can also be written as y = A.sin2π(t/T-x/λ)
where y is the transverse displacement of the particle at a distance x at time t and A is the amplitude of the particle. The speed of the particle can be found out by partial differentiation of y with respect to t.
∂y/∂t = A.cos 2π(t/T-x/λ)*2π/T
→V = {2πA.cos 2π(t/T-x/λ)}/T
For V to be maximum cos2π(t/T-x/λ) = 1
So Vₘₐₓ = 2πA/T
if A =λ/2π
Vₘₐₓ = λ/T = v, where v is the wave speed.
But if A < λ/2π, then Vₘₐₓ < λ/T
→Vₘₐₓ < v.
So, if the amplitude is less than the wavelength divided by 2π, the particle speed can never be more than the wave speed.
