Sound and Wave - Wave Phenomena

Consider a particle moving without friction on a rippled surface, as shown above. Gravity acts down in the negative h direction. The elevation h(x) of the surface is given by h(x) = d cos (kx). If the particle starts at x = 0 with a speed v in the x direction, for what values of v will the particle stay on the surface at all times?

A.
B.
C.
D.
E. 

(GR9677 #83)

Solution:

Gravity pulls the particle down. 
Particle will stay on the surface if  a_max = g

For a wave, a_max = ω²A = ω²d
(A = d = amplitude)

Velocity of the particle: v =  λ / T 

From the picture: λ = 2π/k

v = 2π/kT  = ω/k
v² = ω²/k²
ω² = v²k²

a_max = v²k²d = g 
v² = g/k²d
v = √(g/k²d)

Answer: D