The string center describes a helix around the cylinder with parametric equation given by
p(t) = ((r+h/2)cost,(r+h/2)sint, alpha t)
the helix pitch is calculated with the condition
p(t+2pi)-p(t)=(0,0,h)
so we have alpha (2pi) = h -> alpha = h/(2pi)
The string is wounded around the cylinder from t=0 to t = t_f and we know that
t_f h/(2pi) = n -> t_f = (2pi n)/h
the string length is given by
l = int_(t=0)^(t=t_f) (ds)/(dt) dt
Here d/(dt) p(t) = (-(h/2 + r) sint, (h/2 + r) cost, h/(2pi)) = ((dx)/(dt),(dy)/(dt), (dz)/(dt))
and (ds)/(dt) = sqrt(((dx)/(dt))^2+((dy)/(dt))^2+((dz)/(dt))^2)
giving
(ds)/(dt) = sqrt((h^2 (1 + pi^2))/(4 pi^2) + h r + r^2)
so finally
l = (npi)/h sqrt((h^2 (1 + pi^2))/(4 pi^2) + h r + r^2)
substituting r=3/pi the result is
l=(n sqrt[36 + h (h + 12 pi+ h pi^2)])/h
