Page 91 - Theory and Design of Air Cushion Craft

The 'wave pumping' concept 75

to regulate the lift power and lift inflow rate to keep the cushion pressure constant.
This condition is also the one which will absorb the greatest volume of air; therefore
we will make an analysis of this case.
When the craft moves along the jc-axis for a distance of dx, then the change of
water volume in the cushion can be expressed by the change of water volume at the
bow/stern of the craft as shown in Fig. 2.23, then

dV= B C[(HI2 + h f)dx - (H/2 + h t)dx] (2.36)
where H is the wave height, h f the bow heave amplitude relative to the centre line of
the waves, h r the stern heave amplitude relative to the centre line of the waves and B c
the cushion beam. Thus

because dx/dt = 0,
dV/dt = B c(h f- /z r)v
where v is the craft velocity relative to the waves.
The wave profile can be expressed by
h = (H/2) sin a
where a =2nx/A, thus

otf = o. r + 2nl clA
/
where h is the wave amplitude, c the cushion length and X the wave length. Therefore
dV B CH
-7- — . (sin a f - sin a r) v

B CH [ . / 27r/ c \ . 1
= —r— sin a r H—— — sin a r v
^ V \ A- / J

£ c//v [/ 2;r/ c t \ . . 2nl c ] .. .„
= —^— cos —^ — 1 sm a r — sin —r— cos a r (2.37)
2 L\ /i / / J
In order to determine the maximum instantaneous wave pumping rate, we take the
first derivative of function d VIdt with respect to a equal to zero, then
17 2nl ,\ . 2nL . 1 rt
cos —: 1 cos a r — sm —— sm a r = 0
da dr 2 l\ A / A J
This expression can be written as
tan a r = (cos(27r/ c/A) — 1 )/sm(2nl c/A) (2.38)

Substituting expression (2.38) into (2.37), the maximum instantaneous wave pumping
rate can be written as
(dV\ B cHv [7 2nl c 1 \ . \/. 22nl c\/( 2nl c \ 1 . 1
s
—T— =—^— cos-^ - 1 ma r - sin —^ / cos -^ - 1 sin a r
V dt / max 2 L\ A / L\ A // V A /J J