Page 512 - Practical Design Ships and Floating Structures

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ro (rn), R(m) : auxiliary variables in trochoidal wave equation”
yo (m) : depth of a flow line below water surface (hull bottom) at the stem end
To maintain the wave system, amount of kinetic energy which is equal to integration of dE,, over
water depth from water bottom (yo = 4) to water surface (yo = 0) has to be supplied to the wave
system per a wave length. This meets with wave resistance R, . This relation can be described as
follows:
n
RkVO = @,” lT)dyo (2)
-h
where
R, (kgf) : resistance due to following waves behind a transom stern
vo(m/s) : shipspeed
T(s) =2 7~ / w : period of the following wave, where w = (g / R)”2
h(m) : water depth

If we suppose that the water flow layer at the water depth from a level (yo = -yc) to water surface
(yo = 0) at the stern end breaks, then we can estimate resistance due to not broken and remained
following waves which come from the water flow layer at the water depth from water bottom
(yo = -h) to the level (yo = -yc) at the stem end using Eqn. (2) as follows:
-Y.
R, = (1 1 van) / W Y O
-h
- Yr
= (p B,,g’” /2v0n) I(rn2 IR”Z)(e2Yn’R -(ro /R)2e4yn’Rb&0 (3)
-h
where
R,(k&) : resistance due to the remained following waves
B, (m) : mean breadth of immersed transom stem end plane
n(-) : coefficient representing three dimensional effect, where is used n = 3 which is the value used in
the equation for forward-oriented wave breaking resistance R,,, in our previous papers’*2.
y, (m) : thickness of water flow layer which breaks
Integration in Eqn. (3) is convenient to be conducted separated into two parts as follows, since R and
ro have constant values R, and ros respectively over the water depth from yo = -h to yo = - 6 :

-Y, -6 -Y,
I= I+ I (4)
-h -h -6
where
rod (m),R,(m) : values of ro and R at yo = -6
6(m) : boundary layer thickness at the stem end
h (4 : water depth (supposed to be infinite)
Finally, we can obtain the following equation:
-Ye
R, = (PB,~~‘~ /2v0n)[ krOz lR”2){e2yo’R -(yo /R)2e4ya/Rk&~
-6
+(rOb2R6”* /4){2e-ZS’R6 -(res /Rd)2e46’Rs )I (5)