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dx !" coshk(z + D)
u = = = #$ cos(kx % #t)
dt !x o sinhkD
(10.4.39 a, b)

dz !" sinhk(z + D)
w = = = #$ sin(kx % #t)
dt !z o sinhkD

These are very non linear equations. However, for small displacements of each fluid

element from its rest position , i.e. for small η we can write,
0

x = x + !, z = z +" (10.4.40)
o
o
where the displacements, ξ and ζ are order η i.e. of order the amplitude of the motion.
0
Thus keeping only linear terms in (10.4.39 a, b) yields the much simpler set,
d! coshk(z + D)
= "# o cos(kx $ "t)
dt o sinhkD o
(10.4.41)
d% sinhk(z + D)
= "# o sin(kx $ "t)
dt o sinhkD o

with solutions for the displacements,

coshk(z + D)
! = "# o sin(kx " $t),
o o
sinhkD
(10.4.42 a, b)
sinhk(z + D)
% = # o cos(kx " $t)
o o
sinhkD

The fluid elements execute periodic orbits that are closed ellipses (in linear theory) since

! 2 + " 2 = 1,
a 2 b 2
(10.4. 43 a, b, c)
coshk(z + D) sinhk(z + D)
a = # o , b = # o ,
o o
sinhkD sinhkD

Chapter 10 31