η 2 η 2
η 2 η 2
$
$
$
$
2 + 2 sin (2α )
-
1 + 1 +
x y
x y
η η
η1 η2
$$
$$
cos(2α ) = 0
- 2 1 2 +
(131)
y y
x x
Rearranging terms gives
η η
η1 η2
$$
$$
2 1 2 +
x x
y y
1
α = arctan
(132)
2
η η
η1 η1
2
2
2
2
$
$
$
$
+
2 + 2
-
+
x y
x y
The magnitude of the horizontal component of the velocity will have its minimum and
maximum values at this value of α and at α+ π/2. The value of |v| is calculated at both of
these angles; the larger value is the maximum velocity over all times.
The pressure is obtained from the linear form of the Bernoulli equation;
Φ P
+ + gz = constant
(133)
t ρ
Note that the pressures associated with velocities (dynamic head, v2) are ignored in this
linear form. The expression for Φ in Equation 125 is substituted into this expression and
terms are rearranged to obtain
(
)
P = - ρgz + ρgℜ ηe - iωt Z + constant
(134)
$
The maximum pressure over a wave cycle occurs when the term ℜ ( ηe- iωt ) is equal to
$
H/2. The constant is chosen such that the hydrostatic pressure is equal to zero at z=0;
thus
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