108
CHAPTER 4. HYDRODYNAMIC MODELS
or
(27r)2or _ f
7l2
/st
(4.46)
where the definition for specific weight of water (7 = pg) has been used.
The wavelength at which the surface tension effect reaches a certain
percentage of the gravity contribution can be evaluated by rearranging
Eqn. 4.46 as
L = 27t./-|(4.47)
V 7At
The following example illustrates how to evaluate surface tension effects.
Example 4.2. Surface Tension Effects in Short-Wave Models
What are the wave period and water depth values below which surface tension
effects contribute more than 1% to the wavelength?
Using Eqn. 4.47 with the fractional value fst = 0.01, and typical values for fresh
water at 20 °C of 7 = 9.79 kN/m3 and a — 0.074 (TV • m)/m2, we can solve for the
wavelength where surface tension effects contribute 1%, i.e.,
_ 2^ /0-074 (TV • m)/m2
\ 9.79 kN/m3(0M)
— 0.173 m = 17.3 cm
The wave period can be estimated by using the linear theory deep water approximation for wave celerity given by Eqn. 4.43 when the hyperbolic tangent approaches
unity. Substituting C = L/T, and rearranging gives
T = L I g — 4- —
y 2tt
pL
-1/2
Substitution of the calculated wavelength, recognizing that the surface tension term
will be 0.01 times the magnitude of the gravity term, yields the deepwater wave
period, i.e.,
T = (0.173 m) ( (. A1)(9-806n./?)(0.173m)\~'/2 =
\ 2ir
]
The shallow water limit is found by assuming the deep water wave period is
conserved through the shoaling process. Using the shallow water approximation for
wave celerity (given by Eqn. 4.43 when the value of the hyperbolic tangent approaches
the value of its argument and C is replaced by L/T) gives the expression
2?rT2
gL
27r
pL
CHAPTER 4. HYDRODYNAMIC MODELS
or
(27r)2or _ f
7l2
/st
(4.46)
where the definition for specific weight of water (7 = pg) has been used.
The wavelength at which the surface tension effect reaches a certain
percentage of the gravity contribution can be evaluated by rearranging
Eqn. 4.46 as
L = 27t./-|(4.47)
V 7At
The following example illustrates how to evaluate surface tension effects.
Example 4.2. Surface Tension Effects in Short-Wave Models
What are the wave period and water depth values below which surface tension
effects contribute more than 1% to the wavelength?
Using Eqn. 4.47 with the fractional value fst = 0.01, and typical values for fresh
water at 20 °C of 7 = 9.79 kN/m3 and a — 0.074 (TV • m)/m2, we can solve for the
wavelength where surface tension effects contribute 1%, i.e.,
_ 2^ /0-074 (TV • m)/m2
\ 9.79 kN/m3(0M)
— 0.173 m = 17.3 cm
The wave period can be estimated by using the linear theory deep water approximation for wave celerity given by Eqn. 4.43 when the hyperbolic tangent approaches
unity. Substituting C = L/T, and rearranging gives
T = L I g — 4- —
y 2tt
pL
-1/2
Substitution of the calculated wavelength, recognizing that the surface tension term
will be 0.01 times the magnitude of the gravity term, yields the deepwater wave
period, i.e.,
T = (0.173 m) ( (. A1)(9-806n./?)(0.173m)\~'/2 =
\ 2ir
]
The shallow water limit is found by assuming the deep water wave period is
conserved through the shoaling process. Using the shallow water approximation for
wave celerity (given by Eqn. 4.43 when the value of the hyperbolic tangent approaches
the value of its argument and C is replaced by L/T) gives the expression
2?rT2
gL
27r
pL