3.2. REQUIREMENTS OF SIMILITUDE
59
First notice that the term
is dimensionless, so the ratio of the parameter
between prototype and model should be invariant in an undistorted model. This is
easily shown by taking the ratio as
which equals unity when the vertical scale (Nh) and the wavelength scale4 (Nj_) are
the same (as they are in a geometrically undistorted model). Because the ï-rrh/L
term is the same in prototype and model, then the hyperbolic tangent will also be
the same.
4 The ratio N^ is used to represent the wavelength scale to distinguish it from the
geometric length scale, NlThe scale relationship between the length and wave period scales is found from
the prototype-to-model ratio of wavelength, i.e.,
or
This relationship is written in terms of scale factors as
Nl = Ng • Nt2
The gravity scale ratio in the above expression is, for all practical purposes, equal to
unity. Substitution for Ng in the kinematic relationship, and noting the wavelength
scale, Nj_, is the same as the generic length scale, Nl. we see that kinematic similarity
in gravity wave motion requires
Nt = y/ N l
The above relationship constitutes a criterion of similitude because it is constrained by the mathematical relationship for wave motion.
Note: Example 2.10 in Chapter 2 gave the equation of motion for a simple
pendulum as (neglecting air resistance)
T = 2K(6m)yForming the prototype-to-model ratio and noting that
is dimensionless, and
therefore invariant between model and prototype, yields
59
First notice that the term
is dimensionless, so the ratio of the parameter
between prototype and model should be invariant in an undistorted model. This is
easily shown by taking the ratio as
which equals unity when the vertical scale (Nh) and the wavelength scale4 (Nj_) are
the same (as they are in a geometrically undistorted model). Because the ï-rrh/L
term is the same in prototype and model, then the hyperbolic tangent will also be
the same.
4 The ratio N^ is used to represent the wavelength scale to distinguish it from the
geometric length scale, NlThe scale relationship between the length and wave period scales is found from
the prototype-to-model ratio of wavelength, i.e.,
or
This relationship is written in terms of scale factors as
Nl = Ng • Nt2
The gravity scale ratio in the above expression is, for all practical purposes, equal to
unity. Substitution for Ng in the kinematic relationship, and noting the wavelength
scale, Nj_, is the same as the generic length scale, Nl. we see that kinematic similarity
in gravity wave motion requires
Nt = y/ N l
The above relationship constitutes a criterion of similitude because it is constrained by the mathematical relationship for wave motion.
Note: Example 2.10 in Chapter 2 gave the equation of motion for a simple
pendulum as (neglecting air resistance)
T = 2K(6m)yForming the prototype-to-model ratio and noting that
is dimensionless, and
therefore invariant between model and prototype, yields