38
CHAPTER 2. DIMENSIONAL ANALYSIS
which is a set of two equations with 5 unknowns. Therefore, it will be necessary to
specify 3 of the k-values, and solve for the remaining pair.
The first pi term should contain the dependent variable to the first power, so
make fc2 = 1- Also, we would rather not have depth and fetch appearing in the same
parameter, so set
= k5 = 0- Solving for the remaining values gives k3 = 1 and
k3 = -2, and the first pi term is
H, = g'H.' IT’ h°X°=g -£For the second pi term, we don't want the dependent variable to appear, so let
k2 = 0, and we want to include depth but not fetch. Therefore, let k4 = 1 and
k3 — 0. This results in ki = 1 and k3 = —2, as before, and the second pi term is
n2 = g* l H° U~2 3 * * * * hl X° = £
In summary, calculation of a complete set of dimensionless products (pi
terms) proceeds in the following manner:
1. Determine the important independent problem variables
related to geometry, material properties, and external effects.
2. Set up a matrix listing the exponent of each fundamental
unit contained in the dimensions of each variable.
3. Determine the number of dimensionless products required
to form a complete set. In practically all cases the number
will be n—r, where n is the number of original independent
variables and r is the number of fundamental dimensions
encompassed by the variables.
Using the same logic for the third pi term, and being sure to include the fetch,
gives
n3 = g1 Hs° U~2 h° X1 = 9 ~
From this analysis, we conclude that significant wave height is related to depth,
windspeed, and fetch length by a relationship of the form
gHs _ F / gA gX\
U2
\U2' U2 )
or
H - 2Ï p
g
\U2’ U2 )
CHAPTER 2. DIMENSIONAL ANALYSIS
which is a set of two equations with 5 unknowns. Therefore, it will be necessary to
specify 3 of the k-values, and solve for the remaining pair.
The first pi term should contain the dependent variable to the first power, so
make fc2 = 1- Also, we would rather not have depth and fetch appearing in the same
parameter, so set
= k5 = 0- Solving for the remaining values gives k3 = 1 and
k3 = -2, and the first pi term is
H, = g'H.' IT’ h°X°=g -£For the second pi term, we don't want the dependent variable to appear, so let
k2 = 0, and we want to include depth but not fetch. Therefore, let k4 = 1 and
k3 — 0. This results in ki = 1 and k3 = —2, as before, and the second pi term is
n2 = g* l H° U~2 3 * * * * hl X° = £
In summary, calculation of a complete set of dimensionless products (pi
terms) proceeds in the following manner:
1. Determine the important independent problem variables
related to geometry, material properties, and external effects.
2. Set up a matrix listing the exponent of each fundamental
unit contained in the dimensions of each variable.
3. Determine the number of dimensionless products required
to form a complete set. In practically all cases the number
will be n—r, where n is the number of original independent
variables and r is the number of fundamental dimensions
encompassed by the variables.
Using the same logic for the third pi term, and being sure to include the fetch,
gives
n3 = g1 Hs° U~2 h° X1 = 9 ~
From this analysis, we conclude that significant wave height is related to depth,
windspeed, and fetch length by a relationship of the form
gHs _ F / gA gX\
U2
\U2' U2 )
or
H - 2Ï p
g
\U2’ U2 )