2.2. PRINCIPLES OF DIMENSIONAL ANALYSIS
33
and
pgL2/nV= f—Y (^)=Re1 Fr~2
\ p J \ V2 /
If a physical problem involves several variables having the same dimensions, then the ratio of any two variables forms a dimensionless product.
For example, wave height (tt), wave length (Z), and water depth (/i) all
have dimension of length; therefore, it is possible to form the dimensionless
parameters H/L (wave steepness), h/L (relative depth), and H/h (heightto-depth ratio). Note, however, that
H
T
H
h
h
L
which indicates that it takes only two dimensionless products to form a
complete set of products from the variables H, L, and h.
The primary reason for being concerned about the number of dimensionless products that can be constructed from a set of variables stems from
Buckingham’s theorem.
A part of the Buckingham’s theorem states, If an equation is
dimensionally homogeneous, it can be reduced to a relationship
among a complete set of dimensionless products.”
(Langhaar 1951).
In other words, if an equation relates variables of a process in such a way
that the form of the equation does not depend on the fundamental units
of measurement, then the equation can be rearranged to be a relationship
between all the dimensionless products that can be formed from the given
variables.
Example 2.4. Fetch-Limited Wave Forecasting
The variables thought to be important in predicting significant wave height (Hs)
growth in fetch-limited conditions are gravity ( and fetch length (X). Therefore, an equation describing this fetch-limited wave
growth would be of the form
Hs = f(g,U,h,X)
The five variables can be formed into a complete set of three dimensionless products.
One such set contains the products gHs/U2, gh/U2, and gX/U2. Therefore, by
33
and
pgL2/nV= f—Y (^)=Re1 Fr~2
\ p J \ V2 /
If a physical problem involves several variables having the same dimensions, then the ratio of any two variables forms a dimensionless product.
For example, wave height (tt), wave length (Z), and water depth (/i) all
have dimension of length; therefore, it is possible to form the dimensionless
parameters H/L (wave steepness), h/L (relative depth), and H/h (heightto-depth ratio). Note, however, that
H
T
H
h
h
L
which indicates that it takes only two dimensionless products to form a
complete set of products from the variables H, L, and h.
The primary reason for being concerned about the number of dimensionless products that can be constructed from a set of variables stems from
Buckingham’s theorem.
A part of the Buckingham’s theorem states, If an equation is
dimensionally homogeneous, it can be reduced to a relationship
among a complete set of dimensionless products.”
(Langhaar 1951).
In other words, if an equation relates variables of a process in such a way
that the form of the equation does not depend on the fundamental units
of measurement, then the equation can be rearranged to be a relationship
between all the dimensionless products that can be formed from the given
variables.
Example 2.4. Fetch-Limited Wave Forecasting
The variables thought to be important in predicting significant wave height (Hs)
growth in fetch-limited conditions are gravity ( and fetch length (X). Therefore, an equation describing this fetch-limited wave
growth would be of the form
Hs = f(g,U,h,X)
The five variables can be formed into a complete set of three dimensionless products.
One such set contains the products gHs/U2, gh/U2, and gX/U2. Therefore, by