non-dimensionalizing a non differential equation

Mar 2020
4
0
africa
I am familiar with non-dimensionalizing a differential equation; however, I am not familiar with a method to non-dimensionalize a non-differential equation. I just need to arrive to a non-dimensionalized formula. I have attempted it as shown here.

The equation i am trying to non-dimensionalize is as follows.

U_{orb} = \frac{\pi H_{rms}}{T_{p}sinh(2kh))}

please help
 

topsquark

Math Team
May 2013
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1,052
The Astral plane
I'm not sure what you are trying to find. Are you looking for the unit of \(\displaystyle U_{orb}\) or are you trying to verify the unit for \(\displaystyle U_{orb}\)?

I'm also not fond of the way you are writing this. It's confused you to the point where you don't have the correct argument for your sinh function. (You wound up dropping the L and h.) It's best to just drop out the variables and leave the units. For example:
\(\displaystyle U_{orb} \times \frac{T_p}{H_{mo}} = \frac{ \pi H_{rms} }{ T_p ~ \sinh \left ( 2 \frac{2 \pi }{L} h \right )}\)

would be written as
\(\displaystyle \left [ U_{orb} \right ] \times \frac{ [T] }{ [L] } = \frac{ [L] }{ [T] }\)
(I'm using [T] for temperature units and the sinh function has no units. The unit for \(\displaystyle \left [ U_{orb} \right ] \) I am presuming is unknown.)

So the unit for \(\displaystyle \left [ U_{orb} \right ] \) would be
\(\displaystyle \left [ U_{orb} \right ] = \frac{ [L] ^2 }{ [T] ^2 }\)

Is this what you are looking for?

-Dan
 
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Mar 2020
4
0
africa
I'm not sure what you are trying to find. Are you looking for the unit of \(\displaystyle U_{orb}\) or are you trying to verify the unit for \(\displaystyle U_{orb}\)?

I'm also not fond of the way you are writing this. It's confused you to the point where you don't have the correct argument for your sinh function. (You wound up dropping the L and h.) It's best to just drop out the variables and leave the units. For example:
\(\displaystyle U_{orb} \times \frac{T_p}{H_{mo}} = \frac{ \pi H_{rms} }{ T_p ~ \sinh \left ( 2 \frac{2 \pi }{L} h \right )}\)

would be written as
\(\displaystyle \left [ U_{orb} \right ] \times \frac{ [T] }{ [L] } = \frac{ [L] }{ [T] }\)
(I'm using [T] for temperature units and the sinh function has no units. The unit for \(\displaystyle \left [ U_{orb} \right ] \) I am presuming is unknown.)

So the unit for \(\displaystyle \left [ U_{orb} \right ] \) would be
\(\displaystyle \left [ U_{orb} \right ] = \frac{ [L] ^2 }{ [T] ^2 }\)

Is this what you are looking for?

-Dan
Hi Dan,

Thank you for the response.

What I am looking for is \(\displaystyle U_{orb} \) to be completely dimensionalized. so initially what I did to non dimensionalize it was the following:

\(\displaystyle U_{orb} \times\frac{T_p}{H_{mo}}\)


the unit of Uorb is \(\displaystyle \frac{m}{s }\)
the unit of T_p is \(\displaystyle s\)
the unit of H_rms is \(\displaystyle m\)
the unit of h is \(\displaystyle m\)
the unit of L is \(\displaystyle m\)
 
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topsquark

Math Team
May 2013
2,533
1,052
The Astral plane
Hi Dan,

Thank you for the response.

What I am looking for is \(\displaystyle U_{orb} \) to be completely dimensionalized. So initially what I did to non-dimensionalize it was the following:

\(\displaystyle U_{orb} \times\frac{T_p}{H_{mo}}\)


the unit of Uorb is \(\displaystyle \frac{m}{s }\)
the unit of T_p is \(\displaystyle s\)
the unit of H_rms is \(\displaystyle m\)
the unit of h is \(\displaystyle m\)
the unit of L is \(\displaystyle m\)
Sorry, I'm still not getting what you are after. Could you please define what you mean by "completely dimensionalized?" I was taking it to mean that you are looking to find what dimensions \(\displaystyle U_{orb}\) has.

Given your table is it true that \(\displaystyle U_{orb} \times \frac{T_p}{H_{mo} }\) is unitless. However, the RHS of your equation comes out to have units of m/s, so the units don't match.

Be aware that, though we need the units to match on both sides of an equation, you also have to make sure that the equation is physically correct as well. For example, \(\displaystyle v^2 = v_0^2 + ax\) has the correct units, but the equation itself is invalid. So you need to take care in how you are making the units match. To be specifc, I don't understand (physically) what multiplying \(\displaystyle U_{orb} \times \frac{T_p}{H_{mo} }\) means.

-Dan
 
Mar 2020
4
0
africa
Sorry, I'm still not getting what you are after. Could you please define what you mean by "completely dimensionalized?" I was taking it to mean that you are looking to find what dimensions \(\displaystyle U_{orb}\) has.

Given your table is it true that \(\displaystyle U_{orb} \times \frac{T_p}{H_{mo} }\) is unitless. However, the RHS of your equation comes out to have units of m/s, so the units don't match.

Be aware that, though we need the units to match on both sides of an equation, you also have make sure that the equation is physically correct as well. For example, \(\displaystyle v^2 = v_0^2 + ax\) has the correct units, but the equation itself is invalid. So you need to take care in how you are making the units match. To be specifc, I don't understand (physically) what multiplying \(\displaystyle U_{orb} \times \frac{T_p}{H_{mo} }\) means.

-Dan
"completely dimensionalized" means I want a unitless number. U_orb is the orbital velocity of a propagating wave. While T_p and Hrms are the wave height and period of said wave. What I am trying to achieve is a dimensionless orbital velocity.
 

topsquark

Math Team
May 2013
2,533
1,052
The Astral plane
"completely dimensionalized" means I want a unitless number. U_orb is the orbital velocity of a propagating wave. While T_p and Hrms are the wave height and period of said wave. What I am trying to achieve is a dimensionless orbital velocity.
The only way that I can grok this (and have a meaning) would be to use a "reference speed," call it v, to measure \(\displaystyle U_{orb}\) against. Then you can use \(\displaystyle \frac{U_{orb}}{v}\) as a unitless quantity. Without knowing what exactly you are doing, this is the best that I can come up with.

-Dan
 
Mar 2020
4
0
africa
hi dan,

Thank you for the response.

What I am looking for is \(\displaystyle U_{orb} \) to be completely dimensionalized. so initially what I did to non dimensionalize it was the following:

\(\displaystyle U_{orb} \times\dfrac{T_p}{H_{mo}}\)


the unit of Uorb is \(\displaystyle \dfrac{m}{s }\)
the unit of T_p is \(\displaystyle s\)
the unit of H_rms is \(\displaystyle m\)
the unit of h is \(\displaystyle m\)
the unit of L is \(\displaystyle m\)

Hi Dan,

Thank you very much for your recommendations.

I used a reference velocity called the eulerian velocity to try nondimentionalise it. the following are the results:

the U_orb for wave hight of 3(blue) and 1 (orange) for 3 cases and 3 experiments (x-axis is calibrating factor and y-axis is U-orb)

Figure_2.png

and the following is the result after nondimentionaliseing using a reference velocity:

Figure_1.png

Would you say it was a successful nondimentionalizeing?
 
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