Math Is Fun Forum

Jonathan terry

Guest

Catenary

Hi,

I work on a tugboat and we tow a barge from Jacksonville, FL to San Juan PR.  I am trying to find a formula to solve for the depth of the tow wire below the surface of the water.

Given the total length of the towing wire and the distance between the back of the tug boat and the front of the barge, how deep below the surface of the water is the lowest point of the tow wire?

Assume length of wire is 2100' and distance between tug and barge is 1650'

Thanks

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Catenary

Hi;

I once worked under the assumption that a hanging cable was a parabola?! Actually, I got quite a good result using that assumption... I would now agree that a hanging cable is a catenary, why do you assume that a submerged cable is also one? But I have heard that it is an approximate catenary and maybe that is good enough...

I had to make a few assumptions and I used the definition of a catenary given here

https://en.wikipedia.org/wiki/Catenary

I came up with that the cable hangs 575.63788199 ft at the center below the water.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Catenary

hi Jonathan terry

Welcome to the forum.

bobbym:  Wow! I know everything in the US is bigger than elsewhere but how deep is the river?

If a weighted wire hangs between two fixed points, it will hang in a catenary curve.  But your end points are not fixed.  The tug is exerting a sideways force on the wire, which will lessen the maximum depth.  I'm not sure that a catenary is still valid in that case.  I would expect the current to affect the result too.  There are so many variables here that you might get a more reliable result by attaching a depth device and doing an experiment.  Do you get wear at the lowest point when the wire runs aground?

Further thought:   That's a mighty long tow length.  Isn't the wire a hazard to other shipping.  How do you avoid a smaller vessel coming between tug and barge?  And what happens at a bend in the river?

I know there is theory about wires hanging in a catenary curve but I don't know the underlying proof.  I'll do some research and see if I can make some progress with this.

later edit:  here's what the physicists say https://www.physicsforums.com/threads/c … le.762801/

And these guys http://www.ybw.com/forums/archive/index … 94512.html recommend a longer length to avoid the risk of breaking so maybe that answers my question about length.

The Wiki page has the proof and it looks good for a wire under tow.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Catenary

Hi;

I just computed the math answer possibly. I agree, the problem like most math problems...It has very little to do with reality.

bobbym:  Wow! I know everything in the US is bigger than elsewhere but how deep is the river?

That only applies to Texas. But he is supposedly towing from Jacksonville to Puerto Rico. Some very deep water around them parts, the Puerto Rican trench is the deepest part of the Atlantic Ocean. Of course, it is loaded with alien bases down there and this type of alien only thinks of humans as delicious blood bags. They see that tow line and it is like someone just rang the dinner bell.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Catenary

Arrhh.    I get it.  Should have looked on a map first.  I imagined this was a river trip; not across the ocean.  I realised that FL was Florida and assumed that PR was some other State.  Excuse my ignorance.  Red face. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Catenary

It will be a Catenary with g_effective=(s-1)/s*g where s is specific gravity of cable material to take care of buoyant force on the cable.

Last edited by thickhead (2017-03-07 05:55:23)

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