# Rope waves with a twist

In the picture you see a person walking a slackline. A slackline is a tensioned flatband of polyester. Typical tensions are between 1 kN to 15 kN depending on the length of the line. The lines are very thin (about 1 mm - 2 mm I would say), but the width is usually an inch (2.5 cm). There are other materials than polyester in use, but the properties are very similar. Under the high tension the lines are very "bouncy", so they stretch a lot and have relatively high efficiency in returning the bounce energy.

My question is about the oscillations you can see on the rope. For simplicity let us assume that no person is walking on the line, the oscillations still appear then, I just couldn't find a better photo.

The oscillations are cause by the wind. There are two kinds, transverse waves and rotational/twist waves which are visible since the cross-section of the line is not circularly symmetric. I know that the driving force for the transverse waves can be explained by a Kármán vortex street (see also this question and answer about the Tacoma bridge). I was wondering how that relates to the driving force for the rotational waves?

In particular:

• Is the principal the same as for the transverse waves? There is a few reasons why I can't really see that working: the rotational waves in this case are sometimes so large that the flat part goes vertical (i.e. against the wind) or even rotates by multiple revolutions.
• The most important one: Why is the wavelength of the rotation waves (i.e. the separation of the nodes as can be seen in the picture) so much lower than for the transverse waves? This is an observation that is not obvious from the picture, but usually there are only like 3 or 4 oscillations nodes for the transverse ones and like 30 for the rotational ones.
• The question will get too broad if I ask more questions about this, so let's focus on the above. I am generally interested in this phenomenon and there are other questions I can't quite answer (e.g. when it is not very windy there aren't any oscillations and sometimes when the wind is weak oscillations come and go. So why is there a "critical wind speed" at which they start resonating up?).

Some hints/guesses:

• probable driving forces are:

• the flag instability
• buckling, from the horizontal bending of the band caused by wind drag (try to bend a steel rule in its plane, it will buckle because it's much easier to bend orthogonally to its plane)
• the wavelength is linked to the intensity of the restoring force when the band is deformed. E.g. in the Melde string, higher tension gives higher wave velocity and thus greater wavelength for a given frequency. In a tensioned band, it is quite easy to observe that twist is easier to perform than bend.

• Thanks for your answer and sharing your insight! The qualitative answers already look pretty good, I was hoping for a bit more detail though. – Wolpertinger May 1 '16 at 14:36
• I hope my answer will stimulate our fluid mechanics experts :-) – L. Levrel May 1 '16 at 17:35
• you don't really deserve the bounty for this lazy answer ;-) but it still gave good insight and before the bounty gets lost, i might as well give it to you. would still be interested in a more detailed answer though. – Wolpertinger May 7 '16 at 13:03
• Not laziness, I can assure you. Physics guesses as I stated, I'm no fluid mechs expert. Thanks for the bounty! – L. Levrel May 7 '16 at 20:42

I'm having a problem visualizing the transverse waves with 3 or 4 nodes you mention. All videos I've found show standing waves with only two nodes, the slackline moving up and down between the anchor points (or between an anchor point and the person walking the slackline). The rotation waves I saw (maybe torsion oscillation is a better name) also had only two nodes. The "nodes" in the picture aren't actual nodes, they are points where the twist angle corresponds to the viewing angle. If the middle of the line makes (a bit more than) seven complete rotations, you'll have 14 positions on the left and on the right where the rotation is a multiple of 180 degrees, giving you 28 visual points. Their number indicates the amplitude of the torsional oscillation rather than the number of nodes.

As far as I can see, the number of nodes and therefor the wavelengths (for standing wave and torsional oscillation) are equal; the same is true for their frequencies: In the video 50 Meter Slackline im Wind, I count 10 full periods in 4.29 seconds for both the vertical and torsional oscillation. The movement of the clouds indicates that the wind is coming from the right, and the line moves upward when it rotates counter clockwise, downward when rotating clockwise, which suggests that the vertical oscillation may be caused by the Magnus effect.

Whether or not the vertical movement (the bending of the line) causes the torsional rotation, or whether the rotation amplifies itself, I'm not sure, but there clearly is positive feedback: "Skybow" kites are based on the phenomenon, these are basically long narrow ribbons attached to swivels, freely rotating (reaching speeds of more than 10000 rpm). The original ones are flat symmetric ribbons that will turn in either direction, so their rotation is not based on a design feature (like a curved surface). See the blog by one of the designers for more.

A formula for the frequency of the torsional oscillation can be found in "On the Vibration of a Ribbon of Metal or Other Materials when excited by Air Currents" (see google, I'm restricted to two links). $$f=\frac{1}{2L}\sqrt{\frac{T}{\mu}+\frac{\tau}{\mu}16\frac{b^3}{a}}$$ $\mu$ = linear density; T = tension; $\tau$ = "rigidity" (shear stress); 2b = thickness; 2a = width; L = length

Ignoring the rigidity component , this becomes: $$f=\frac{1}{2L}\sqrt{\frac{T}{\mu}}$$ Which is identical to the frequency for a standing wave (fundamental harmonic) in a string of length L, showing that the frequencies are indeed the same.

• +1 and Thanks for your answer! I've never considered that they might not be nodes, but I thought they were. For the rotational once you might be right, but I am certain that the transverse once have nodes, especially if there isn't a lot of tension on the line. – Wolpertinger May 24 '16 at 8:57