How does a steady wind flow generate sound?

When a wind blow through sharp edge, say, edge of a paper, you can see the vibration of the paper and hear the sound.

For this type of oscillation, it should be a damped oscillation with external driving force $\mathcal{F}(x)$:

$\ddot{x}+2\lambda\dot{x}+\omega_{0}^{2}x=\mathcal{F}(x,t)$

with the driving force in the form of sinusoidal $\mathcal{F}(x,t)\sim F\sin(\omega t)$ (It should be a 3D version though). However, if the air flow is really steady, the external force should be constant so there is no oscillation at all from this equation.

A simple guess is that the motion of the paper influence the streamline of air flow so that the pressure gradient provides the exact sinusoidal driving force. This mechanism is reasonable because the propagation of sound wave has the oscillation relationship between the displacement and the pressure. A question here is how the air flow sustain the energy loss due to the damping.

It is also not easy to think of the initial condition as well as external force for these oscillation. So

1) What exactly is the physical mechanism to generate sound when a 'steady' air flowing through a sharp edge?

2) Another similar question is how the driving force act in the resonance in musical instrumental such as pipe? A 'steady' flow of air is also provided somewhere to the pipe, but air oscillates inside the pipe.

Edit: From the answers, people suggest few mechanism for different situations. Few more questions:

3) For static object, Kármán vortex street can form behind the object. So is the sound frequency the same as frequency of vortex generation, or the same as the expansion of the vortex? Sound is a spherical wave propagating outward, so identify the sound frequency should locate the point of generation.

4) Where is sound generated in the situation (1)? Is the sound frequency the same as the frequency of the vibrating paper?

Fluid mechanics is a difficult topics, but there are approximation for special cases trying to explain the underlying mechanism, such as the flag fluttering cited by j.c.

-

This is an structure-fluid interaction type of problem, where the elasticity of the boundary interacts with the fluid turbulence to produce an amplyfying effect. In essect, "steady flow" only exists in text books and not in real nature. Even the most viscous flow has turbulence and non-laminar effects. – ja72 Nov 23 '10 at 19:35
– Vortico Nov 23 '10 at 20:35

4 Answers

There are really two parts to your question. First, how does the wind affect the motion of the paper, and how does that motion then couple to the behavior of the wind?

For the case of a flag fluttering due to the wind (which may or may not be applicable depending on what you have in mind), the physics of the instability leading to fluttering has been worked out in a fairly celebrated paper of 2005 by Argentina and Mahadevan. They argue that:

[...] in a particular limit corresponding to a low-density fluid flowing over a soft high-density flag, the flapping instability is akin to a resonance between the mode of oscillation of a rigid pivoted airfoil in a flow and a hinged-free elastic plate vibrating in its lowest mode.

I suspect this paper and probably some of the papers that cite it would be the right place to look. As you can see, this part of the question that you've asked is of fairly high interest in current research.

Second, how does this motion generate sound? This is also an interesting question but I know less about this. I would suppose that some resonant vibrational modes of the sheet of paper that you're asking about (those involved in the flapping modes described by Argentina and Mahadevan) generate the sound. But I don't know very much about sound generation or acoustics.

-

::fires up a groovy surf-rock bassline:: Good, good, good citations! – dmckee♦ Nov 23 '10 at 7:32
The flag fluttering is different from a paper edge cos they have different boundary condition of the fixed point. Good and interesting paper though. – hwlau Nov 24 '10 at 4:25

It is always about a resonance between the vibrating element and a pressure mod in some resonator (in case of sharp edge, the edge itself oscillates).
I believe the oscillations itself are produced by the turbulent vortices forming Kármán street behind the reed or edge.

EDIT: I have even found a reference: http://arxiv.org/abs/physics/0008053v1

EDIT2: In detail, the process starts with static edge in laminar flow; speed of flow increases and finally the Reynolds number exceeds critical value in which streamlines start to disengage from edge first forming vortices in Karman street (this is called Strouhal instability) and then a totally random turbulent flow. The vortices interact with the edge "poking" it quite randomly (Karman street has some frequency, but I think it does matter only for some narrow cases); now the process is driven by resonances. The edge has some resonance frequencies, the air around can also have them if enclosed in some container, in case of some instruments the musician's lips are also involved -- all those for some kind of mechanical filter that amplifies certain frequencies (instruments) or ranges of frequencies (random setups). Finally, those mechanic vibrations induce pressure weaves in air that we hear as a sound, again either clear as in case of instruments or a noise as in case of trees, roofs and other stuff.

-
I have taken a look for the reference. It only give the simple calculation of the frequency, but not the mechanism. – hwlau Nov 22 '10 at 17:44

1
@hwlau Yes; this is because fluid dynamics is too complex to make an analytical calculation. I'll try to edit my answer to give more details. – mbq♦ Nov 22 '10 at 19:58

The full description of viscous fluid flow (i.e. the Navier-Stokes equations) is non-linear and can be sensitively dependent on initial conditions. What this means in practical terms it that you can't always count on your intuition.

The evolution of the Kármán vortex streets linked by mbq from laminar flow around an obstacle are are a classic demonstration. (And easy enough for a determined middle school student to create in the garage for a science project, though things are likely to get wet...)

Any way, once the wind starts doing non-linear things, it can generate periodic stresses, and from that you get the whistling or humming noise we all know and love. Add a resonant cavity and you can amplify nice pure tones which is the short-short version of how this works in wind instruments.

-

The Karman vortex streets seems occurs around a static object. How about if the paper edge is vibrating? Is the sound come from the vortex or the virbrating sheet? Or put it simpler, is the frequency of the generating vortex the same as vibrating sheet. – hwlau Nov 22 '10 at 17:43
@hwlau: It depends. I believe a flute is driven by pressure oscillations in the wind stream, but reeds work off the resonant response of the reed. It's a complicated subject, and the motion or vibration of the obstacle must be taken into account. Closed form solutions are generally impossible and the numeric discipline is called Computation Fluid Mechanics and is very challenging. – dmckee♦ Nov 22 '10 at 18:54

This is a similar problem to the karman vortices developing around cables/pipes when wind blows around them. When each vortex sheds and releases from the boundary layer there is a reactive aerodynamic "lift" developed which might move the structure. When the structure returns it forces the next vortex to shed aplifying the effect. It is like pushing a pendulum. If you push on the right time, even a tiny force can have a large effect.

Search cable galloping, aeolian vibrations and structural wind noise. Note that with cables the wind might excite the 160-th basic harmonic of the system, and not the first few as you might expect. Thats because the internal turbulence in wind resonates based on the Strouhal number is it might be of the order of 40 Hz-160 Hz.

-

Simple question: If it is turbulence, how can it generates the periodic motion for n-th harmonics. Or my assumption wrong? – hwlau Nov 24 '10 at 4:27
– ja72 Nov 24 '10 at 5:02
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9339147210121155, "perplexity_flag": "middle"}