A whistle is a device that makes sound from air blown from one end forced through a small opening at the opposite end. They are shaped in a way that allows air to oscillate inside of a chamber in an unstable way. The physical theory of the sound-making process is an example of the application of fluid dynamics or hydrodynamics and aerodynamics. The principles relevant to whistle operation also have applications in other areas, such as fluid flow measurement.

Types Wilson, et al., in their study of human whistling (see below), pointed out the importance of including the symmetry or asymmetry of the unstable flow in addition to the feedback classes listed below. Because of the close relationship of flow symmetry to the sound field generated, their concept was included here as part of the sound source description (monopolesymmetric and dipoleasymmetric).

Monopole

Whistles that generate sound through fluctuations of mass flow across a boundary are called monopole-like sources. The figure on the right is an example of a small sphere whose volume is oscillating. For this type of source, the sound is emitted radially, so the sound field is the same in every direction and decays with the inverse square of the distance. The sound power can be expressed as:

where is the sound power in watts for a monopole is the density of the medium is the speed of sound in meters per second St is the Strouhal number is vortex shedding frequency in hertz is the characteristic length in meters (in this case, the amplitude of oscillation) is the average flow speed in meters per second

Dipole

Whistles that generate sound through fluctuations of momentum or stress and strain of a force exerted on the surrounding medium are called dipole-like sources. The figure on the right is an example of a small rigid sphere that is moving back and forth in a given direction. This results in a non-uniform sound field. The sound power can be expressed as:

Feedback categories Aerodynamic whistles rely on the instability of flows and feedback that enhances the instability. There are several ways that feedback can occur.

Category I The sound from a whistle in this category is primarily a by product of source motion. The back reaction is insufficient to strongly control source motion, therefore whistles are not in this category.

Category II The back reaction of the medium is a determinant of source motion. Unstable fluid motion or the sound generated by it can feedback to the source and control it. What is required is: a source of steady power; an amplification mechanism that can convert the steady power to time-varying power; a disturbance, which supplies the oscillations to be amplified; a means of generating sound or other oscillatory fluid Brownian motion; a means for feedback of that oscillatory motion as a disturbance to the input of the amplifier. Whistles are in this category. There are several ways to describe the feedback process.

Class I The feedback is essentially incompressible; the speed of sound, although finite, is sufficiently large that it can be considered infinite. This action may be called hydrodynamic feedback. There are a number of class I devices.

Class II The feedback is compressible and does not depend on the speed of sound. This action may be called quasi-compressible feedback.

Class III The feedback is compressible and depends on the speed of the sound. This may be called acoustic feedback.

The figure on the right shows a block diagram of these feedback mechanisms. All aerodynamic whistles operate under one of the classes.

Stages

Feedback in whistles is nonlinear mechanics or chaos. Because of the non-linearity, it is possible to have more than one frequency at a given flow rate. The difference depends on whether the flow rate is achieved by increasing the flow or by decreasing the flow. The possibility of multiple states is called a stage and is shown schematically in the figure on the right. As the flow speed increases (Reynolds number, Re) the frequency slowly climbs (nearly constant Strouhal number, St) but then the frequency jumps up abruptly to a higher stage. As the flow speed is later decreased, the frequency slowly decreases but then jumps down abruptly to a lower stage. This pattern is called a hysteresis loop.

Higher stages are associated with more vortices in that distance, hinting that this distance can be an important characteristic dimension. In several whistles, three stages have been identified. Overblowing in some musical wind instruments causes stage I to jump to stage II.

Flow instability

Flow instability is the engine for whistles. It converts steady energy to time-dependent energy.

An example is shown in the figure on the right with a water jet.Chanaud, R. C., MS Thesis, University of California, Los Angeles, 1960. The laminar two-dimensional jet amplifies small disturbances at the orifice to generate a vortex street. For this case, the flow speed, in terms of Reynolds number, was graphed against the disturbance frequency, in terms of the Strouhal number for a variety of disturbance amplitudes to reveal the region of instability as shown in the figure on the left. The value of D in the figure represents the ratio of the lateral disturbance displacement to the nozzle width; the disturbances were minute.

One important source of instability in a fluid is the presence of a velocity gradient or shear layer with an inflection point. In a whistle, the instability starts at some point in the three-dimensional region and then moves along some path in that region as the local variables change. This makes a comprehensive understanding of whistle instability mechanisms very difficult.

Scaling Whistles come in all shapes and sizes, but their operation can be unified through the concepts of dynamic and geometric similarity using dimensional analysis. Nature knows nothing of the specific measurement systems we use; it cares only about ratios between the various forces, time scales, and several dimensions. To compare them, we need to take into account the established ratios that are relevant to whistle operation.

The similarity is best exposed by determining a speed U, that is, characteristic of the dynamics, and a dimension L, that is, characteristic of the geometry. If these values are used in dimensionless numbers, such as those listed below, much understanding of the phenomenon can be achieved.

Strouhal number

f is vortex shedding frequency in hertz L is the characteristic length in meters U is the average flow speed in meters per second

The first number is the ratio of unsteady inertial forces to steady inertial forces. The number was named in honor of Vincenc Strouhal, who first deduced the relationship between the vortex shedding frequency around a cylinder and the flow speed. The characteristic variables were the cylinder diameter L and the speed U of the flow over it.

This number permits relationships to be developed between the different sizes and speeds. This equation may be referred to as a fluid mechanical Strouhal number in comparison with the second version, which may be referred to as the acoustic Strouhal number. The first version is used for the dynamic similarity of the fluid motion in whistles, while the second version is used for the dynamic similarity of the acoustic motion in whistles. Many whistles require the use of both numbers.

Mach number

It is the ratio of the steady speed to the speed of sound. The number was named in honor of Ernst Mach, who first studied (among other things) supersonic flow and shock waves. This number describes the range between flows that can be considered incompressible and flows that compress.

Reynolds number

It is the ratio of the steady inertial forces to the steady viscous forces. The number was named in honor of Osborne Reynolds, an engineer who did pioneering studies on the transition of laminar to turbulent flow in pipes.

Rossby number

It is the ratio of linear velocity to tangential velocity for swirl flows. The frequency is characteristic of the rotation rate of the flow. The number was named in honor of Carl-Gustaf Rossby, a meteorologist who first described the large-scale motions of the atmosphere in terms of fluid mechanics.

Dimensionless force

The ratio of the actual dynamic force to the steady momentum and steady current density.

Dimensionless volumetric flow rate

The ratio of the dynamic volumetric flow rate to the steady volumetric flow rate.

Monopole-like whistles In these whistles, the flow instability is symmetric, often resulting in periodic ring vortices, and the sound generation is associated with fluctuations in volumetric/mass flow rates.

Hole tone (teapot whistle, bird call)

The steady flow from a circular orifice can be converted to an oscillatory flow by adding a downstream plate with a circular hole aligned with the orifice. Small perturbations in the flow at the hole feedback to the orifice cause a variable volumetric flow rate through the downstream hole because of the symmetry of the feedback. The disturbance in the jet is a symmetric vortex ring that moves at some speed slower than the mean jet speed until it encounters the hole, and some fluid is forced through it, resulting in a monopole-like sound field in the half-space outside. The figure on the right shows a schematic of the geometry.

To invoke dynamic similarity, the characteristic speed in a study was chosen to be the average speed U of the jet at the orifice and the characteristic length was chosen to be the orifice diameter δ. Tests were made at five spacing distances h/δ from the orifice. Two scaling laws were used: the Strouhal number was graphed as a function of the Reynolds number. The results are shown in the figure on the right.

The frequency of the tone is determined by how often a vortex encountered the hole while moving at some speed u less than the initial jet speed. Since the jet slowed as it proceeded toward the hole, the speed of the vortex slowed with it, so the frequency and Strouhal number were greater at closer spacing. The Strouhal-number data showed clearly the almost linear relation between frequency and initial jet speed. In four of the distances tested, there were jumps between stage I and stage II. The hysteresis loops are clear indications of the complex nature of the jet instability gain structure.

The uniformity of the measured sound field for this whistle confirmed its monopole-like nature. Measurements of speed dependence of the sound level showed it to be very close to U4, further confirming the monopole nature of the source. At these speeds and spacing, the feedback was normally class II, but reflecting surfaces as far away as 3 meters and with proper phasing, controlled the tone, converting the feedback to class III.

The hole tone is also known as the teapot whistle. They found that above a Reynolds number of 2000, hole tone operation occurred with symmetric vortex evolution and a constant Strouhal number with Reynolds number. At lower speeds, the cylindrical volume responded as a Helmholtz resonator. Baron RayleighStrutt, J. W. Baron Rayleigh, The Theory of Sound, MacMillan and Co. 1877. was aware of this whistle; it was called the bird call then. In Australia, there is the Tenterfield fox whistle and the traditional fox whistle that appear to operate as hole tones.

Corrugated pipe whistle

This whistle has dozens of popular names. Corrugated tubes or pipes are often used in various applications because they are easier to bend than traditional pipe. Steady flow through a corrugated pipe at low Reynolds numbers results in a fluctuating volumetric flow rate that generates a monopoly-like sound field at the pipe exit. Examples of such pipes are shown in figure on the right.