How long does a tuning fork ring

There is another important characteristic of waves depicted in the animation above. A careful inspection of the particles of the air represented by dots reveal that the air molecules are nudged rightward and then move back leftward to their original position. Air molecules are continuously vibrating back and forth about their original position. There is no net displacement of the air molecules. The molecules of air are only temporarily disturbed from their rest position; they always return to their original position.

In this sense, a sound wave like any wave is a phenomenon which transports energy from one location to another without transporting matter. For more information on physical descriptions of waves, visit The Physics Classroom Tutorial.

Detailed information is available there on the following topics:. Physics Tutorial. For comparison, we plot the sound pressure level for the same air domain when the tuning fork is held up. The difference is quite stunning with very low sound pressure levels in all parts of the air above the table except for in the vicinity of the tuning fork.

This matches our experience with tuning forks as shown in the original YouTube video. Sound pressure levels for the tuning fork when held up. So far, we have not touched on the original question: Why does the frequency double when the tuning fork is placed on the table? One possible explanation could be that there is such a natural frequency, which has a motion that is more prominent in the vertical direction.

For a vibrating string , for example, the natural frequencies are integer multiples of the fundamental frequency. This is not the case for a tuning fork.

If the prongs are approximated as cantilever beams in bending, the lowest natural frequency is given by the expression. This frequency is a factor 6. It cannot be involved in the frequency doubling. However, there are other mode shapes besides those with symmetric bending. Could one of them be involved in the frequency doubling? This is unlikely for two reasons. The first reason is that the frequency doubling phenomenon can be observed for tuning forks with different geometries, and it would be too much of a coincidence if all of them have an eigenmode with exactly twice the fundamental natural frequency.

The second reason is that nonsymmetrical eigenmodes have a significant transverse displacement at the stem, where the tuning fork is clenched. Such eigenmodes will thus be strongly damped by your hand, and have an insignificant amplitude. One such mode, with a natural frequency of Hz, is shown in the animation below. Since it is only experienced when we press the tuning fork to the table, the double frequency vibration has a strong axial motion in the stem. Also, we can see from a spectrum analyzer you can download such an app on a smartphone that the level of vibration at the double frequency decays relatively quickly.

There is a transition back to the fundamental frequency as the dominant one. The dependency on the amplitude suggests a nonlinear phenomenon. The axial movement of the stem indicates that the stem compensates for a change in the location of the center of mass of the prongs.

Without going into details with the math, it can be shown that for the bending cantilever, the center of mass shifts down by a distance relative to the original length L , which is. The important observation is that the vertical movement of the center of mass is proportional to the square of the vibration amplitude. Also, the center of mass will be at its lowest position twice per cycle both when the prong bends inward and when it bends outward , thus the double frequency.

The stem has a significantly smaller mass than the prongs, so it has to move even more for the total center of gravity to maintain its position. The stem displacement amplitude can thus be estimated to 0. This should be seen in relation to what we know from the numerical experiments above.

In reality, the tuning fork is a more complex system than a pure cantilever beam, and the connection region between the stem and the prongs will affect the results. For the tuning fork analyzed here, the second-order displacements are actually less than half of the back-of-the-envelope predicted 0.

Still, the axial displacement caused by the second-order moving mass effect is significant. Furthermore, when it comes to emitting sound, it is the velocity, not the displacement, that is important. So, if displacement amplitudes are equal at Hz and Hz, the velocity at the double frequency is twice that at the fundamental frequency. Since the amplitude of the axial vibration at Hz is proportional to the prong amplitude a , and the amplitude of the Hz vibration is proportional to a 2 , it is necessary that we strike the tuning fork hard enough to experience the frequency-doubling effect.

As the vibration decays, the relative importance of the nonlinear term decreases. This is clearly seen on the spectrum analyzer. The behavior can be investigated in detail by performing a geometrically nonlinear transient dynamic analysis.

The tuning fork is set in motion by a symmetric impulse applied horizontally on the prongs, and is then left free to vibrate. It can be seen that the horizontal prong displacement is almost sinusoidal at Hz, while the stem moves up and down in a clearly nonlinear manner.

The stem displacement is highly nonsymmetrical, since the Hz contribution is synchronous with the prong displacement, while the Hz term always gives an additional upward displacement. Due to the nonlinearity of the system, the vibration is not completely periodic. Even the prong displacement amplitude can vary from one cycle to another. The blue line shows the transverse displacement at the prong tip, and the green line shows the vertical displacement at the bottom of the stem.

If the frequency spectrum of the stem displacement plotted above is computed using FFT, there are two significant peaks at Hz and Hz. There is also a small third peak around the second bending mode. Frequency spectrum of the vertical stem displacement. In addition to the aesthetic examples found in a concert hall, mechanical resonators have become increasingly important for a wide variety of advanced technological applications, with such diverse uses as filtering elements in wireless communications systems, timing oscillators for commercial electronics, and cutting-edge research tools which include advanced biological sensors and emerging quantum electro- and optomechanical devices.

Rather than producing pleasing acoustics, these applications rely on very "pure" vibrations for isolating a desired signal or for monitoring minute frequency shifts in order to probe external stimuli. For many of these applications it is necessary to minimize the mechanical loss.

However, it had previously remained a challenge to make numerical predictions of the attainable Q for even relatively straightforward geometries. Researchers from Vienna and Munich have now overcome this hurdle by developing a finite-element-based numerical solver that is capable of predicting the design-limited damping of almost arbitrary mechanical resonators.