# Frequency of a Tuning Fork

Question: Which of the following affect the frequency of a tuning fork?

• Tine stiffness
• Tine length
• The force with which it's struck
• Density of the surrounding air
• Temperature of the surrounding air

Answer Attempt: Based on the formula for the frequency, I know that tine stiffness (or density) affects it, and so does the tine length. I believe the temperature and density of air can have a slight affect as well. What about the force with which it's struck?

• Counter-question: assuming small oscillations, does the frequency of a pendulum depend on the force you impart to it or the properties of the air?
– user10851
Apr 7, 2013 at 18:01

Nope. The frequency of the tuning fork doesn't depend on the force with which it's struck. Every tuning fork has its own desired frequency. And hence, people usually say, "Hey - take that fork, the one with 432 Hz (an example) on it...".

The oscillations usually get damped out after some period. But, the frequency still remains the same... Usually, the frequency of a tuning fork depends only on the property of its material.

If you have a look at the Wikipedia article on frequency, the frequency turns out to be

$$F \propto \frac{1}{l^2}\sqrt{\frac{EI}{\rho A}},$$

where $F$ is the frequency, $l$ is the length of the tines, $E$ is the Young's modulus of the material (which is related to stiffness), $I$ is the second moment of area of the tines (which is related to inertia), $A$ is the cross-sectional area of the tines and $\rho$ is the density of the material.

This clearly shows that the parameters are all the properties of the material from which the fork is made, as well as its shape.

The frequencies of vibration of a tuning fork can be affected by the force used to strike it. Depending upon the magnitude, direction, and the number and location of the points of application, vibrational modes other than the fundamental mode could be excited. Each of these vibrational modes have a different frequency. The mode frequencies are characteristic, but not all necessarily need to be excited every time. For a tuning fork, the fundamental mode, the clang mode, the asymmetric in-plane mode, the out of plane bending mode, and the asymmetric out of plane modes are shown here as animations.

The frequency engraved on the tuning fork is for the fundamental mode. As a device designed to act as pitch reference, the other modes (of a high-quality fork) should not be easy to excite strongly, and should damp out quickly. But you should be able affect how much the auxiliary modes are excited by how you strike the fork.

All physical harmonic oscillators will change their frequency versus the amplitude of oscillation, (even precision pendulum clocks). If the tines of a tuning fork are struck very hard the frequency will drop because the tines have farther to move and; though the internal restoring force also increases with displacement, it will not be enough to exactly keep the frequency steady. Thus cheap soft metal oscillating things go "boing" or "twang". Those onomatopoeias mimic the characteristic frequency rise as the oscillator's sound quickly dies out, the metal swings through smaller and smaller arcs and so the restoring force of the metal becomes more and more linear versus displacement distance. That effect is called "overdrive" or running an oscillator in the "non-linear regime" but of course that effect is always present to some degree, if you look close enough, in any parameter regime. At the level of instrument tuning accuracy that effect will not be bad enough to cause unacceptable pitch accuracy, since it is well below about 0.1 Hz or 1 cent in the forks I have studied; music sounds OK within that accuracy. But for a homemade tuning fork that effect could be bad. I note that contrary to the previous opinions on this webpage above; the density of the surrounding air, or wood, or whatever medium the vibrations are carried away from the tuning fork in, will only decrease the frequency a tiny amount, maybe more so the higher the medium's density, or perhaps the rate of energy flow away from the fork directly dictates the "pull" of the fork's frequency down, but there will be some drop as the environment of the fork contacts it and interacts with it's oscillation. However usually that will not be at a level high enough to concern musical uses. That is because the tine motion has a high amount of stored energy due to the metal's mass and velocity, and strength. Such a powerful motion can only be changed by a low density medium like air, at a very subtle level, below the concern of music. But in the case of a temperature inequality between the fork and the air: as that air heats the fork, the air changes the properties of the fork's metal and effects the frequency indirectly by that slower thermal route, not by the air changing the pitch itself as suggested by people above on this page. I note: the effect of helium changing frequencies itself mentioned by the other posters above is not correct, when inhaled it changes the resonance of the vocal tract. Such a phenomenom does not concern tuning forks and so its pitch will be virtually the same in helium as in air (ignoring the microscopic effects I mentioned).

I believe that the force doesn't affect the frequency of the sound generated, based on how guitar strings work. Addressing each point:

• You have different strings of different densities which give different tones, so yes, tine stiffness is a factor.
• You press the string at different positions to give different tones, so yes as well for the length.
• I never noticed any change in the tone whether the strings are stuck stronger or weaker. The volume is different, not the tone.
• If you really mean the frequency of the sound the fork produces, then I would say yes based on how people can use carbon dioxide to make their voice sound higher pitched.
• Since density of air is immediately related to temperature, I would say yes to this as well.

EDIT: Since it seems that the question is not about the sound the tuning fork produces, the last two options are not valid.

• Thanks for your feedback. I agree with everything you've said. Unfortunately that "selection" of answers is incorrect (according to my multiple choice lab question), so that's why I was asking here! Apr 7, 2013 at 14:56
• @YadaY Okay, that means that the density of air are wrong, and what the question really asks is the frequency of the tuning fork itself, as opposed to the sound it produces! Which means, A and B are the correct answers. Apr 7, 2013 at 15:18
• @YadaY: Glad to be of assistance :) Apr 7, 2013 at 16:19