Does the frequency of a tuning fork depend on the strength by which it is struck?
If yes, then how was the actual frequency(stated on the tuning fork) calculated? Was there a defined or average human strength
I found some resources at this site itself to understand this but it was still confusing
Does the frequency of a tuning fork depend on the strength by which it is struck?
No. To a first approximation, a tuning fork’s frequency is determined by its mass, stiffness, and shape, which are fixed. The approximation is good for weak strikes; if you strike it really hard, the approximation becomes worse and it can affect the frequency, so be gentle with your tuning fork.
It’s kind of similar to a swingset. You can push your child gently or forcefully in the swing, but the swing “wants” to swing at a frequency determined by its length and the strength of Earth’s gravity. Objects tend have “natural” frequencies at which they oscillate, determined by intrinsic properties like their mass and their restoring forces.
Another example is a pendulum (which is like a swingset). For small amplitudes of oscillation, its frequency is independent of the amplitude. You can start a pendulum swinging, count time with it, and as the swings slowly diminish they will keep marking the same time intervals.
I think that as far as wave mechanics go, energy (strength = energy) put into a system usually correlates with amplitude, not frequency. I'd support this claim by using logic; a tuning fork would be useless if energy dictated frequency, because every time you hit the tuning fork you'd hear a different frequency, and you can't guarantee that you'll hit the tuning fork with the same strength every time. The structure of the fork, I'd assume, has more to do with frequency than strength does. Although, when hit with exceeding strength the tuning fork may 'overextend' which could effect the rate at which it oscillates, causing a minor change in frequency, but the result would be relatively negligible from the perspective of the user.
Measurements on the topic for this forum can be done by holding a $8 tuning fork in front of a mic, then analyze the sound file with the free program "SpectraLabs" using the "Waterfall" display and using the built in Goertzel frequency extraction algorithm. That will give accuracy to around 0.01Hz. This website does not allow me to post evidence like data files or graphs or sound files, so you will have to do it on your own if you want to see. On a graph from my experiment on a hand held fork I measured in 2015 I showed that over the ~minute the fork was oscillating the pitched dropped approximately linearly from 524.091Hz to 524.082Hz, end of factual information. Now I will explore that result with the much less useful philosophy technique appropriate to this website: I've searched for the frequency shift versus amplitude of 261Hz to 523Hz forks in the milliHertz range, but it is very difficult to measure because competing effects at that level: even doppler effects from the motion of my hand holding the fork in front of the microphone creates frequency shifts of a few centiHertz, and energy of the moving tines converts into heat in the metal which could change the frequency, any temperature gradient in the room pulls the pitch at the level of 1000ppm per 10 degrees Fahrenheit. If the fork is held in a vice to end doppler effects then it dies out too quickly. If I hang it from a string then it swings from the striking to start its oscillation. I could start it ringing with an electromagnet oscillating at the fork's frequency, but haven't done that yet, afraid it will be hard to get large enough amplitudes. But so far I know the pitch change with (normal) amplitudes is below 0.1Hz and below 1 cent. However it must of course be there at some level. 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 be lower 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.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 (thus pendulum clocks were run in evacuated cabinets, the lower the energy drain from an oscillator the better its stability; the opposite is a system that dies out quickly with a "chirp" of ambiguous frequency). 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. Thus tuning forks work well for setting intonation of musicians. The forks are often stamped with a frequency precise to 0.1 Hz, and that is in fact around the limit to their reliability, which gets encumbered by a multitude of effects below that level. (Technical note: above in many places I tried to avoid using the concept of " oscillator's Q" by substituting for it "how fast the oscillations die out". I actually mean "Q of the oscillator". rather than "die out rate", but most people do not know of "Q" and so would have needed to be introduced by a long discussion. The two are normally correlated. Tuning forks have a very high "Q" of around 100, which is exactly the reason they go "hum" instead of "boing" when they are struck and why they are good frequency references, the point of all this talk. Quartz crystals have Q of about 10,000, Rubidium clocks about 100 million, and Cesium clocks have Q of over a billion, thus they are all even better references than tuning forks.)
