# How do I determine the physical size of a quartz crystal of a given shape and vibrational mode needed to approximate a specific resonant frequency?

So, one of my D&D characters has a cut/relief-carved quartz crystal amulet (medallion shaped) that's both an arcane focus (crystal) and a holy symbol (amulet), and I thought it would be a good idea for it to resonate within the audible range, given that quartz crystals can make good, pure-tone mechanical resonators. However, I can't find much on what size a quartz resonator needs to be for a given shape, vibrational mode, and frequency.

In my case, I'm thinking this would be a flexural mode resonator for practicality and probably audibility/volume as well, made in a cylindrical disk shape (with relief carvings on both sides and a hole in it for wearability) to serve as an amulet, and with a resonant frequency somewhere in the audible range (preferably somewhere from a few hundred Hz to 15kHz). Considering that a reasonable maximum size would be somewhere around 25mm in diameter and 5mm thick, what actual size would be needed to reach the audible frequency range?

• I believe the resonant frequency has a lot to do with the speed of sound in the resonant medium (5.8km/s), if the size is a multiple of 1/4 of a wavelength then boom, resonance (although you want 1/2 a wavelength so it only vibrates in the middle). Based on the size of this old military 2kHz quartz crystal your amulet would probably be... amulet sized. The catch is that thicker resonators are stiffer, i.e. they move less when resonating, i.e. they're going to make less noise (more movement = louder noise) Dec 21, 2019 at 21:21
• @Samwise -- what would the minimum thickness be for an amulet-sized quartz disc to be robust enough to withstand an adventurer's lifestyle? Dec 21, 2019 at 21:28
• "Minimum thickness": very thin, supposing that the adventurer knows a good jeweller who can mount the crystal in a suitable frame. Dec 21, 2019 at 22:07
• @AlexP ah, interesting point re: the frame. What would such a frame look like? I was originally thinking it'd be a frameless setup, but I don't think there'd be any objection to a frame being a thing Dec 21, 2019 at 22:10
• Ideally, the frame would be in the shape of Euterpe, the muse of music. Dec 21, 2019 at 22:12

In the general case, you can't, because there's no closed form of computing the vibration modes, except for the very simple and very uniform geometries. Even then some 'weird' function may need to be invented - for example, the vibration modes for a circular membrane will use the Bessel functions - which are defined as "they are the elementary solutions for the circular membrane vibration equation, all the other vibrations mode will be a combination/superposition of them"

To make the matter worse for your case, the carving in an amulet will drastically modify these vibration modes. What appears yo you as just a small indentation will modify the timbre of the vibrating element, by tuning down some harmonics. With a careful adjustment of such "engraving", one can tune down the fundamental frequency and enhance the first harmonic (harder to do then the reverse, but still possible).

As an example in which even simple vibrating elements are tuned, go no further than tongue based instruments, in which the vibration is the one of a "tongue" with a clamped end and the other free (for example slit drums - some with an exceptionally well tuned sound).
Some may tell you it not a big deal to tune it, that it's actually a brain dead simple to make and tune one. Yet, if you really want a fine one, you are going to have surprises of the unpleasant kind.

Even harder when you get on making something from solid surfaces - it will take days for someone already an expert to make one.

Bottom line - it's hard because the vibration modes for anything else but simple shapes and homogeneous materials are not entirely captured by science in easy formulas that are amenable to engineering. This is where artistry, craftsmanship (and maybe magic) have lots of room to play.

• Yeah, I'm trying to simply get the approximate fundamental frequency for a given size and mode, we can disregard the carvings for that approximation even I reckon Nov 13, 2021 at 2:34
• Highly depends on the shape. With circular shapes (and no holes), you go with the Bessel functions that I linked above (take the fundamental mode). For elliptic shape, you go with the equivalent Mathieu functions. The rectangular membranes are easier. Nov 13, 2021 at 2:54
• fr sufficient audible sound it also has to have sufficient change in its shape or have some amplifier - that's why a plate may sound, aka glass dish or a glass cup, being a sort of membrane, but same size chunk of same material won't produce that much sound. yeah, the question is anything but trivial, but interesting. I think the answer lacks any example which does not require clicking links, I mean a glass rod will make sound, even if it won't work for op's sizes but it is a starting point at least. nice youtube link btw. Nov 13, 2021 at 13:29

A modern 20kHz tuning fork crystal, as used in electronics, is housed in a cylinder 6mm in length and 2mm in diameter. Assuming the frequency scales linearly with size, you would probably get something up to the size of a thumb depending on desired frequency.

• This answer fails to take into account that tuning-fork crystals are named such because they are physically cut in a fork-like shape, and vibrate accordingly as a result. This presentation from a crystal manufacturer is a good summary of how they work. Mar 26, 2020 at 22:19
• I think you are right on the money with the tuning fork crystals, and Shalvenay has a perfect link for the background of those things. Looking at that material, a hollow, thickwalled disk ( for protection) with one or more internal forks and a small-aperture hole for listening would be ideal. The building of such a magnificent contraption from one piece would surely be only possible with elven magics and dwarven cunning, but if one were to manufacture it as two pieces, and later glue it, even a gifted artificer might be able to produce it.... Feb 24, 2021 at 20:48
• The cylinder has very little to do with the frequency. If you cut the cylinder open you'll see the crystal takes up a very small amount of the cylinder. The resonance frequency of a quartz crystal is based on its size, shape, and the electric charge applied to it.
– stix
Feb 24, 2021 at 21:05