Marimba resonator reconditioning

[bobcube]

I have a concert marimba where the resonator tubes are stopped at the bottom by wooden plugs. Over the years, several of these have shrunk and dropped out.
I will probably make new plugs by 3D modelling.

I have searched the web for some way of calculating the likely position of the plugs in each of the affected resonator tubes, but can't seem to get a sensible result.

Indeed the related thread on this subject, which used the formula 1/4 wavelength - 0.61r, where 'r' is the tube radius, givse a result that's nowhere near any of the existing (working) resonators.

It's quite difficult to tune by ear for various reasons, so can anyone provide a simple formula which I can use to calculate the approximate position of a stopper in a tube,  for a given note frequency in a tube of known diameter?
 

[Jim McCarthy]

Well that formula - 1/4 wavelength -0.61r is as close as you can get in my experience. A couple of things to think about if it doesn't seem to match reality. 1. measure the actual frequency of your bars - bars are not always in tune so can drift away from resonators. 2. Be sure to consider both tuning standard (ie A440? 442?) and temperature/humidity. Remember W=V/F where F=frequency and V=velocity of sound - but that will differ depending on the air temp and humidity. Plugs can have a "seal point" that is not quite as it seems. Ideally we are looking for a perfect flat resonator floor that seals at the exact boundary -  but that is often not the case. That top surface of the plug closest to the bar is where we are measuring to of course - but if the shape is slightly concave or convex or has a bump or if the seal point is a bit further down in the middle of the plug... it will make the direct measurement inaccurate.

My best advise if you are making new plugs, is to make adjustable ones so you can move it till you get it working well with the bar. The simplest way is to use a holesaw to cut out 2 SLIGHTLY undersize disks of 3/4" plywood - find an O ring that is pretty much a perfect OD fit to the ID of your tube (thicker is better like about 3/16" or 1/4" rubber diam) Insert and glue in a nut to your top disk - bolt with a couple of locked nuts from the bottom so tightening the bolt squeezes the O ring to lock position and enough bolt protruding to use as a handle. That's about the simplest and cheapest way.

[bobcube]

Thanks for the reply Jim.

I've chewed my pencil, and discovered I made a maths error. Tsk tsk..!

All my calculations were in millimeters, but I overlooked converting the wavelength from cm to mm. Which obviously gave me a much smaller measurement!

For the record, a couple of currently OK keys give the following results
           current resonator stopper depth     formula result     wavelength   interna tube radius
G#4                        182 mm                            194 mm                83 cm                22.5 mm
A#4                        164 mm                            172 mm                74 cm                22.5 mm

Tha marimba seems to have A=444 hz, so slightly above UK concert pitch, which may partly explain the difference between measured and formula result?

But, from what you say, it seems necessary to do the final tuning by ear, unfortunately.