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  1. Greetings Free Reed enthusiasts. I posted here about a year ago explaining an analysis I was undertaking on how the Western free reed works, starting from fundamental principles in Fluid Mechanics. I’m happy to announce that I’ve made progress to the point where resulting calculations reveal much about the physical working of this sound source, and I’d like to present some results here. Very briefly, the analysis starts with the Euler-Bernoulli wave equation for the vibration of the tongue and combines it with a physical model that describes the external forces acting on the tongue. These forces include the bellows pressure force, the viscous friction acting on the skin of the tongue produced by relative air flow, the aerodynamic “form drag” acting on the tongue, and the effect of the mean airflow through the slot. The model does not include inertial forces caused by air on the tongue, though it can be shown that these forces are small, especially if the vibration frequency is not too large. There are many different characteristics of this vibration, far too many to present here, but without getting into too much background or detail, I can present some interesting results. Many concertina players say they can hear a difference between the sound of brass tongues and steel tongues, and many of us have wondered how different materials would sound. In the case of Brass vs Steel, I uploaded a .docx file that viewers can view on a cloud storage site, with details below. This file contains harmonic amplitudes plotted on a log scale, in the same way that a Fourier spectrum is viewed. The amplitudes are normalized with respect to the fundamental. In order to understand what these harmonics are, consider first that the vibrating tongue can vibrate in different bending modes, each with its own frequency. All free reed vibration in musical instruments involve virtually only the first bending mode, in which the tongue bends in a smoothly curved fashion, with only one node, where vibration is zero, at the rivet. The fundamental frequency of the musical tone we hear is the same as the frequency of the first bending mode, also called a fundamental. These details are sometimes confused. The second bending mode, which can occur under spurious conditions, has a node at the rivet and a node roughly a third of the length away from the free end, and the tongue bends in snakelike fashion. Unlike a string, whose modes of vibration are easily excited and are spaced at frequencies that are closely integer multiples of the fundamental, the second mode of the tongue vibration is roughly six times the frequency of the first mode, with succeeding modes separated by growing frequency gaps. Now, within a given mode of vibration, the tongue can also vibrate with frequencies that are multiple integers – harmonics - of the bending mode frequency. Let’s focus on a tongue vibrating in its first bending mode, the usual case. Let’s also consider an isolated tongue, without the slot, vibrating as a simple cantilever. If the excitation is “gentle” enough, or when any initial “complicated” vibrations die out, the motion will be very close to that of a sine wave. This is why tuning forks sound so pure. However, when the tongue is placed over the slot and pushed by a bellows pressure, the excitation is not so gentle. The external forces on the tongue during its vibration change suddenly, depending where the tongue is in its cycle. These sudden, changing forces induce harmonics in the general motion, although compared to the fundamental motion, they are usually small. You can imagine, instead of the relatively gentle sinusoidal swinging, sudden minute lurches and hindrances superimposed on the motion. Although these adjustments are relatively small, they can be measured, and who can really say how much affect they have on air vibration, and thus on the sound we hear? And if you think about it, how else could brass make a different musical tone than steel, if not by the way it vibrates? We expect that somewhere in the frequency spectrum of the tongue motion, there are clues to why brass and steel sound differently. The plots in the .docx file compare the harmonic amplitudes of tongue motion between the usual ASTME 1095 spring steel and Brass 260, characterized only by their Young’s Modulus and density. Each plot is for a different bellows pressure, indicated on the plot. The log scale means there’s a factor of ten in displacement amplitude separating each integer on the vertical scale. For low bellows pressure, the first harmonic greatly dominates, indicating relatively pure sinusoidal motion. The 2.5 orders of magnitude on the plot is a factor of about 316 in amplitude between the fundamental and first overtone. Also, differences between steel and brass are unnoticeable. As bellows pressures increase, higher harmonics play larger roles, and the difference between brass and steel becomes evident. With minor exception, steel often dominates brass in the higher harmonics, especially for the lowest of those harmonics where differences occur, around the 4th or 5th. However, the first three harmonics or so never really show much difference. Because of the relative amplitudes, we are more likely to hear the differences in the 4th to 6th harmonics more than those in the higher harmonics. From these plots we see a demonstration of the “brighter” sound of steel, or the “mellower” sound of brass, with such differences becoming more pronounced at larger bellows pressure. But why does that occur? The answer to that lies in the details of the physical model, to be uncovered by further investigation. In short, it’s due to the nonlinear ways in which energy dissipation acts in the system. The URL for the .docx file is below, and any visitor should be able to view it. I’m not releasing it for download because I intend to publish these results in the future and I’d rather wait for public release. https://app.box.com/s/zchj1y5d4l65fttok3n4sofupzx2gsio Best regards, Tom www.bluesbox.biz
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