Nobel Lecture: The dilemma of attribution

I suspected that there were some members of the live audience who were somewhat apprehensive about sitting through the morning’s physics lectures. After all, there were three guys there to talk about one minus sign. If it were just two people and a plus sign, +, one could talk about the | and the other about the —. However, to my mind, this year’s awards represent or symbolize not just a minus sign but a large body of significant advances in our understanding of fundamental physics and are the work of not just three people but a great many scientists, stretching out over many years and many countries. This is really a prize for that whole community

The plucked string: an example of non-normal dynamics

Motion of a single Fourier mode of the plucked string is an example of transient, free decay of coupled, damped oscillators. It shares the rarely discussed features of the generic case, e.g., possessing a complete set of non-orthogonal eigenvectors and no normal modes, but it can be analyzed and solved analytically by hand in an approximation that is appropriate to musical instruments' plucked strings.Comment: 18 pages, 3 figures; revised to include reference to additional examples of transient growth and non-normal dynamics and expanded to include the most general coupling of vertical to horizontal string modes and not just the simplest example presented earlie

Banjo Break Angle Tension Modulation as Parametric Oscillation

The motion of the floating bridge of the banjo, in conjunction with the break angle of the strings over that bridge, produces string tension modulation that is first order in the amplitude of the string motion. This note refines a previous suggestion regarding the impact on the frequencies of the strings’ and bridge’s motion. For a given mode frequency pair of string and bridge, the resulting tension modulation produces a new, additional motion characterized by the sum and difference of the original ones.Strictly speaking, this corresponds to canonical “frequency modulation” only in the limit of modulation slow compared to the string frequency. The more general result is precisely an example of what is known as “parametric oscillation,” first analyzed by Rayleigh. The qualitative impact of tension modulation on banjo timbre remains as suggested previously. It is only the precise math and physics that warrants this correction

The Resonator Banjo Resonator, part 2: What makes them really crack?

A simple experiment quantifies the difference between the sound production of a banjo with and without a resonator back. Driven by a small tweeter mounted inside the pot, for frequencies above about 4500 Hz, the produced external sound is 6 to 10 dB louder with the resonator than without. With the banjo played in any normal fashion, this gives a negligible contribution to the overall volume. However, that difference is clearly a reflection of the universally recognized resonator sound, in close analogy to plosive consonants in human speech. No direct correlation is observed between the head-resonator separation and the spectrum of the enhanced response. This suggests that direct reflection off the back is not a primary contributor to the resonator/openback difference, leaving differences in overall absorption as the major suspect.Comment: 19 pages, 6 figures, 4 mp3 file

Banjo Drum Physics - theoretical preliminaries

The interaction of a drum's head with its enclosed air is presented in the simplest possible form appropriate to the questions and issues that arise in understanding the timbre of the banjo. The inherent air-head impedance mismatch allows treating the head as driver of the air and the air's effect, in turn, as back reaction. Any particular question can then be addressed with a calculation in simple wave mechanics. The analysis confirms and quantifies the notion that internal air resonances enhance the response of the head at its and their frequencies. However, the details of just how are fairly complicated.Comment: 31 pages, 13 numbered figures + 2 on page

Physics of the Bacon Internal Resonator Banjo

The internal resonator banjo, patented and first sold by Fred Bacon around 1906, remains something of a cult favorite and is still produced by some independent luthiers. According to enthusiasts, the characteristic design elements produce a sound that is mellower, richer, and of greater complexity and presence than without them. Aspects of that sound are studied here, comparing instruments that are otherwise identical and identifying physics mechanisms that are likely responsible.Comment: 24 pages, 21 figures, 3 linked mp3 sound file