Parabolic curves in a Helmholtz solution for a bowed string

José L Brun,Javier B Gómez

doi:10.1088/1361-6404/aae8b5

Abstract

If one is not familiar with the physics of the violin, it is not easy to guess, even for an experimental physicist, that the so-called Helmholtz motion can be obtained as a solution to the one-dimensional wave equation for the motion of a bowed violin string. It is worth visualising this aspect from a graphical perspective without recourse to ordinary Fourier analysis, as has customarily been done. We show in this paper how to obtain the shape of the Helmholtz trajectory, that is, two mirror-symmetric parabolas, in the ideal case of no losses from internal dissipation and no viscous drag from the air and the non-rigid end supports. We also show that the velocity profile of the Helmholtz motion is also a solution of the one-dimensional wave equation. Finally, we again derive the parabolic shape of the Helmholtz trajectory by applying the principle of energy conservation to a violin string.

Highlights

The university physics curriculum traditionally pays little attention to acoustics in general, and even less to musical acoustics in particular
Many physicists and most musicians intuitively associate any wave on a string with the sinusoidal wave of textbook physics, the vibrations of a bowed string are in practice treated very differently, as Helmholtz solutions
As the bow is drawn across the strings of a violin, the motion of the bowed string is surprising, as a superb YouTube video shows [8]

Summary

Introduction

The university physics curriculum traditionally pays little attention to acoustics in general, and even less to musical acoustics in particular. We refer here to music for physicists (not to be confused with physics for musicians, which is a very noble enterprise) [1] Notwithstanding this general lack of attention, music is sometimes heard in lecturers’ offices, and musical instruments sometimes feature in students’ laboratory experiments, possibly. Many physicists and most musicians intuitively associate any wave on a string with the sinusoidal wave of textbook physics, the vibrations of a bowed string are in practice treated very differently, as Helmholtz solutions. It is worth noting that the movement of a plucked string, for example on a guitar, can be treated as a Helmholtz solution, this is quite different from the bowed case considered in this paper [9]. Regardless of the differences in the movement of bowed and plucked strings, their fundamental tone is the same (if the strings have the same length, )

The idealised string: simple Helmholtz motion

The Helmholtz motion as a solution to the wave equation

The elastic potential energy

The kinetic energy

Conclusion