# Machine Shop 2 and Acoustics Tutorial

Machine Shop II

Acoustics Tutorial

## Machine Shop II

Welcome to the third lab in E5, where you will begin to fabricate parts for your group's instrument in the Machine Shop in the basement of Papazian (next to Hicks).

A reminder of the lab groups :

 Lab A Lab B Lab C Group A1: Pickup and whammy bar Omari Faakye Nick Kitten Carl Shapiro Johnny Tompkins Group B1: Master program Madeleine Abromowitz Pam Costello Justin Hughes G Patrick Lisa Schumacher Group C1: Right-hand mechanism Julia Luongo Cyrus Stoller Andrew VanBuren Meghan Whalen Group A2: Left-hand mechanism Jennie Hatch Tane Remington Sofia Saiyed Raúl Santos Group B2: Left-hand interface David Burgy Yannick Lanner-Cusin Levi Mahan Travis Rothbloom Bo Sun Group C2: Right-hand interface Ariel Horowitz Ross Pustell Lorenzo Ramirez Samantha Tanzer Group A3: Left-arm positioning Matt Allen John Leonardy Zsolt Terdik Ben Zhong

## Acoustics Tutorial

At the end of class on Tuesday I began to talk about the basic physics of how our musical instruments work. Very briefly, all musical instruments produce periodic vibrations which register as "pitch" to human listeners. The more periodic(repetitious) the waveform, the stronger the sense of pitch. If the frequency components that make up the sound are related by integer multiples (harmonics), then the waveform will be very periodic. If the frequency components are inharmonic, then a "clang" component to the sound will be heard, and the sound will have less of a sense of what note is being produced (think of snare drums vs. tom-tom drums vs. kettledrums).

For strings fixed and tubes open at both ends, the boundary conditions on both ends are identical and the one-dimensional wave equation has a solution consisting of multiples of half-wavelengths. Each mode of vibration occurs at a frequency given by f = nv/2L, where n is an integer n = 1, 2, 3, 4, ... corresponding to the harmonic number (where the lowest note n=1 is the fundamental frequency). L is the length of the string or air column and v is the wave speed (v = 343 m/s for air at 25°C, or v = sqrt(T/rho) for a string of mass-per-unit-length rho and tension T). The fundamental frequency is what is usually perceived as the pitch of the instrument, while the relative amplitudes of the harmonic modes affect the timbre (pronounced "TAM-ber") or tone color of the instrument.

As I mentioned on Tuesday, the location of the point at which the string or air column is excited or stopped affects which of the possible modes (and their corresponding frequency components) are allowed to remain in the vibration recipe. Thus a pressure node (negligible fluctuation in acoustic pressure) occurs wherever a hole is placed in air-filled tube, and a displacement node occurs wherever a player's finger rests lightly on a string. Only modes with nodes at these points are allowed to exist; the lowest-frequency one (usually) determines the perceived pitch. Conversely, the pluck or bow-contact point of a string eliminates from the vibration recipe all those modes with nodes there. For instance, plucking or bowing a string 1/5 of the way along its length eliminates the 5th harmonic (because it has a node there), as well as the 10th, 15th, 20th, 25th, etc. harmonic (because they too have nodes there).

Click here for information on the relationships between different notes in the equal-tempered scale.

 Comments or Questions?  Send me email Carr Everbach Engineering Department Swarthmore College