Physics in the Arts Lecture S3 Strings I Bach s Cello Suite No 1 in G Major Yo yo Ma The Plan This week strings Next week spring break no class no labs April 6 8 sound waves and pipes April 13 15 superposition beats harmony and musical scales April 20 22 Fourier analysis musical instruments sound perception timbre loudness pitch April 27 29 Exam review Miguel and exam 3 Musical Notes from Vibrating Strings Wikipedia Strings Magazine Wikipedia qualitystrings com Transverse and Longitudinal Waves Transverse Wave Oscillations perpendicular to wave velocity Water waves Electro magnetic Waves britannica com Transverse and Longitudinal Waves Transverse Wave Oscillations perpendicular to wave velocity Water waves Electro magnetic Waves britannica com Longitudinal Wave Oscillations in direction of wave velocity Sound Iowa PBS pressure waves Transverse and Longitudinal Waves Transverse Wave Oscillations perpendicular to wave velocity Water waves Electro magnetic Waves britannica com Longitudinal Wave Oscillations in direction of wave velocity Sound Iowa PBS pressure waves Demo Ropes and Slinky Transverse Pulse Along Taut String Pulse velocity v Force F Transverse Pulse amplitude Rope length L Transverse Pulse Along Taut String Pulse velocity v Force F Transverse Pulse amplitude Rope length L Pulse velocity is determined by the rope and the force tension applied v F L Force measured in Newton s F Linear mass density L N kg m s2 mass of rope Length L of rope in units kg m Pulses Travel in Waves From last time Simple Harmonic Oscillation in time Period T time s t Pulses Travel in Waves From last time Simple Harmonic Oscillation in time Period T time s t Waves also oscillate in space Wavelength distance m x Pulses Travel in Waves From last time Simple Harmonic Oscillation in time Pulses are combinations of waves with di erent periods and wavelengths velocity is same for all v T Period T time s t Waves also oscillate in space Wavelength distance m x Pulses Travel in Waves From last time Simple Harmonic Oscillation in time Period T time s t Waves also oscillate in space Wavelength distance m x Pulses are combinations of waves with di erent periods and wavelengths velocity is same for all v T sin 360 t sin 360 t T T x x Right moving Left moving Standing Waves Left moving wave gets re ected where it is attached and becomes right moving Interference of incoming and re ected waves produce standing waves https physics bu edu duffy HTML5 transverse standing wave html This work by Andrew Duffy is licensed under a Creative Commons Attribution NonCommercial ShareAlike 4 0 International License Standing Waves Left moving wave gets re ected where it is attached and becomes right moving Interference of incoming and re ected waves produce standing waves sin 360 t T Left moving Right moving sin 360 t x x sin 360 t T 2 cos 360 x T https physics bu edu duffy HTML5 transverse standing wave html This work by Andrew Duffy is licensed under a Creative Commons Attribution NonCommercial ShareAlike 4 0 International License Standing Waves Left moving wave gets re ected where it is attached and becomes right moving Interference of incoming and re ected waves produce standing waves sin 360 t T Left moving Right moving sin 360 t x x sin 360 t T 2 cos 360 x T https physics bu edu duffy HTML5 transverse standing wave html This work by Andrew Duffy is licensed under a Creative Commons Attribution NonCommercial ShareAlike 4 0 International License More complicated standing waves look like this Rope length L Figure credit Carrie Francis More complicated standing waves look like this Figure credit Carrie Francis A single belly has length If there are n 1 2 3 4 bellies then 2 Rope length L 2 n L More complicated standing waves look like this Figure credit Carrie Francis A single belly has length If there are n 1 2 3 4 bellies then 2 Rope length L 2 n L So only discrete wavelengths exist 2L 2L 2 2L 3 2L 4 Frequencies of standing waves f v 1st 2nd 3rd 4th 5th partial Fundamental partial overtone partial overtone partial overtone partial overtone 1st 2nd 3rd 4th Figure credit Carrie Francis Frequencies of standing waves f v f1 v 2L 1st 2nd 3rd 4th 5th partial Fundamental partial overtone partial overtone partial overtone partial overtone 1st 2nd 3rd 4th Figure credit Carrie Francis Frequencies of standing waves f v 1st 2nd 3rd 4th 5th partial Fundamental partial overtone partial overtone partial overtone partial overtone 1st 2nd 3rd 4th Figure credit Carrie Francis f1 v 2L f2 2 f1 f3 3 f1 f4 4 f1 f5 5 f1 and so on Frequencies of standing waves f v 1st 2nd 3rd 4th 5th partial Fundamental partial overtone partial overtone partial overtone partial overtone 1st 2nd 3rd 4th Figure credit Carrie Francis f1 v 2L f2 2 f1 f3 3 f1 f4 4 f1 f5 5 f1 and so on Frequencies of standing waves are just the integral multiples of fundamental frequency f1 The Pitch of a String When plucking a string most audible frequency is usually the fundamental frequency but many others can still be heard L yamaha com Voicing Formula f1 v 2L 1 2L F L L L qualitystrings com Three Ways to change the Pitch Voicing Formula f1 v 2L 1 2L F L 1 Adjust the tension Doubling the tension increases pitch by factor 2 1 4 Three Ways to change the Pitch 2 Adjust the string length Voicing Formula f1 v 2L 1 2L F L Doubling the length decreases pitch by factor 2 L L qualitystrings com Three Ways to change the Pitch 2 Adjust the string length Voicing Formula f1 v 2L 1 2L F L Doubling the length decreases pitch by factor 2 L L Guitar strings all have more or less the same length but vibrating portion of string is changed by pressing frets qualitystrings com guitarworld com Three Ways to change the Pitch 3 Adjust the linear mass density Could change material or simply the string s diameter Voicing Formula f1 v 2L 1 2L F L Three Ways to change the Pitch 3 Adjust the linear mass density Voicing Formula f1 v 2L 1 2L F L Could change material or simply the string s diameter L material density unit kg m3 string s cross section area unit m2 string Area 4 diameter 2 cross section diameter D diameter D Three Ways to change the Pitch 3 Adjust the linear mass density Voicing Formula f1 v 2L 1 2L F L Could change material or simply the string s diameter L material density unit kg m3 string s cross section area unit m2 string Area 4 diameter 2 cross section diameter D diameter D Doubling the string diameter quadruples its linear mass density So pitch decreases by factor 2 yamaha com The low E