Physics in the Arts Lecture S4 Strings II J S Bach Violin Concerto in E Major demo guitar and SpectrumView Exercise 1 A guitar string has diameter D 0 011 0 028cm www stringsbymail com The mass density of stainless steel is 8 g cm3 8000 kg m3 What is the linear mass density of the string tablet Exercise 1 solution L 4 D2 4 8000 kg m3 0 028 0 01m 2 0 00049 kg this is 0 49 g m m Exercise 2 Our electric guitar is a taut string with tension F 19 6 lbs kg kg 9 81 m s2 86 N www daddario com What is the wave speed along the string What is the frequency of the fundamental mode if the length is L 0 64m tablet Exercise 2 solution wave speed v F L 86N 0 00049 kg m 418 m s fundamental mode f1 v 2L 418 m s 2 0 64m 327 1 s 327Hz This is indeed around the correct frequency of the high E string Idealized string Real strings are stiff in nitely thin Force F only restoring force from tension Neglecting damping we get exact harmonic modes f f1 2 f1 3 f1 4 f1 Real string string cross section diameter D stretched compressed diameter D Sti ness Restoring force against bending Real strings are stiff Sti ness Restoring force against bending stretched compressed Including this e ect the wave speed depends slightly on the wavelength Frequencies of higher partials are a bit higher than in ideal string Inharmonicity diameter D fn nf1 nf1 3 128 D4E L2F n2 elastic modulus in E property of material Pa N m2 Fletcher Rossing The Physics of Musical Instruments The thicker the string the worse this gets Real strings are stiff For our high E string from earlier D 0 011 0 028cm F 86 N L 0 64m Stainless steel E 200 000 000 000 N m2 Inharmonicity of 7th partial f7 7f1 7f1 0 00043 If the low E string were just four times thicker we would nd f7 7f1 7f1 0 11 Real strings are stiff For our high E string from earlier D 0 011 0 028cm F 86 N L 0 64m Stainless steel E 200 000 000 000 N m2 ideal high E ideal low E sti high E The di erence is completely inaudible sti low E This sounds horrible This is the most important reason why the low strings need to be wound Plucking a String What combination of fundamental frequency and overtones is heard depends on how a string is plucked Plucked in middle Plucked closer to bridge Plucked very close to bridge fundamental 1st overtone 2nd overtone 3rd overtone 4th overtone 5th overtone 0 25 0 5 0 75 1 0 25 0 5 0 75 1 0 25 0 5 0 75 1 Warm tone mostly fundamental Full tone includes higher modes Bright tone many higher modes https tidal com track 2178486 u Paul Chambers 1935 1969 https tidal com track 1277485 u Larry Graham Which bass playing sounds warmer Which brighter Missing Partials Modes that have nodes at plucking location get suppressed Larry Graham Plucked in middle Plucked 1 3 to bridge 0 25 0 5 0 75 1 0 25 0 5 0 75 1 fundamental 1st overtone 2nd overtone 3rd overtone 4th overtone 5th overtone Lightly Touching a String If one plucks a string and lightly touches it somewhere the string is muted heavily damped Touching it at location of nodes of standing waves the corresponding modes aren t damped Figure credit Carrie Francis Lightly Touching a String If one plucks a string and lightly touches it somewhere the string is muted heavily damped Touching it at location of nodes of standing waves the corresponding modes aren t damped By touching a string at speci c positions one can isolate overtones Figure credit Carrie Francis demo guitar and SpectrumView