Stringed instruments L = String length V = Speed of a wave on a string F = String vibration frequency = .5 V / L D = String diameter A = String cross-sectional area Q = String density T = String tension (Newtons) P = Tensile stress = T / A Pmax = Maximum string tensile stress before breaking = Tensile strength Z = Material strength-to-weight ratio = Pmax / Q Pi = 3.14159 K = Pi L^2 F^2 The speed of a wave on a string is V^2 = T / (Q*A) Hence T = Q D^2 Pi L^2 F^2 = Q D^2 K Tension = Density * Diameter^2 * K where K = Pi L^2 F^2. For example, for a viola C string, you can vary tension, density, and diameter, but you can't vary K. K is fixed by the frequency and length of the string. Transform to dimensionless units. T = t * 100 Newtons Q = q * 10000 kg/meter^3 D = d * .001 meters K = k * .0001 t = q d^2 k String Freq Length Characterisic k (Hz) (mm) tension (N) Violin G 195.6 320 50 1.231 Viola C 130.4 388 60 .804 Cello C 65.2 690 140 .636 Bass E 41.2 1060 160 .599 Guitar E 82.5 650 140 .903 Bass guitar E 41.2 860 220 .394
Maximum frequency of a string T = Q D^2 F^2 Pi L^2 --> F^2 = 4 L^-2 T / (Q*A) The maximum frequency of a string is determined by the strength-to-weight ratio Z. Z = Tensile strength / Density MaximumFrequency^2 = .25 L^-2 Z Tensile Density Z/10^6 strength g/cm^3 GPa Nylon .045 1.15 .04 Gut .2 1.5 .13 Magnesium alloy .4 1.8 .22 Titanium alloy .94 4.5 .21 Steel alloy ~ 2 7.9 .25 Tungsten .55 19.25 .029 Kevlar 3.6 1.44 2.5 Zylon 5.8 1.5 3.9 Carbon nanotube 7 .116 60.3 Technology not yet developed A space elevator requires a material with Z > 100
Maximum frequency (Hertz) Gut Steel Zylon Carbon nanotube Violin 563 781 2960 12100 Viola 465 644 2440 3160 Cello 261 362 1370 5620 Bass 170 236 895 3660 Guitar 277 385 1519 5973
Density Density Price g/cm^3 $/g Water 1.0 Gut ~ 1.5 Synthetic ~ 2.5 Aluminum 2.8 <.01 Titanium 4.5 .01 Steel 7.9 <.01 Nickel 8.9 .01 Silver 10.5 .6 Tungsten 19.2 .05 Gold 19.3 24 Rhenium 21.0 6 Platinum 21.4 88 Iridium 22.4 13 Osmium 22.6 12 Densest element Tungsten is the only dense metal that is not expensive.
Tungsten string diameters Density = 19.25 g/cm^3 Diameters in mm. Freq Violin Viola (Hz) (mm) (mm) E 660 .14 .12 A 440 .20 .18 D 293 .31 .28 G 196 .46 .42 C 130 .69 .62 F 86.9 1.03 .93 Bb 57.9 1.55 1.40 Eb 38.6 2.33 2.10 Ab 25.8 3.49 3.15 Freq Cello (Hz) (mm) A 220 .32 D 147 .48 G 97.8 .71 C 65.2 1.07 F 43.5 1.60 Bb 29.0 2.47 Eb 19.3 3.61 Freq String Bass (Hz) Bass Guitar (mm) (mm) G 97.8 .50 .72 D 73.3 .66 .96 A 55.0 .88 1.28 E 41.2 1.18 1.70 B 30.9 1.57 2.27 F# 23.2 2.09 3.02 C# 17.4 2.79 4.03 G# 13.1 3.72 5.37 Freq Guitar (Hz) (mm) E 330 .22 B 248 .30 G 196 .38 D 147 .50 A 110 .67 E 82.5 .90 B 61.9 1.20 F# 46.4 1.60 C# 34.8 2.13 G# 26.1 2.84 A string can be made an octave lower by doubling the diameter. Diameter * Frequency = Constant
Guitar Tension = 140 Newtons String Freq Material Density Diameter (Hz) (g/cm^3) (mm) G 782.2 Zylon 1.5 .34 D 586.7 Zylon 1.5 .45 A 440.0 Zylon 1.5 .60 E 330.0 B 247.5 G 195.6 D 146.7 A 110.0 Tungsten 19.3 .67 E 82.5 Tungsten 19.3 .90 B 55.0 Tungsten 19.3 1.34 F# 36.7 Tungsten 19.3 2.01 C# 24.4 Tungsten 19.3 3.03
Characteristic tension Height of Height of String Tension (N) top string bottom string (mm) E A D G C (mm) (mm) Violin 320 80 50 45 45 3.2 5.2 Viola 388 65 55 55 55 4.8 6.2 Cello 690 160 130 130 130 5.2 8.2 Bass 1060 160 160 160 160