Physics 109 STRINGS Wave Speed on a String Formula v FT v FT mL Lm linear mass density Effects Tension FTFT Increase tension wave speed increases Decrease tension wave speed decreases Increase mass per length wave speed decreases Linear Density Change by Material Thickness diameter String Thickness and Density Diameter examples d 6d 6 A 9 A 9 d 12d 12 A 36 A 36 Key idea Doubling thickness cross sectional area quadruples So density mass length also quadruples Standing Waves on a String Interference creates nodes no motion and antinodes bellies max motion Formed by reflection at fixed ends Frequency in the hundreds of Hz Only specific wavelengths fit on the string Modes Partials Overtones 1st Mode Fundamental 1st Partial One antinode two nodes ends Frequency f1f1 2nd Mode 1st Overtone 2nd Partial Two antinodes Frequency f2 2f1f2 2f1 3rd Mode 2nd Overtone 3rd Partial Three antinodes Frequency f3 3f1f3 3f1 General fn nf1fn nf1 More modes shorter wavelengths higher frequencies Nodes and Bellies Nodes points of no movement Bellies points of maximum movement Mode shapes determine where these occur Pluck vs Touch Plucking The string must move where you pluck it Activates all modes that have a belly at the pluck location Any mode with a node at the pluck point is not activated Touching lightly Makes that point a node amplitude 0 Silences modes that would normally have a belly there Used to produce harmonics Pluck Positions and Activated Modes Pluck at End Activates all modes Strong fundamental Pluck at 12L21 L Activates modes without a node at L 2L 2 Silences even modes 2 4 6 Pluck at 14L41 L Activates modes without a node at L 4L 4 Silences modes 4 8 12 Wave Speed Example Guitar A string L 64 cm 0 64 mL 64cm 0 64m FT 129 NFT 129N 6 5 g m 0 0065 kg m 6 5g m 0 0065kg m a Wave speed v 1290 0065v 0 0065129 b Mass of string segment m L 0 0065 0 64m L 0 0065 0 64 Frequency Example Given Find L 0 64 mL 0 64 m FT 129 NFT 129 N 0 0065 kg m 0 0065 kg m Fundamental frequency f1f1 1st overtone f2f2 Uses formula fn nv2Lfn 2Lnv Activated odd modes 1 3 5 Silenced even modes Pluck at L 3L 3 Activates all modes except multiples of 3 Silences 3 6 9 Pluck at L 8L 8 Modes Activated by Plucking Positions Pluck at L 2L 2 Silences modes that have a node at 1 81 8 Modes 8 16 24