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Oscillating Strings a String Can Oscillate As a Harmonic Oscillator It Has Mass (Inertia) When Stretched Taut It Has a Stable Eq

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Oscillating Strings a String Can Oscillate As a Harmonic Oscillator It Has Mass (Inertia) When Stretched Taut It Has a Stable Eq Music Oscillating strings A string can oscillate as a harmonic oscillator It has mass (inertia) When stretched taut it has a stable equilibrium position When displaced from that equilibrium position, it experiences a restoring force due to the tension in the string The frequency of oscillation is independent of amplitude The strength of the restoring force is set by Tension Length The strings inertia is set by Mass per unit length A vibrating string – fixed at both ends A string can vibrate in different patterns Each pattern is called a mode of oscillation String can vibrate in a single arc – fundamental vibrational mode An antinode in velocity occurs in the middle of the string A node in velocity occurs at each end of the string The frequency of this vibration Increases with tension (tight strings oscillate quickly) Decreases with length (long strings oscillate slowly) Decreases with mass per length (heavy strings oscillate slowly) A vibrating string – fixed at both ends – harmonics Higher-order modes of vibration String can vibrate as two half-strings – second harmonic Two antinodes Three nodes (both ends and the middle) Oscillation frequency is twice the fundamental frequency String can vibrate as three third-strings – third harmonic Three antinodes Four nodes Oscillation frequency is three times the fundamental frequency Driving the oscillation Pluck the string – transfer energy to the string all at once Bow the string – transfer energy to the string gradually Shaking the string at the right frequency transfers lots of energy Less damping (longer oscillation) makes this more effective A bow grabs and releases the string at just the right frequency Often strings vibrate in a mixture of modes, giving the instrument its character Sound Travels as a density wave in air – alternating patterns of compressions and rarefactions Air at uniform density is in a stable equilibrium When disturbed pressure imbalances produce restoring forces Inertia of the air resists acceleration In open air the disturbance travels through the air as a traveling wave A confined system supports standing waves Traveling waves Oscillations are local, but disturbance travels Longitudinal wave – Oscillations are in the same direction that the disturbance travels – Sound Transverse wave – Oscillations are perpendicular to the direction that the disturbance travels – Waves on a stretched string Wavelength – distance between two high pressure regions measured at one instant in time Frequency – number of times the pressure oscillates per unit of time measured at one point in space Wave velocity – speed and direction of disturbance wave speed = wavelength x frequency Producing sound Strings don’t move much air – air flows around the string – not much sound Surfaces move a lot of air – air can’t flow around the surface – lots of sound Many instruments couple strings to surfaces to produce more sound Surfaces vibrate in different modes – two dimensional patterns, which can be much more complicated Confining air in a pipe The air in a pipe can support standing waves Open – Open pipe Pressure node (velocity antinode) at both ends Fundamental – pressure antinode in the middle Wavelength is twice the pipe length Overtones at 2x, 3x, 4x, … the fundamental frequency Open – Closed pipe Pressure antinode (velocity node) at closed end Pressure node (velocity antinode) at open end Vibrates like half an open – open pipe Fundamental – wavelength is four times the pipe length Overtones at 3x, 5x, 7x, … the fundamental frequency .
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