Harmonics

Strings as Harmonic Oscillators Its mass gives it inertia Its mass gives it inertia Its tension and curvature give it a restoring force Its tension and curvature give it a restoring force It has a stable equilibrium It has a stable equilibrium Its restoring force is proportional to displacement Its restoring force is proportional to displacement

Modes of Oscillation Fundamental Vibration (First Harmonic) String simply vibrates up and down String simply vibrates up and down Frequency of vibration ( pitch) is Frequency of vibration ( pitch) is proportional to tension proportional to tension inversely proportional to length inversely proportional to length inversely proportional to density inversely proportional to density

A series of frequencies that includes the fundamental frequency and integral multiples of the fundamental frequency. A series of frequencies that includes the fundamental frequency and integral multiples of the fundamental frequency. Harmonic Series λ = 2L, f 1 λ 2 = L, f 2 = 2 f 1 λ 3 = 2/3 L, f 3 = 3 f 1

To Calculate the harmonic series of standing waves on a vibrating string f n = n (ν/2L) frequency harmonic number

Air Column as Resonant System A column of air is a harmonic oscillator A column of air is a harmonic oscillator Its mass gives it inertia Its mass gives it inertia Pressure gives it a restoring force Pressure gives it a restoring force It has a stable equilibrium It has a stable equilibrium Restoring forces are proportional to displacement Restoring forces are proportional to displacement f0f0 2f 0 3f 0

Air Column Properties An air column vibrates as a single object An air column vibrates as a single object Pressure node occurs at center of open column Pressure node occurs at center of open column Velocity antinode occurs at ends of open column Velocity antinode occurs at ends of open column Pitch (frequency of vibration) Pitch (frequency of vibration) inversely proportional to column length inversely proportional to column length inversely proportional to air density inversely proportional to air density Length

Air Column Properties Just like a string, an open air column can vibrate at many different frequencies (harmonics). Just like a string, an open air column can vibrate at many different frequencies (harmonics). f n = n (ν/2L)

A pipe with 1 closed end only generates odd numbered harmonics A pipe with 1 closed end only generates odd numbered harmonics pressure nodes occurs at sealed end pressure nodes occurs at sealed end Air Column Properties λ = 4L λ 3 = 4/3 L λ 5 = 4/5 L

Calculating the harmonic series of standing waves on a closed pipe f n = n (ν/4L) frequency harmonic number