1. Harmonics Physics Chapter 13-3 Pages 494-503

2. A. Standing waves on a vibrating string Fundamental frequency – lowest frequency of vibration of a standing wave  Symbolized as f 1 Harmonic series – series of frequencies which are multiples of the fundamental frequency  f 2, f 3, f 4, …

3. Equation v = fλ f = v / λ Fundamental frequency of a string fixed at both ends  f 1 = v/λ 1 = v/2L L = string length

4. Harmonics (multiples of f 1 ) FrequencyWavelength f 2 = 2 f 1 λ 2 = L f 3 = 3 f 1 λ 3 = 2/3 L f 4 = 4 f 1 λ 4 = ½ L

5. Harmonic Series of standing waves on vibrating string f n = n (v/2L) n = 1, 2, 3, … n = harmonic # v = speed of the waves on a string L = string length f = frequency

6. * If you put a finger down on a string, now only part is vibrating and a new fundamental frequency is created - Many fundamental frequencies can be produced on a single string - Table 13-3 page 495

7. B. Standing waves in a column of air Standing waves can be set up in a tube of air  examples: organ pipes, trumpet, flute  Some move down the tube, some reflect back up forming a standing wave

8. Harmonic series of a pipe open at both ends f n = n (v/2L) n = 1, 2, 3, … **All harmonics possible Open ends are antinodes and allow free range of motion (*different than a string) Can change f 1 by making the column of air longer or shorter Simplest standing wave in pipe = ½ λ (length of the pipe)

9. Harmonic series of a pipe closed at one end  Examples: trumpet, saxophone, clarinet - the shape of the instrument will affect the harmonics Movement of air is restricted at closed end creating a node - Open end is an antinode - Only the odd harmonics are possible - Simplest standing wave = ¼ λ f n = n (v/4L) n = 1, 3, 5, … ** Pitch determine by the fundamental frequency ** 2 nd harmonic is 1 octave above the fundamental frequency

10. C. Timbre (sound quality) - Quality of a steady musical sound - Different mixtures of harmonics produce different sound quality - Instruments have a characteristic timbre

11. D. Beats – interference of waves of slightly different frequencies traveling in the same direction Appears as a variation in loudness from soft to loud to soft Waves combine due to superposition  Constructive (in phase) and destructive (out of phase) interference Beats per second corresponds to differences in frequency between the waves / sounds