HarmonicsPhysics Chapter 13-3

Pages 494-503

A. Standing waves on a vibrating string

Fundamental frequency – lowest frequency of vibration of a standing wave

Symbolized as f1

Harmonic series – series of frequencies which are multiples of the fundamental frequency

f2, f3, f4, …

Equationv = fλ f = v / λ

Fundamental frequency of a string fixed at both ends

f1 = v/λ1 = v/2L L = string length

Harmonics (multiples of f1)Frequency Wavelengthf2 = 2 f1 λ2 = Lf3 = 3 f1 λ3 = 2/3 L

f4 = 4 f1 λ4 = ½ L

Harmonic Series of standing waves on vibrating string fn = n (v/2L) n = 1, 2, 3, …

n = harmonic #v = speed of the waves on a string

L = string lengthf = frequency

* 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

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

Harmonic series of a pipe open at both ends

fn = 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 f1 by making the column of air longer or shorter

Simplest standing wave in pipe = ½ λ (length of the pipe)

Harmonic series of a pipe closed at one end

Examples: trumpet, saxophone, clarinet - the shape of the instrument will affect the harmonicsMovement 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 λfn = n (v/4L) n = 1, 3, 5, …

** Pitch determine by the fundamental frequency** 2nd harmonic is 1 octave above the fundamental frequency

C. Timbre (sound quality)

- Quality of a steady musical sound- Different mixtures of harmonics produce different sound quality- Instruments have a characteristic timbre

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