13.3

Harmonics A vibrating string will produce standing waves whose frequencies depend upon the length of the string. Harmonics Video 2:34

In the lowest frequency of vibration, one wavelength will equal twice the length of string and its called the fundamental frequency (f 1 ). For f 1, 1λ = 2L One wavelength = 2*length of string Fundamental FrequencyHalf of a wavelength

Harmonics A Harmonic series is a series of frequencies that include the fundamental frequency and multiples of that frequency. 1st harmonic = f 1 2 nd harmonic = f 2 = 2*f 1 3 rd harmonic = f 3 = 3*f 1 Etc…

Harmonics The second harmonic is the next possible standing wave for the same string length. This shows an increase in frequency, and a decrease in wavelength. f 2 =2f 1 λ 2 = L Second Harmonic = 2*fundamental frequency

Harmonics As the harmonic increases the frequency increases and wavelength decreases. Ex: f 3 = 3f 1 λ 3 = 2/3λ 1 f 4 = 4f 1 λ 4 = ½ λ 1 Standing Waves, Fixed at Both Ends Animation

Formula for other harmonics Harmonic Series of standing waves f n = n* V n=1, 2, 3… 2L Frequency = harmonic number x (speed of waves on the string) (2)*(length of vibrating string)

Standing Waves in an Air Column If both ends of a pipe are open, all harmonics are present and the ends act as antinodes. This is the exact opposite of a vibrating string, but the waves act the same so we can still use the same formula to calculate frequencies. fn = n* V n=1, 2, 3… 2L Frequency = harmonic number x (speed of waves on the string) (2)*length of vibrating air column)

Standing Waves in an Air Column If one end of the pipe is closed, only odd harmonics are present (1, 3, 5, etc). This changes the formula: f n = n* V n=1, 3, 5… 4L Frequency = harmonic number*(speed of waves on the string) (4)*length of vibrating air column)

Example What are the first three harmonics in a 2.45 m long pipe that is open at both ends? Given that the speed of sound in air is 345 m/s. L= 2.45 m v= 345 m/s f n = n*v/2L 1 st harmonic: f 1 = 1*(345 m/s)/(2*2.45 m) = 70.4 Hz 2 nd harmonic: f 2 = 2*(345 m/s)/(2*2.45 m) = 141 Hz 3 rd harmonic: f 3 = 3*(345 m/s)/(2*2.45 m) = 211 Hz

Example What are the first three harmonics of this pipe when one end of the pipe is closed? Given that the speed of sound in air is 345 m/s. L= 2.45 m v= 345 m/s f n = n*v/4L 1 st harmonic: f 1 = 1*(345 m/s)/(4*2.45 m) = 35.2 Hz 3 rd harmonic: f 3 = 3*(345 m/s)/(4*2.45 m) = 106 Hz 5 th harmonic: f 5 = 5*(345 m/s)/(4*2.45 m) = 176 Hz

Why do different instruments sound different? Timbre is the quality of a steady musical sound that is the result of a mixture of harmonics present at different intensities. This is why a clarinet and a trumpet can play the same pitch but they sound different. Harmonics Applet

Beat When two waves of slightly different frequencies travel in the same direction they interfere. This causes a listener to hear an alternation between loudness and softness and is called beat.

Beat Formation of Beats Applet The frequency difference between two sounds can be found by the number of beats per second.