Sound Harmonics

Standing Waves on a Vibrating String  On an idealized string, the ends of the string cannot vibrate They should both be nodes  So the longest possible wavelength of the string would be twice the length of the string This lowest frequency allows us to obtain the fundamental frequency of the string  Fundamental frequency – the lowest frequency of a standing wave  Fundamental frequency = wave speed / wavelength 1 = wave speed / (2 * string length)  f 1 = v/λ 1 = v/2L

Standing Waves on a Vibrating String  Harmonics are integral multiples of the fundamental frequency So the second harmonic has a frequency equal to twice the fundamental frequency and so forth  Harmonic series of standing waves on a vibrating string  Frequency = harmonic number * (speed of waves on the string)/(2*length of vibrating string) f n = nv/2L; n = 1, 2, 3, 4,...

Standing Waves on a Vibrating String

Standing Waves in an Air Column  Standing waves can be present in a tube of air, such as the pipes of a pipe organ or the inside of a trumpet or saxophone The presence of harmonics depends on the openness of the air column  If both ends of a pipe are open, all harmonics are present  Harmonic series of a pipe open at both ends  Frequency = harmonic number * (speed of sound in the pipe)/(2*length of vibrating air column) f n = nv/2L; n = 1, 2, 3, 4,...

Standing Waves in an Air Column

 If one end of the pipe is closed, only the odd harmonics are present  Harmonic series of a pipe closed at one end  Frequency = harmonic number * (speed of sound in the pipe)/(4*length of vibrating air column) f n = nv/4L; n = 1, 3, 5,...  Most wind instruments are essentially like a pipe with one end closed because the mouth and reed (in woodwinds) closes one end The often curved nature of the instruments cause differences from the that of the true pipe

Standing Waves in an Air Column

 Harmonics account for timbre  Timbre – the quality of a steady musical sound that is the result of a mixture of harmonics present at different intensities  Musical instruments do not produce a single wave, but many at different intensities They interfere with one another  The resulting wave is more complex than a simple sine wave

Standing Waves in an Air Column  Therefore, musical instruments produce richer, fuller sounds than something like a tuning fork Likewise an entire orchestra produces a fuller sound than a single instrument  Each note whether from a tuning fork or an orchestra is made up of repeating patterns Periodic  There are twelve notes in the chromatic musical scale The thirteenth note has a frequency twice the first note  These thirteen notes make up an octave

Standing Waves in an Air Column

Beats  The interference of slightly different waves produces beats When the waves constructively interfere, the sounds are louder  When the crests line up, the waves are said to be in phase When they destructively interfere, the sounds are softer  When they completely cancel each other and no sound is heard they are said to be out of phase

Beats  Beat – interference of waves of slightly different frequencies traveling in the same direction, perceived as a variation in loudness  The number of beats per second corresponds to the difference between frequencies This does not tell you which wave has the greater frequency

Beats