Music Physics 202 Professor Vogel (Professor Carkner’s notes, ed) Lecture 8

Intensity of Sound  The loudness of sound depends on its intensity, which is the power the wave delivers per unit area: I = P/A  The units of intensity are W/m 2  The intensity can be expressed as: I = ½  v  2 s m 2  Compare to expression for power in a transverse wave  Depends directly on  and v (medium properties)  Depends on the square of the amplitude and the frequency (wave properties)

Intensity and Distance  Consider a source that produces a sound of initial power P s  As you get further away from the source the intensity decreases because the area over which the power is distributed increases  The total area over which the power is distributed depends on the distance from the source, r I = P/A = P s /(4  r 2 )  Sounds get fainter as you get further away because the energy is spread out over a larger area  I falls off as 1/r 2 (inverse square law)

Inverse Square Law Source r 2r A 1 =4  r 2 I 1 = P s /A 1 A 2 =4  (2r) 2 = 16  r 2 = 4A 1 I 2 = P s /A 2 = ¼ I 1

The Decibel Scale  The human ear is sensitive to sounds over a wide range of intensities  To conveniently handle such a large range, a logarithmic scale is used known as the decibel scale  = (10 dB) log (I/I 0 )  Where  is the sound level (in decibels, dB)  I 0 = W/m 2 (at the threshold of human hearing)  log is base 10 log ( not natural log, ln)  There is an increase of 10 dB for every factor of 10 increase in intensity

Sound Levels  Hearing Threshold  0 dB  Whisper  10 dB  Talking  60 dB  Rock Concert  110 dB  Pain  120 dB

Human Sound Reception  Humans are sensitive to sound over a huge range  A pain level sound is a trillion times as intense as a sound you can barely hear  Your hearing response is logarithmic  A sound 10 times as intense sounds twice as loud  Thus the decibel scale  Why logarithmic?  Being sensitive to a wide intensity range is more useful than fine intensity discrimination  Similar to eyesight  Your ears are also sensitive to a wide range of frequencies  About 20 – Hz  You lose sensitivity to high frequencies as you age

Generating Musical Frequencies Many devices are designed to produce standing waves e.g., Musical instruments Frequency corresponds to note e.g., Middle A = 440 Hz Can produce different f by changing v Tightening a string Changing L Using a fret

Music  A musical instrument is a device for setting up standing waves of known frequency  A standing wave oscillates with large amplitude and so is loud  We shall consider an generalized instrument consisting of a pipe which may be open at one or both ends  Like a pipe organ or a saxophone  There will always be a node at the closed end and an anti-node at the open end  Can have other nodes or antinodes in between, but this rule must be followed  Closed end is like a tied end of string, open end is like a string end fixed to a freely moving ring

Sound Waves in a Tube

Harmonics  Pipe open at both ends  For resonance need a integer number of ½ wavelengths to fit in the pipe  Antinode at both ends L = ½ n v = f f = nv/2L  n = 1,2,3,4 …  Pipe open at one end  For resonance need an integer number of ¼ wavelengths to fit in the pipe  Node at one end, antinode at other L = ¼ n v = f f = nv/4L  n = 1,3,5,7 … (only have odd harmonics)

Harmonics in Closed and Open Tubes

Beat Frequency  You generally cannot tell the difference between 2 sounds of similar frequency  If you listen to them simultaneously you hear variations in the sound at a frequency equal to the difference in frequency of the original two sounds called beats f beat = f 1 –f 2

Beats

Beats and Tuning  The beat phenomenon can be used to tune instruments  Compare the instrument to a standard frequency and adjust so that the frequency of the beats decrease and then disappear  Orchestras generally tune from “A” (440 Hz) acquired from the lead oboe or a tuning fork