Important Formulas and Facts
(that you should know from Unit #2)
Sound Intensity Level and Loudness
SIL=10 LOG (I / Io) SIL is measured in decibel (dB);
Io = 1.0 pW/m2 = 1 x10-12 pW/m2;
I =Average Intensity
I = 0.0012 p2 I = Intensity [pW/m2]; p = peak pressure [N/m2 = Pa].
Range of Audible frequency f = 20 Hz to 20 kHz (approximately & on average).
Threshold of Hearing Io = 1.0 pW/m2 = 1 x10-12 W/m2; SIL = 0 dB.
Threshold of Pain Ipain = 1.0 W/m2 = 1 W/m2; SIL = 120 dB
Just Noticeable Difference (JND)
When 75% correct responses occur in Two-alternative Forced Choice (2AFC) test.
Musical dynamics pp~ 50 dB; p~ 60 dB; mp~mf~ 70 dB; f~ 80 dB; ff~ 90 dB; fff~ 100 dB.
Loudness Loudness is measured in Sone. A sone is the loudness that will
produce the same psychophysical response as a sound with a SIL of 40 dB at 1000 Hz.
Loudness Level Loudness level is measured in phon. A phon is equal to the SIL (in dB)
at 1000 Hz that is perceived to be equal to the test tone.
The Flectcher-Munson Diagram illustrates this point.
Intensity and distance Sound Intensity (in the absence of reflections) decreases as the square
of the distance.
I1 / I2 = ( d2 / d1) 2
SIL2 - SIL1 = 2 LOG (d2 / d1); 6 dB for each doubling of distance.
Adding Sound Waves Sound can be added coherently (if phase difference is defined) or
incoherently (if phase difference is random or changing rapidly).
Coherent:
Icoherent = .0012 (p1 +n p2)2; n is the phase factor.
Incoherent:
Iincoherent = .0012 (p12 + p2 2 )= I1 + I2
Wave Properties
There are three ways to bend a wave:
Reflection, refraction and diffraction.
Reflection rule Reflection is bouncing off of an change in the medium.
Angle of incidence = angle of reflection.
Refraction Bending of a wave because of a change in velocity across wave front.
Direction of bend is toward slower side.
Diffraction Bending of a wave around and obstacle.
Longer wavelengths bend more than short ones
Doppler Effect Change in frequency and wave length due to motion of source or
observer (to higher frequency when approaching, to lower with receding).
fObserver = fSource [(v - vObserver ) / ( v - vSource)]
Beats Beats are the variation in the intensity of two sounds whose frequencies
are close to each other due to their coherent addition.
fsum = (f1 + f2)/ 2, the average.
fbeat = (f1 - f2), the difference.
Interference Interference is the coherent addition of waves in a space.
When the waves are exactly in phase with each other the intensity is maximized; when they are exactly out of phase the intensity is minimized.
Reverberation
Intensity equation I2 = I1 10-6 t / TR,” exponential decay”
Sabine equation Reverberation time given by:
TR = 0.16 V/Se;
Reverberation TR [sec] is the time for the sound to decay by 60 dB (1/1,000,000).
V is the volume of the room [m3].
Se is the effective surface area of the room [m2].
Se = α1 S1+ α 2 S2+ α 3 S3+ α 4 S4+ etc.;
α is the “absorption coefficient” of the material on the walls, floors and ceiling and is given in tables.
For good listening music rooms need TR > 2 sec, for speaking, <2 sec is best.
Musical Scales
Diatonic Scale Based on ratio of whole numbers.
Octave ratio:: 1:2.
The Perfect Fifth:: 2:3
The Major Third:: 5:4 and so on.
Chromatic Scale Has semitones with frequency ratios near 9:8 or 10:9.
Must be retuned for each key.
Equal or Even Tempered Scale
Each half step or semitone is exactly the same with a frequency ratio equal to the twelfth root of 2: 12√2 = 1.05946; about 106%.
Cents A cent is 1/100 of a semitone, equal to a frequency ratio of
120√2 = 1.0005778.
Strings
Marsenne Equation
f = (1/2L)√ (T/ μ);
f = fundamental frequency [Hz].
L = the length of the string [m].
T = tension on the string [N].
μ = mass density of the string [kg/m]
Harmonics fn =n f, where n is an integer
Dr. Matteson’s rule number 26:
The distance between a node and the adjacent antinode is 8/4.
(Quarter wavelength with cheese)