# Formula sheet & numerical constants

```Equation Sheet for Final Exam
Equations
Energy: kinetic energy KE = (1/2)mv2; potential energy PEgrav = mgh &amp; PEspr = (1/2)Ky2
Work and Energy: Work = Force  distance; W = Fd; Work done = KE
Conservation of energy: KE + PE = constant = Etotal = (1/2)m(vbot)2 = mghtop
Periodic motion: frequency f (Hz); period T (s); f = 1/T; amplitude A; PEmax = (1/2)KA2
Mass on spring: Fsp = - Ky, f = (1/2π)√(K/m); Pendulum: f = (1/2π)√(g/l)
Relations involving v, the wave speed:
- wave relation: v =   f
- sound wave in air: v = (331.3 + 0.6 tC) m/s -&gt; at 20C v = 343.3 m/s
- wave on a string: v = √(T/), T is string tension [N] and  is mass/length [kg/m]
Wave propagation: reflection (inc = refl); refraction (speed varies); diffraction (size~)
Wave interference: due to path length difference, (d2 – d1), constructive d = 0, , 2…,
destructive, d = 1/2, 3/2…, or
due to frequency difference, f2  f1, ffused tone = (1/2)(f2 + f1); fbeats = f2 – f1
Doppler effect: receiver(R) moving &amp; source(S) stationary: fR = fS(v+vR)/v
receiver(R) stationary &amp; source(S) moving: fR = fS v/(v-vS)
Harmonic Series for Standing waves: harmonic number n, fundamental is n = 1
On a string: fn = nf1, n = 2L/n, fn = v/n = (n/2L)√(T/), for n = 1,2,3…
Open air-filled pipe: fn = v/n = (n/2L)v, for n = 1,2,3…
Closed air-filled pipe: fn = v/n = (n/4L)v, for n = 1,3,5…
Sound: Pressure p(N/m2) and Intensity I(W/m2); I = (p2)/()  (p2)/(400)
Intensity level LI (dB); LI = 10 log (I/I0), where I0 = 1.0  10-12 W/m2
Lp = 20 log (p/p0), where p0 = 2.0  10-5 N/m2
Relative level LI (dB); LI = LI2 - LI1 = 10 log (I2/I1); for I2/I1 = 2n, LI = n  3dB
Adding Intensities LI = 10 log [(I1 + I2)/I0],
Power P(W), Intensity I(W/m2), and Area(m2); Power = Intensity  Area, P = I  A
Point Source; Intensity at distance r from a source of power P; I = P/(4r2)
Loudness (phons), Pitch and Timbre; Tempered scale; fn = f0  (1.0595)n
Fourier Spectrum: for complex wave of period T, f1 = 1/T, fn = nf1, and An = amplitude
for triangular wave, odd harmonics only with amplitudes A n = A1/n2
for square wave, odd harmonics only with amplitudes A n = A1/n
Combining Tones: for f2  f1, beating phenomenon, ffused tone = (1/2)(f2 + f1); fbeats = f2 – f1
for f2 &gt;&gt; f1, sidebands, (f2 &plusmn; f1)
Hearing: range of hearing 20 Hz to 20 kHz with maximum sensitivity for 1000-4000 Hz
Speech and Singing: acoustic model of vocal tract, fn = n(v/4L) for n = 1,3,5,..
String Instruments: Standing waves of harmonic number n (the fundamental is n = 1)
fn = nf1, n = 2L/n, fn = v/n = (n/2L)√T/, for n = 1,2,3…
Brass Instruments: closed cylindrical pipe, fn = v/n = n(v/4L), for n = 1,3,5…
Percussion Instruments: for church bells, fm ≈ 0.11 f1 (2m+1)2
Electronic Circuits: Voltage V(volts), Current I(amps), Resistance R(ohms)
Ohm’s Law V = I R, Electrical Power P(watts) = VI = I2R = V2/R
Sound Recording: Microphone sensitivity SV and output V, 20 log V = SV + LP - 94 (all dB)
AnalogDigital, # of N-bit digital “words” = 2N, step size = (V)/(2N-1)
Signal-to-Noise S/N(dB) = 20 log(Vsignal/Vnoise) = 20 log(2N)
Dynamic Range in dB = 20 log(Amax/Amin)
Nyquist sampling theorem: Sampling rate  2  Signal Frequency
Room Acoustics: Direct and reflected sound have times from source to observer t = d/v
Reverberation Time RT = 0.161(V/A), with V = volume in m3 and
A = Siai = sum of (each surface area Si )(its absorption coefficient ai)
Sound Energy: Sound Energy = Sound Intensity  Area  Time of Exposure
Conversion Factors and Constants
Conversion Factors
1 kg
= 1000 gm
1 m = 100 cm = 39.37 in
1 km
= 1000 m
1 in
= 2.54 cm
1 ft =
12 in = 0.305 m
1 mile = 5280 ft = 1610 m
1s
= 1000 ms
1 hour = 3600 s
1 N = 1 kgm/s2
1 J = 1 Nm = 1 kgm2/s2
Constants
Acceleration due to gravity
g = 9.8 m/s2
Density ( = Mass/Volume)
Water = 1000 kg/m3
Nylon = 1140 kg/m3
Steel = 7700 kg/m3
Abbreviations
mega M = 10 6
kilo
k = 103
milli
m = 10-3
micro  = 10-6
Figures
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