Chapter 6
Harmonics
The Sound of the Big Bang
Even when all Earthly and astronomical sources of radio waves are screened out, some static remains on the most sensitive radios. This static is caused by radiation left over from the big bang, the explosion that created the universe.
© Microsoft Corporation. All Rights Reserved./NASA/WMAP Science Team
Microsoft ® Encarta ® 2006. © 1993-2005 Microsoft Corporation. All rights reserved.
Frequency
We perceive frequency as “higher” or “lower” sounds. The frequency of a sound is the number of cycles, or oscillations, a sound wave completes in a given time. Frequency is measured in hertz, or cycles per second. In these examples, the frequency of each higher wave is double that of the one below, producing the same note at different frequencies, from 110.00 Hz to 880.00 Hz. Waves propagate at both higher and lower frequencies, but humans are unable to hear them outside of a relatively narrow range.
© Microsoft Corporation. All Rights Reserved.
Microsoft ® Encarta ® 2006. © 1993-2005 Microsoft Corporation. All rights reserved.
The masks of God, Oriental mythology 1976 by Joseph Campbell “ For the inaudible music of the spheres, which is the hum of the cosmos, in being becomes audible, through music it is the harmony, the meaning of social order, the harmony of the soul, itself discovers therein its accord.”
Period of audible sound waves 1 x 10-3
Speed of sound V=√B/P in a solid VA1=√V/P
A molecule travels at 401 m/s
Sound travels at 1,000 feet per second
Sound intensity level β=10 log (1/10) decibels
Sound waves speed 330 m/s in air 343 m/s at 20oc
Fundamental frequency Hz ƒ1 = V/2L = 200m/s / 2(1.0m) = 100 Hz
Ultra sound PR = (P1-Pe/P1-P2)2 x 100
Harmonic motion ( Kinematics) χ = A cos (wt+φ)k/m χ= 4π2χ/T2
Beats ƒ2T –ƒ1T=1 T=2π√m/k ƒbeat 1/T = ƒ2-ƒ1
Diatomic music scale
Hz. Note
256 C
288 D
320 E
341 F
384 G
427 A
480 B
512 C
Resonant frequency reaches a maximum when the frequency of the driving force equals the natural frequency. The forced vibration wave length of emitted sound wave equals the circumference.
Bulk/density/temperature
V= √B/P V=(331m/s)√1 +T/273
Spherical wave Intensities
I1=Pav/4πrf I2=Pav/4πrg I1/I2 = rg/rf
Wave varies I/r2
V=Δx/ΔT V=ƒλ
Intensity of a wave
1≡ 1/A ΔE/ΔT
Heat capacity – systems of coupled oscillators Cp∫wmaxo N(w) CE(w)dw
N(w) = frequency spectrum CE(w) = heat capacity of frequency W
Wmax = high frequency cut off. In wave motion the phase and group velocities
Vp= w/k, Vg = dw/dk, k=2π/λ.
Acoustic and optical modes
Acoustic and optical modes in two or more different types of atoms m1us=c1(υs+1/2+υs-1/2 – 2us) m2us+1/2 =c1(us+1+u2-2υs+1/2) w2=c1(m1+m2/m1m2)±[cf(m1+m2/m1m2)2-4cf/m1m2 sin =1/2ka]
Work and energy (joule) j
ƒ1/ƒ2 = m1a1/m2a2 N-m =kg –m2/s2
Renaissance energy
ƒo =1/2π √LC
Thermionic emissions
1ƒ≈ BV2E –K/V
Line integral in closed loop
∫cV x dl or φV x dl annulus π(R2-r2) between 2 concentric circles
Truncated icosahedrons
Truncated icosahedrons – the use of light metrics harmonics to form a closed cage model, shrink-wrapped around metal ions. Second harmonics is used to make Bucky balls ~30mJ of Q – switched (Nd x YAG). The only plausible structure to satisfy all sp 2 valences is only a spheroid structure. All valences are satisfied and the molecules appear to be aromatic. A singularity is a perfect combination of energy flowing together in harmony and is not dependant on speed or mass. Pythagorean concept of harmonicas as a grand unified theory of the universe has been obscured for more than two millennia. Technically this is called the spiral of fifths. The seventh octaves and the twelve fifths differ by 1.0136 called the comma of Pythagoras. Mathematically the ratio 2 to 1 doubled seven times gives a frequency 128 times higher, but if you proceed on this basis of the ratio of 2 to 3 which is 1.5 and you go up the twelve fifths. These give you 129.75 and if you divide 129.75 by 128 you get the decimal 1.0136. In the text Katatome Kanonos (Division of the Canan) says the number 531.441 is greater than twice 262,144. The explanatory statement is six sequioctave intervals (multiple 262,144 by two to get 524,288, divide 531,144 by that number gives 1.013643265) this proves that melodic space is irrational (pure notes). In 1584 the Chinese invented a special system to cope with it known as “Equal temperament” (flat notes).
Spherical harmononics
Spherical harmononics may be used to describe the normal modes of oscillation of spherically shaped objects. Each harmonic has two identifying indices that distinguish it from other spherical harmonic waveforms. On the surface of a vibrating sphere certain nodal circles appear where the surface is at rest. The number of these nodal circles for a given spherical harmonics is called its order n;n is one of the indices used to identify the normal mode. . The second identification index corresponds to the number of nodal circles, which pass through the poles of the vibrating sphere. A general property of a nodal circle does not pass through the poles of the sphere, and then it must lie in a plane parallel to the spheres’ equator, parallel 1=6 m=0, both 1=6 m=3, vertical 1=6 m=6 P=(r,Θ,φ) (Cartesian coordinates). A boundary condition is when a region of infinite compression occurs, the boundary layer under continuous changing pressure, will accelerate at an infinite rate under the action of the force. The impedance in sound waves stress P and the velocity of displacement δξ /δt have a relationship to the waveform ξ =ξ0 exp {i(wt – kx)} (applied to induced, voltage to current). Complex impedance is when voltage and current are oscillating (not in phase) there is power transferred ½ (V1* + V*1) = ½ 11*(z+z*). Matching phase is when trying to maximize the power transfer, access a boundary or a series of boundaries. The source of impedance Z i feeding a load of impedance Z2, electric-circuit theory shows the power developed in the load p=1/2 11*(Z2 + Z*2) = 1/2V V* (Z2 +Z*2) { 1/Z1+Z2 + 1/Z*1 + Z*2}2 Maximum is when Z*1 = Z2. Sub harmonic oscillations are oscillations of a non-linear system with periods that are integral multiples of the period of the driving force The oscillator changes with a frequency of w/n, with n=an integer. A non-linear oscillator undampened the equation of motion is w2¨q +q + єq3 =ƒ cos 3τ A harmonic (non-linear) perturbation theory (Lindstedt – Poincare) undampend driven a harmonic oscillator ƒ=1, є = 1/10, w= 1. Standing waves occur when ever oscillations are confined to a finite space
Harmonics # integration q0 q1 q2 sum
1 2.3555 2.37126 -0.01389 C1 2.35737 +C1
3 0.04563 - 0.04167 .00366 .04533
5 0.00083 - - .00073 .00073
>5(sum) 1.17x10-5 - - - -
Coupled oscillations are bringing into unison, the “Kuramoto model.” Experiments on the math shows with a large number of electronic oscillations on a microchip the coupled oscillation would bring them into unison in two neat phase transitions. The Coulomb field φ = φ0 +φ1 + φ2 + … The quantities which depend only on the polar angles are called generalized spherical harmonics. For every index n there are in all 2n+1 independent spherical harmonics. Yn Arbitrary parameters can be determined from the charge distribution producing the field φ(r) = ∞n=0∑ 1/rn +1 Yn(0,α) φ(r) =∫ p(r1)/│r-r1│ dv1 and/or = ∑ ei/│r-r1│ it has the potential of a given distribution of a charge. A field of rotationally symmetric quadrupole, the angular dependence is given by the second zonal spherical harmonic.P2 (cosΘ)=(3 cos2 Θ-1)/2 (in a uniform quadruple electric field there is neither a force nor a torque) Coulombs’ law is the force of electrical attraction and is proportional to the product of the electrical charge and inversely proportional to the square of the distance between them.
Fourier’s theorem
Any periodic function ƒ(x) can be expressed as the sum of a series of sinusoidal functions which have wavelengths which are integral submultiples of the wavelength λ of ƒ(x)
ƒ(x) = C0+C1 cos (2πx/λ + α1) + C2 cos (2πx/λ2 + α2) + … + Cn cos (2πx/λn + αn) + … the n’s are called the order of the terms which are harmonics, amplitude Cn and phase angle αn.
Sound
Humans hear 16 Hz to 20,000 Hz or 20 KHz.
- tolerable 1.0 x 1012 decibel
- pain 120 (dB)
- deafening 100 (dB)
- damage 140 (dB)
The maximum safe levels for different exposures;
- 15 min 100 (dBa)
- 2 min 109 (dBa)
- 28 second 115 (dBa)
- 0.11 seconds 139 (dBa)
- 30 min 97 (dBa)
- 24 hours 80 ma level in (dBa)
- 8 hours 85 (dBa)
- 4 hours 88 (dBa)
- 1 hour 94 (dBa)
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