MiniTest 2 Lecture Summary

Simple Harmonic Motion: a special vibration

• Hooke's Law has brought us full circle back to displacement. This means that the motion of a mass bouncing on a spring has to be very special!
  • We tried a position vs. time graph consisting of straight line motion and turn-arounds. This did not satisfy the "circle of physics".

• The cosine function (or more generally, sinusoidal functions) does satisfy the circle of physics.

Sinusoidal motion is very important, called Simple Harmonic Motion.

• Any sort of vibration/oscillation needs a force pulling back towards a center. Such a force is a restoring force, and the center is the point of equilibrium.
• Hooke's law describes a linear restoring force, so called because a graph of F vs Δx is a straight line.
• SHM important because:
  1. Hooke's law describes the way many things deform.
  2. Even if it doesn't, it is almost always a good approximation for small displacements.
  3. Sinusoidal functions are easy to deal with mathematically.
  4. Sinusoidal functions "sound nice", less harsh.  Sound of SHM is called a "pure tone.
• Parameters of SHM
  • Period and frequency already covered.
  • Peak-to-peak-amplitude is the total top-to-bottom height of the SHM motion.
  • Amplitude is the top to middle height of the SHM motion.
  • Last characteristic one might need to describe a sine wave: how it is shifted along the time axis.  Instead of referring to the time, it is handy to refer to "the fraction of one cycle" by which it is shifted.
    • To do this, we define phase, a quantity that is proportional to time, with one cycle being 360° of phase.  (° is degrees, very closely related to the degrees used to measure angles.)  We will always choose to have 0° at a peak of the SHM.
    • Then you can specify what phase the function is at when t = 0s. That is called the starting phase or initial phase or the phase constant.
  • For SHM, you can calculate the displacement (vertical position) at any point if you know the phase, with A cos(φ).
    • In fact, the complete equation for SHM on a time axis is Δx = A cos[(360°/T) t +φ₀].  We looked at a spread sheet that showed this function.

A = A_pp/2

Δφ∝Δt

Δx = A cos(φ)

Δx = A cos[(360°/T) t +φ₀]

• Special Properties of SHM
  • SHM special property 1: the maximum speed is π/2 larger than the peak-to-peak-average velocity.
    • Since property 1 is really just a property of the slope of the cosine function, it also works for the maximum and peak-to-peak-average acceleration, as obtained from the velocity graph.  The maximum acceleration is also π/2 larger than the peak-to-peak-average acceleration.
      • We found that an ant on a tuning fork does not achieve very high speeds, but experiences accelerations 30 times greater than the acceleration due to gravity!
  • SHM special property 2: gave the equation relating the frequency of vibration to the mass and the spring stiffness constant. We showed how this is qualitatively consistent with expectations relating to mass and spring strength.
  • SHM special property 3: the amplitude of SHM doesn't affect the frequency or period. Even though a larger amplitude means traveling farther, it also involves larger forces and speeds.
• Describing sounds physically
  • Connecting physics to experience: Amplitude relates to volume, frequency related to pitch, period relates to pitch in an inverse way, and starting phase has no bearing on how something sounds.
  • Duration, how long a sound lasts, is a separate concept.  In particular, it is not the same as period.
  • Note that property 1 is crucial for making music. Any real vibration actually dies away with time, but property 1 means that the frequency (pitch) will not change as it dies away.
  • There is another characteristic of sounds: timbre, or sound quality.  For instance, two intruments (e.g., flute and trumpet) could play the same pitch at the same loudness, but the sounds would be different.  We would like to quantify this, but it cannot be associated with a simple scale of numbers.

Complex Sounds

• Not only are vibrations associated with the production of sound, they are also needed for the detection of sound (e.g., ear drums and microphone diaphragms).  We'll need to look at detection vibrations to analyze different sound timbres.
• Sound timbre depends on the shape of the displacement vs. time graph of the detecting vibration.  This graph can be far from SHM-like, perhaps due to sounds combining.
• When two sounds are detected simultaneously, how does the detector move?  Its displacement graph is the "moment-by-moment" sum of how it would have moved due to only one of the sounds.  This is called superposition.
  • When adding (superposing) two graphs carefully by hand, you must at least add them at the maximum and minimum of both graphs.

v_avg,p-p = 4A /T

v_max = (π/2) v_avg,p-p

v_max = 2π A /T

f = 1/(2π) sqrt(k/m)

• By superposing SHMs, we can make complex vibration graphs.
  • Fourier's theory: any wave shape can be made by superposition of cosine waves.
  • Technically, for each component cosine we need to specify (1) amplitude, (2) frequency, and (3) initial phase.
  • Ohm's 2nd Law: the phase has almost no effect on the timbre.  Timbre is determined only by which frequencies are present, and the amplitude of each.  This law is not completely true, but a decent approximation.
• Spectrum: a graph of amplitude versus frequency, on which we can show which component cosine waves are needed to make a complex sound.
  • Each dot or peak represents a whole sinusoidal vibration.  NOTE: This does not say that each peak represents one cycle of a sinusoidal vibration.  Each peak represents an entier sine curve, extending as far as you like in either direction.
  • Real spectra look like graphs as a function of frequency, often including narrow peaks. When the peaks are there, we want to focus on them. Therefore, we will represent such spetra with vertical lines (which represent the peaks).

• Fourier Transform: a mathematical method to figure out which cosine waves are necessary to make any particular complex sound.
• Fast Fourier Transform (FFT): a particularly efficient way to do the math.
• Example spectra
  1. a sinusoidal wave has a spectrum that looks like a single peak.  The spectrum does not depend on how many cycles there are, or what the duration is.
  2. Noise is technically defined as a sound whose spectrum has no peaks. There may still be regions of frequency with more or less intensity, however.
  3. Many sounds have clear peaks in their spectrum.
     • Any peak in a spectrum is called a partial, and these are numbered from the lowest frequency (starting with 1).
     • The lowest frequency peak is called the fundamental.
     • Other peaks are called overtones and these are also numbered from the lowest frequency (starting with 1).  This means that partial 2 is overtone 1, partial 3 is overtone 2, etc.
  4. Any periodic graph with period T:
     • The only allowed contributing periods are T, T/2, T/3, ... so that the only allowed frequencies in the spectrum are f, 2f, 3f, ...
     • The terminology for peaks described above can still be used.
     • Because the peaks have a special relationship, then can also be called harmonics.
     • Harmonics are numbered according to their frequency: what multiple of the fundamental's frequency is it?  So the fundamental is the 1^st harmonic.  The n^th harmonic is the one with frequency n f₁, where f₁ is the fundamental's frequency.
  5. A periodic wave is not required to have peaks at all the harmonic frequencies.  If a harmonic is missing, all higher-frequency peaks will have a harmonic number that is different from the partial number.
     • For instance, to build a square wave you use only the odd harmonics; all the even harmonics are missing.
     • Note that this square wave spectrum is tricky.  The peaks appear to be evenly spaced, just like a regular harmonic spectrum.  However, a close look shows that the origin-to-fundamental spacing is half of the other peak spacings.
     • [Only in text: Rossing 7.10] To make a square wave, the amplitudes have a special pattern too: 1, 1/3, 1/5, 1/7, ...
• When two sounds are combined by superposition, their spectra combine by simply merging the peaks from each input spectrum.  This is because Fourier analysis is the opposite of superposition; you reverse the superposition in order to figure out peaks for the spectrum.
  • If the two sounds happen to both have a partial at the same frequency, their cosine waves will add together by superposition.
  • Two sinusoidal oscillations are in phase if their phases match at any given time.
  • Two oscillations are out of phase if their phases are 180° different at any given time.
  • Two in-phase oscillations reinforce each other, so amplitudes add.  But out-of-phase oscillations actually cancel each other, leading to a smaller amplitude.
  • If two combining spectra have partials at the same frequency, and you don't know the phase relationship (since spectra don't normally show that), best guess is that they combine to be just a bit larger than the larger of the two inputs.

f_n = n f₁

A_comb= A₁ +A₂  (in phase case)

A_comb= |A₁ -A₂|  (out of phase case)

• Synthetic listening (hearing multiple partials as one complex sound) vs. Analytic listening (hearing multiple partials as separate sounds): it's all in the mind.  Synthetic is also sometimes called Holistic.
• If two simultaneous tones have very similar frequencies, we hear the volume get alternately louder and softer, and effect called beats. Beats are explained by physics, not physiology. The waves of the two sounds are going in and out of phase.
• Formulas for the apparent pitch and the frequency of the beats were given.

Energy, Power, and Intensity

• Amplitude is related to loudness, but a more useful concept is energy.
• Energy is the ability to do something
• Most obvious is energy of motion: Kinetic Energy (KE)
• Less obvious is "stored" energy: Potential Energy (PE) (examples: lifted mass, stretched spring)
• If you quantify energy properly, it turns out that energy is conserved, which makes it very useful.
• When energy is transfered from one thing to another in a visible way, physicists call that work.  This definition sometimes agrees with the English meaning of the word, and sometimes doesn't.
• Stretch a spring, and then stretch it farther; the second time adds work both because the spring is pulling harder and simply because you moved it farther. That results in a square in the PE equation for springs.
• Harder to give a good justification for the square in the KE equation.
• Unit of energy is named the Joule.  From the two equations, we saw that this could be thought of as either N*m, or kg*m²/s².

f_heard = 0.5 (f₁ +f₂)

f_beat = |f₁ -f₂|

PE = 0.5 k (Δx)²

KE = 0.5 m v²

• s=d/Δt
• s_sound=340 m/s
• f = 1/T
• Δ(anything)=(thing)_final-(thing)_start
• s_avg=d/Δt
• v_avg=Δx/Δt
• a_avg=Δv/Δt
• F = m a

• A = A_pp/2
• Δφ∝Δt
• f_n = n f₁
• f_heard = 0.5 (f₁ +f₂)
• f_beat = |f₁ -f₂|

• π = 3.1415
• g = 9.81 m/s²
• f = N /Δt
• F = –k Δx

• Δx = A cos(φ)
• Δx = A cos[(360°/T) t +φ₀]
• v_avg,p-p = 4A /T
• v_max = (π/2) v_avg,p-p= 2π A /T
• f = 1/(2π) sqrt(k/m)
• A_comb= A₁ +A₂  (in phase case)
• A_comb= |A₁ -A₂|  (out of phase case)
• PE = 0.5 k (Δx)²
• KE = 0.5 m v²