MiniTest 3 Lecture Summary

Date	Lecture topics	Equations
Fri 10/5	Energy, Power, and Intensity Most obvious is energy of motion: Kinetic Energy (KE) [HW7.1] Less obvious is "stored" energy: Potential Energy (PE) (examples: lifted mass, stretched spring)	PE = 0.5 k (Δx)² KE = 0.5 m v²
Fri 10/12	As SHM happens, the same total energy is continually being converted between KE of the mass and PE in the spring. PE max happens at max displacement (i.e., amplitude). There speed and KE are zero. KE max happens at zero displacement (i.e., mass at equilibrium point, speed at max). There PE is zero then. [HW7.1] Did an example using energy conservation to determine the maximum speed of SHM based on amplitude, spring constant, and mass. [HW7.2] Combining energy conservation with SHM max velocity formula actually derives SHM frequency formula. Sound carries energy (as in definition from the dictionary). Energy per unit time (produced, in transit, or delivered) is power (symbol W, unit J/s = Watt = W) [HW7.3] Sound power is also spread out (consider sound down tube), so we need sound power per unit cross-sectional area, which is intensity (symbol I, unit W/m²). [HW7.3, HW7.9, HW7.10] WARNING: "A" is sometimes used for both area and amplitude Human range is, roughly speaking, from 10^-12 W/m² (threshold of hearing) to 1 W/m² (threshold of pain) Symbol for 10^-12 W/m² is I₀. [HW7.8]	E = PE_max = 0.5 k A² W = E/Δt I = W/A I₀ = 10^-12 W/m²
Mon 10/15	Multiple sound sources: Usually, for a bunch of sound sources that each produce the same intensity, the total intensity is proportional to the number of sources.[HW7.7, HW7.8] As sound spreads away from a sound source, energy is generally not lost from the sound. The energy does spread accross a larger area, thus reducing the intensity. Pertinent areas are often spherical in type. Formula for sphere [HW7.9, HW7.10] and hemisphere given. When comparing two things with the same shape, any area is proportional to any length squared. [HW7.4, HW7.5, HW7.6] As sound leaves a source, the same total power passes through any surface enclosing the source. Often the sound spreads out over a larger and larger area. IF the wave crest's shape stays constant, then its area is proportional to the square of the distance traveled. This means (combining previous proportions) that Intensity is inversely proportional to the square of the distance to the source. [HW7.7, HW8.9] WARNING: A proportion is not an equation. You may NOT plug numbers in one side and get a number out for the other side. This works for point sources. for extended sources of sound (rivers, highways) shape of sound is not spherical, and doesn't stay the same as it goes further away.	I_N = NI₁ sphere: A = 4πr² hemisphere: A = 2 πr² A ∝ d² I ∝ 1/r²
Fri 10/19	After the MiniTest, the lecture hall was too hot to concetrate. Reviewed some qualitative facts, then ended class early. This resulted in taking some things out of order, so I'm just going to include them in the following lecture entry.
Mon 10/22	It is a very general relationship that any energy-like quantities (energy, power, intensity) is proportional to the square of any associated amplitude-like quantity (amplitude, peak-to-peak amplitude, maximum speed). [HW8.8] Just as we can now measure "how big" an oscillation or sound is with amplitude or intensity, you can also use either to measure the contributions of partials in a spectrum. We can have amplitude spectra or intensity spectra. [HW7.11] Although a proportion cannot be used to calculate numbers (e.g., get an intensity from an amplitude), it can be used to calculate ratios. If partial B has 1/2 the amplitude of partial A, then on an intensity spectrum partial B will have (1/2)²=1/4 of the intensity of partial A. Note that this will NEVER change the qualitative relationships between the partials. The tallest will remain tallest, etc. Decibels and SIL To conveniently work with sounds of various typical loudnesses, we need to consider intensity on a logarithmic scale, on which each tick mark represents increasing by a multiple (we used tick marks for multiples of 10). [HW7.11, HW8.1] The kind of scale that you are used to (where each tick mark represents increasing by adding) is a linear scale. Estimation between tick marks on a logarithmic scale is difficult, because numbers are not evenly spaced. To handle the range of intensities, we invent the Sound Intensity Level (SIL) scale, which runs from 0dB to 120 dB. When intensity is multiplied or divided by 10, its SIL goes up or down by +/-10 dB. [HW7.15] The word "level" indicates the measurement of something in dB. Symbol for SIL is L_I. Sound Pressure Level (SPL, symbol L_p) is pretty much the same as SIL, at least as far as we at concerned. Sound Power Level (symbol L_W) is different, and will not be considered for now. SIL relates better to subjective experience, intensity relates better to physical intuition. SIL (dB) generally do NOT do what you would intuitively expect for physical calculations! Example of how this is useful: If two sounds have very similar intensities, they will be percieved as having the same loudness. The minimum difference required in order to percieve the difference is called the loudness just noticable difference or l-jnd. The loudness jnd is roughly 1dB, regardless of where you are in the range. Fechner's Law: all human perception is on a logarithmic scale (more or less). Equations for converting between SIL and Intensity are given (see to right). [HW7.16] [HW7.17] They involve the log function [HW7.12, HW7.13] and raising 10 to a power [HW7.14] When comparing two sounds, you want to consider differences between SILs and ratios of anything else (e.g., intensities, amplitudes, powers). [HW8.5, HW8.7] Equations for comparing two sounds are almost identical to those of describing the absolute level of sound.	{E, W, I}∝ {A, A_pp, v_max}² I₀ = 10^-12 W/m² L_I = (10 dB) log(I/I₀) I = I₀ 10^(L_I/10 dB) ΔL_I = L_I₁-L_I₂ = (10 dB)log(I₁/I₂) I₁/I₂ = 10^(ΔL_I/10 dB)
Fri 10/26	Given an intensity spectrum (with intensity versus frequency), convert it to an SIL spectrum by taking the intensity of each partial and applying the SIL equation. [HW8.1] The relationship between amplitude, intensity, and SIL of partials is exactly the same as for separate sounds. The same equations apply. It is often useful to use the comparison equations, comparing two partials to each other. If multiple sources of sound are the same in intensity, then the number of sources is proportional to the intensity. SIL is NOT proportional to number of sources. [HW8.2, HW8.3, HW8.4, HW8.6] Tips for using and working with dB. A good rule: Never multiply or divide SIL (in dB) by anything, except by "10dB" in the conversion equations. Sometimes problems can tell you the comparison between two sound loudnesses without telling you how loud either one is. This can be disconcerting, but we do have equations to handle this situation. If you are given the SIL comparison, it must be the difference L_I1-L_I2. If you are given an intensity comparison, it must be the ratio I₁/I₂. Often the key to problems is to convert all given SIL into intensities, so that we can use known properties (proportional to number of sources, proportional to 1/distance², etc). [HW8.6] Example problem combining (intensity variation with distance) and expressing loudness with SIL instead of intensity. [HW8.9, HW8.10]
Mon 10/29	Structure of the Ear I: Place Theory of Pitch The ear has several parts. Vocabulary words for the class are outer ear, middle ear, inner ear, ear drum, ossicles, cochlea, oval window, basilar membrane, bony shelf, helicotrema. The cochlea, where most of the action takes place, is a fluid filled tapered cone, coiled up like a snail shell. It's about 4 cm long. Down the center of the cochlea, dividing it into 2 chambers, is the basilar membrane. At the narrow end of the cochlea is a small hole between the chambers, called the helicotrema. The basilar membrane is supported by the bony shelf. When pure tones of different frequencies enter the cochlea, different sections of the basilar membrane vibrate. This general idea, that frequencies correspond to places on the basilar membrane, is called the place theory of pitch. Consider something that has a prefered frequency (e.g., a mass on a spring). What happens if we shake it ("drive it") at various frequencies? Vibrations due to an outside source are called driven oscillations. Object moves with the driving frequency, not its prefered frequency. A very low driving frequency leads to a small amount of motion. Very high driving frequency also leads to almost no motion. Shaking near the prefered frequency results in a large amount of motion. That phenomenon is called resonance. When the driving agent is at the frequency that the oscillator prefers, they are said to be in resonance. The prefered frequency is often called the natural frequency or resonant frequency of the object. The basilar membrane has a reverse taper. It is narrow, thin, taut and stiff at the cone opening, and hence has a high natural frequency. It is wider, thicker and slack at the other end, and hence has a low natural frequency. When a sound (with various frequency components) enters the cochlea, a section of the basilar membrane is excited if the sound has a partial near that section's natural frequency. Hence, the cochlea is basically doing a Fourier transform of the sound. The length of the cochlea is like a frequency axis. Thinking of the length of the cochlea as a frequency axis, it is roughly a logarithmic axis, as per Fechner's Law. Thus, frequency scales on graphs are often logarithmic. [HW7.11, HW8.1] The cochlea's Fourier transform is acutally kind of sloppy. Oscillators respond with some motion even if the driving frequency is only close to the natural frequency. Also, because the membrane is all one piece, a single spot can't virate without vibrating nearby pieces. Thus a pure sine wave would excite a region of the membrane that is a few millimeters long. Such a region is called a critical band. The width of a critical band is generally not measured in millimeters, but instead in terms of frequency (thinking of the cochlea as a frequency axis). A critical band width is roughly 1/4 of the central frequency, but never less than 90Hz. The range of intensities detectable by our ears actually depends on the sound frequency. A graph of the limits (threshold of audibility and threshold of pain) has the same axes as a spectrum. [HW8.11] When people do not wish to get into details of how things vary with frequency, the accepted default frequency is 1000 Hz. Similarly, a graph can be made showing what SIL is required to obtain the same percieved loudness at different frquencies. Technically, the word loudness refers to this preception, and is measured using the unit phon. The "loudness" button on many stereos is intended to compensate for low sensitivity at low frequencies. If you turn down the volume below the intended playing level, low frequencies are reduced by the same SIL, which means they are reduced too much in terms of loudness. The loudness button boosts bass to compensate. Hearing level (HL) was measured for the hearing test. This measures a persons hearing threshold (in SIL) relative to the average human threshold (in SIL). [HW8.12, HW8.13, HW8.14]	(SIL individual threshold) = (SIL normal threshold) + HL

Equations

HINT: It is more effective to memorize these as relations between concepts. It is less effective to memorize these as strings of letters.

	Old	New
To memorize:	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\|	W = E/Δt I = W/A I_N = NI₁ A ∝ d² circle: A = πr² I₀ = 10^-12 W/m² {E, W, I}∝ {A, A_pp, v_max}² SIL individual threshold) = (SIL normal threshold) + HL
On Equation Sheet:	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	E = PE_max = 0.5 k A² sphere: A = 4πr² hemisphere: A = 2 πr² I ∝ 1/r² L_I = (10 dB) log(I/I₀) I = I₀ 10^(L_I/10 dB) ΔL_I = L_I₁-L_I2 = (10 dB)log(I₁/I₂) I₁/I₂ = 10^(ΔL_I/10 dB)