Notes from Yost, Chapter 2: Sinusoids, The basic sound. Pp. 11-21. [15 January 2004]
This chapter is short but dense, with a number of formulas, some of which we should try to understand if not memorize. The fundamental point is that we hear "sounds" and that "sounds" result from certain sorts of vibration of objects in our environment, which exert a force on the surrounding medium (usually air). Objects that make sounds are oscillating or vibrating objects, that may repeat the temporal characteristics of their exertions at regular intervals. Objects that vibrate have two essential features which are extremely important in the production of sounds and in their reception: one is that they are elastic, that is, when they are displaced or deformed by the imposition of a force then they tend to return to their initial position; the other is inertia (or mass), which means that some force must be applied in order to induce the initial deformation. As most objects have elasticity and inertia most objects can vibrate (or oscillate) and thus can in principle serve as a sound source. A sound, at least writes Yost, is not necessarily audible to us (it could be audible to another species, for example, though even this is not perhaps a necessary criterion: it depends on the perspective, physicist or psychologist).
Yost then immediately introduces the concept of the Fourier transform, devised by the mathematician Fourier, who showed that any vibration can be synthesized by some collection of simple sinusoids of the right intensity, frequency, and phase: and otherwise, can be analyzed into a constituent group of appropriate sinusoids. An important conception, soon to be met in the text, is that early on in the auditory system the ear performs more or less of a Fourier analysis of any incoming vibration, to separate out the several sinusoids that comprise it. “Sinusoids” are so called because they are “sine waves”; and sine waves are so named because they can be generated by imagining the trigonometric sine function (remember, side opposite/hypotenuse) made up by rotating the radius of a circle anticlockwise, with a “side adjacent” being a horizontal line running through 180 to 0 degrees, the speed of rotation representing the frequency of the sinusoid, the size of the radius representing its amplitude, and the starting phase being the deviation in degrees (or radians) from the starting point at zero degrees. Pendulums describe simple sine waves of displacement over time; the vocal chords display complex sine waves (compounds of sound waves) of displacement over time; as do musical instruments, church bells, hammer strokes, tuning forks, the syrinx of birds, and so forth. The first equation is:
D(t) = A sin(2 Π ft + Θ)
this meaning that the displacement (D) at any time ‘t’ is equal to the maximum amplitude ‘A’ multiplied by the sine of a particular angle, which is given in radians (“2 Π f”, 2 Π being the number of radians in a circle = 360 degrees, and f being the frequency in cps of the sound wave) plus the starting phase, which is Θ, also in radians. When 0 degrees is taken as the right-ward horizontal position on a circle diameter then sine 0 = 0 and D = 0); then a vertical line is at Π /2 radians = 90 degrees and sine 90 = 1 while D is at a maximum (+A); the horizontal radius going to 180 degrees is at Π radians and again sine 180 = 0, and D = 0; and when the radius is vertical and descending then at 3 Π /2 radians or 270 degrees sine 90 = -1 and D is at its least value (-1). So this is the physical description of the sound wave: to a first approximation (or even a second) changes in frequency are heard as changes in pitch, while changes in amplitude are heard as changes in loudness; changes in phase are not heard directly but for binaural hearing differences in phase at the two ears are heard as differences in location. Primarily because all hearing creatures are more sensitive to some frequencies rather than others variation in amplitude cannot be equated directly with loudness — a certain intensity will seem louder if it is at a frequency for which we are very sensitive, 1000 Hz vs. 100 Hz, for example.
Frequency is measured in cycles per second or Hertz (Hz: honoring a 19th century physicist). Exactly lined up with frequency is the notion of the “period” of a sine wave, which is the duration of just one cycle. If the period of a wave is 1 second then it has a frequency of 1 Hz; if the period is 500 milliseconds (ms) then it has a frequency of 2 Hz. To be heard by humans vibrations should have periods no less that about 50 ms (20 Hz) and no more than about .05 ms (20 kHz). So more equations: f = 1/P(period) when P is given in seconds; and P = 1/f. Descriptions of physical stimuli usually speak of the frequency in Hz, rather than periods in time units: “Humans are most sensitive to tones with frequencies of 1000 to 4000 Hz” is the convention, not “Humans are most sensitive to tones with periods of 1 ms to .25 ms. ”Starting phase is given in degrees of angle, with zero phase where 2 Π ft = 0; one period is 360 degrees; a half period is 180 degrees, etc.; and these are the “instantaneous phase angles”. Amplitude is the amount of displacement, of course, and there is a measure called “instantaneous amplitude”; but also peak amplitude, peak-to-peak amplitude, and root-mean-square amplitude which is a running average over one wave, and is calculated by squaring the instantaneous amplitude, integrating it and getting the mean. RMS measures are useful for non-simple periodic sounds. For a sinusoid RMS amplitude is about 0.7 times the peak amplitude.
Figure 2.10 describes an idealized vibration source, consisting of a mass attached to the end of a spring. The mass (the inertial part) is pulled out by a force; the spring (the elastic part) is extended and yields a countervailing force in the opposite direction; when the imposed force is removed the spring pulls back and the mass returns beyond its original position, then oscillates back and forth, in “free vibration.” Due to a concept akin to wind resistance (not quite the same as friction) the oscillations gradually die out, in a series of “damped sinusoids” in which the amplitude gradually fades but the frequency remains the same, as is seen in Figure 2.11.
The supplement has a couple of important concepts embedded in the formulae. One is the importance of mass and its relationship to force: thus, as some famous physicist wrote:
F = m * a; so for a given force, then the greater the mass (m) of an object the less is its acceleration (a) and so the less is its displacement over time (this is Newton’s Second Law)
But also, for stiffness,
F = -s * x; so for a given stiffness(s) the greater is its displacement (x) the greater is the countervailing or restoring force (which is Hooke’s Law)
The reason that these are important (in fact, very important) is not intuitively obvious. It is that these relationships are responsible for the Fourier-like analysis done by the inner ear in separating out the different sinusoidal frequencies along the basilar membrane as Helmholtz had imagined, as they are related to the resonance of objects to particular frequencies (resonance is treated in Chapter 5, basilar membrane tuning in Chapter 7).