Waveforms are patterns of signals that propagate through a medium such as water or air, or in the case of electromagnetic radiation, through empty space with no apparent material medium. (It is still not known how this is possible.) Electrical current, of course, can also propagate as a waveform. We call these patterns “waves” because they are analogous to the rhythmic rising and falling water levels in the world’s oceans.

These waves can be irregular and non-repetitive, as in sound waves and derived electrical patterns produced by the human voice. Often, however, they are repetitive, changes in the pattern taking place relatively slowly if at all over some number of cycles. These waves arise in nature or they may be human-made.

A common type of repetitive signal is the sine wave. Electricity produced by most generators conforms to a sine wave pattern because of the rotary nature of the machine. To see why that is so, we must go back to high school trigonometry and the mathematical definition of the sine as a trigonometric function. (A waveform is a graphic representation of a function.)

Each trigonometric function pertains to the ratio of two specified sides of a right triangle. A right triangle may be placed on an X-Y grid as comprising an oscilloscope display in the time domain.

The angle θ is generated at the intersection of the hypotenuse and side that is a segment of the X-axis. When θ is specified, the other two angles are known because in a right triangle one angle is always 90° and the sum of the three angles is always the same (180°). Therefore, for any angle θ, the proportions of the triangle are the same. If the amplitude of the signal changes, the size of the triangle will increase or decrease accordingly, but the proportions of the triangle are constant for any given value of θ.

As a convention, the two legs of a right triangle are named according to their position with respect to θ. They are opposite and adjacent. The trigonometric ratios are:

• Sine (sin) – opposite/hypotenuse

• Cosine (cos) – adjacent/hypotenuse

• Tangent (tan) – opposite/adjacent

• Cotangent (cot) – adjacent/opposite

• Secant (sec) – hypotenuse/adjacent

• Cosecant (csc) – hypotenuse/opposite

For each trigonometric function of a specified angle θ, there is a unique value, and it can be used in conjunction with other trigonometric functions to find the length of two remaining sides if any one of them is known. These values are readily available in books of tables that were widely used at one time. These values are generated when needed by computers and hand calculators, and they may be obtained from the internet. For example, type into a search engine “cosine of 50 degrees” and the result, 0.64278760968, immediately pops up. Additionally, there is an on-screen calculator that can generate trigonometric functions.

In any discussion of waveforms, it is the trigonometric function of sine that is of overwhelming importance. The moving point on a graph having X-Y coordinates such as a time-domain display in an oscilloscope as generated by the sine function corresponds to this most basic of waveforms. The X-axis is generally defined to correspond to the passage of time while the Y-axis corresponds to amplitude. Looking at the trace, the defining characteristic is clear: the rate of change is greatest when the amplitude is least, at the instant when the trace crosses the X-axis and the value of Y is zero, and the rate of change is least, equal to zero, when the positive and negative values of Y are greatest. This is the essence of the sine wave.

When plucked, a guitar string produces a tone containing significant overtones or harmonics, so it is not exactly a sine wave, but the principle is the same. The vibratory motion and resulting sound are greatest upon being plucked, and they diminish in proportion to the passage of time. The reduction arises from the tension of the string, which is greatest at the peaks, and also to air resistance. To create the sound the string also expends energy in moving the air. In both of these activities, heat is generated and dissipated and the vibration is reduced. The result is a damped wave.

If harmonics are disregarded, the motion of the string and sound made by it take the form of sine waves, with rate of change least at the amplitude peaks. Because the length of the string is slightly longer at these instants, the tension is greater, slowing the motion and causing the rate-of-change to diminish. Because of the increasing resistance to change, the sine wave becomes flatter at the peaks and it is steeper when the amplitude is least.

A tuning fork was designed with a specific agenda in mind. It happens that tuning a musical instrument is best accomplished by comparing its tone to a pure sine wave, and the tuning fork is suitable for this purpose. It was invented in England in 1711 and has served generations of concert musicians and piano tuners who need an accurate reference. (Most modern piano tuners use electronic instruments. One type has a small oscilloscope-type display in *XY* triggered mode that compares by means of a Lissajous pattern the pitch of the instrument being tested to an internally-generated standard. When the display stabilizes, the technician knows that the piano string is accurately tuned.)

The tuning fork, usually made of steel, consists of two prongs that join at a base. When struck against a stationary object, it emits a tone (usually concert A, currently defined as 440 Hz.) The tone is sinusoidal with faint harmonics, known to musicians as overtones. The first harmonic is about 6 ¼ times the fundamental, and because of the high frequency that requires a lot of energy to sustain it, the harmonics fade rapidly leaving a high-quality tone that conforms to a near-perfect sine wave.

When vibrating, the distance between the prongs changes while the handle moves vertically. There is a location where the prongs meet that handle that the metal does not vibrate appreciably, and that is why it can be grasped there without damping the sound wave. Because the oscillations of the two prongs are 180° out of phase, the waveforms cancel one another and the sound is not pronounced. If a sheet of paper is held halfway between the prongs without touching the tuning fork, the sound becomes much louder. Otherwise, the audible sound comes from the handle. When it is touched to a resonance box or wooden tabletop, the apparent volume increases.

Tuning forks are customarily manufactured to exact tolerances, and the frequency is stamped on them. Removing metal by filing it off will alter the pitch. If the prongs are shortened by filing equal amounts of metal off the ends, the pitch is raised. The pitch may be lowered by removing metal from the inside of the base where the prongs join.

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