Vibrations
2 Anatomy of a vibration
Any back and forth motion is called a vibration. Examples are everywhere- kids on playground swings, plucked strings, weights attached to springs, and so on. Many objects that wiggle back and forth- like vocal cords, stereo speakers and tuning forks- cause ripples in the air that are detected by our ears as sound.
Frequency
Vibrations that are regular and predictable- like pendulums- are called periodic. Periodic vibrations are at the heart of music, human speech and almost any other sound where humans perceive pitch. Frequency is the measure of how often cycles repeat- the number of cycles in a given amount of time.
Mathematically, frequency is represented by a lower case script [latex]f[/latex]. In mathematical terms:
[latex]f = \frac{number \: of \: cycles \:} {amount \: of \: time \: for \: those \: cycles}[/latex]
For example, when an audio speaker cone goes in and out 8000 times in 2 seconds, the speaker’s frequency is 8000 cycles divided by 2 seconds, or 4000 cycles per second, or 4000 Hz. Hertz (abbreviated Hz) are the same thing as cycles per second. The Hertz is the official metric system unit for frequency, so you can use metric prefixes (like kilo- and mega-) with Hertz. For example, a radio station at 91.3 MHz creates 91.3 million cycles of radio waves every second!
Units matter!
Throughout the book, you will see green boxes like this one highlighting the important role of units in science.
Every scientific quantity (like 5.4 meters) is a number (like 5.4) multiplied by a unit of measure (like meters). Both the unit and the number matter!
Always include the units! What does “5.4” stand for? A weight in grams? A time in centuries? Without units, it’s impossible to tell!
Learn the math of units! Scientific quantities behave slightly different than numbers in math class- 10.8 ounces does not equal 10.8 grams, but 1 hour does equal 60 minutes. The rules for numbers from math class work for scientific quantities, but you must consider both the number and its unit.
Keep the units in your math! Label your quantities with units and follow the units through your math. This will help you make sense of your answer and sometimes help you spot (and avoid) mistakes.
Period
Another way to think about the “speed” of vibrations is in terms of the vibration’s period. Period is the amount of time it takes for one full cycle of a vibration. Mathematically, period is represented by the lower case Greek letter tau ([latex]\tau[/latex]). Here’s the mathematical definition of period:
[latex]\tau = \frac{amount \: of \: time}{number \: of \: cycles \: during \: that \: time}[/latex]
Period and frequency are inverses of each other- when frequency is high, each cycle of the vibration takes very little time. You’ll notice that equations for frequency and period are reciprocals of each other (like the fractions 3/7 and 7/3). As a result,
[latex]f = \dfrac{1} {\tau}[/latex]
or equivalently,
[latex]\tau = \dfrac{1} {f}[/latex]
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