Vibrations

2 Anatomy of a vibration

Bulova Accutron watch (1971)

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. The most accurate modern clocks rely on atomic vibrations to keep time. Electrical vibrations are the basis of radio signals and microwaves ovens.

The vibrations that cause sound are mechanical vibrations- objects that physically wiggle back and forth, causing ripples in the air that surrounds them. Examples include the paper cone in a audio speaker, the ends of a tuning fork and our vocal cords. The critical component for all these noisemakers is a surface that moves back and forth.

Some vibrations are regular and predictable, like the motion of a pendulum. Regularly repeating vibrations are called periodic. Periodic vibrations are at the heart of music, human speech and almost any other sound where humans perceive pitch.

Frequency and period

For any periodic vibration, perhaps the most characteristic is its frequency. Frequency is how often cycles repeat- the number of cycles in a given amount of time. Mathematically, frequency is represented by a lower case script f.  In mathematical terms:

f = \frac{number \: of \: cycles \:} {amount \: of \: time \: for \: those \: cycles}

Frequencies are almost always expressed in Hertz (Hz). One Hertz is one cycle per second and is the international unit for frequency. The unit honors Heinrich Hertz (1857-1894), a German physicist who studied sound and many other things. The vibrations that cause the sound we hear have frequencies that range from about 20 Hz to around 20,000 Hz.

Period is the amount of time it takes for one full cycle of a vibration. Mathematically, period is usually represented by the lower case Greek letter tau (\tau ).

\tau = \frac{amount \: of \: time}{number \: of \: cycles \: during \: that \: time}

Periods usually are expressed in seconds or seconds per cycle.

Frequency and period are inversely related. Things that repeat quickly naturally have short periods. Compare sunrises and full moons, for example. Sunrises are more frequently than full moons, but days are short, compared to the amount of time between full moons. Expressed mathematically, the relationship between frequency and period is

f = \dfrac{1} {\tau}

or equivalently,

\tau = \dfrac{1} {f}

If you know the frequency, you automatically know the period, and vice versa. A vibration with a frequency of 20 Hz completes 20 full cycles in a second, so each cycle only takes 1/20th of a second.

Some numbers for common vibrations

The table below shows frequency and (corresponding) period for some common “vibrations.”

Frequency (in Hz)

Period (in sec)

 Human heartbeat  ~1 ~1
 Lowest note on the piano ~30 ~0.03
 Highest note on the piano  ~4,000 ~0.0002
 CD sampling rate  ~40,000 ~0.00002
 Cell phone processor  2,000,000,000 ~0.0000000005

Vibrations with long periods (longer than 1 sec) and low frequencies (less than 1 Hz) are sometimes called oscillations.

Frequency and sound

The vibrations that cause sound that human ears can hear have frequencies between about 20 Hz and 20 kHz. Human sensation of pitch is built on frequency. We hear low frequency vibrations as low pitched- think  tuba or foghorn. High frequency vibrations are perceived as high pitched- think piccolo. There is a direct connection between frequency and musical note names. On the piano, each key has a specific frequency that it’s tuned to and the keys are arranged in order of frequency/pitch, with the lowest frequency/pitch at the left end of the keyboard.

Most sounds have frequencies that change. The changes can be slow, like a melody, or fast, like the inflections in human speech or bird calls. These are called quasi-periodic vibrations. If frequency is steady for even a short amount of time- a few milliseconds is enough- the sound is perceived as musical or pitched. Even sounds that don’t seem especially musical, like human speech, often involve quasi-periodic vibration.

Some sounds are caused by vibrations that don’t repeat at all. These are called aperiodic vibrations.  Transients and noise are examples of aperiodic vibrations. Transients don’t last very long- things like a popping balloon or the crack of a whip. Noise refers to random vibrations that last a little while- the object moves back and forth for a long time, but there is no pattern to the motion. Random vibrations cause sounds like static on the radio or air rushing out of a tire or the librarian’s “shhhh.”

Stop to think 1

A bee flaps its wings about 200 times per second, while an eagle flaps its wings about 5 times per second. Which motion has the longer period? Which has the greater frequency?

Amplitude

Diagram of vertical spring-mass system showing amplitude
Amplitude is measured from the middle of the motion to one extreme.

Amplitude describes “how large” a vibration is. Amplitude can be measured in almost any units, depending on what’s fluctuating.

Amplitudes based on location (often called displacement amplitudes) are the easiest to visualize. Displacement amplitudes are perfect for describing spring-mass systems. Imagine a weight that bobs up and down between two locations 6 cm apart. The displacement amplitude is 3 cm.  (Remember that amplitude is measured from the middle to one extreme).

Amplitude and sound

Amplitudes don’t have to be based on location. In sound, it is the air pressure that fluctuates so pressure amplitude is most basic way for describing sound amplitude. Pressure amplitude is measured in Pascals (Pa), the international unit for pressure.

The human sensation of loudness is built mainly on pressure amplitude (though frequency also plays a role). In general, the larger the pressure amplitude a sound has, the louder it sounds. While the official unit for pressure amplitude is the Pascals, pressure amplitude is more commonly expressed in decibels. This is because the pressure amplitudes in everyday sounds varies widely- loud sounds can have pressure amplitudes more than a million times larger than the quietest sounds we can hear. The decibel scale transforms sound pressure amplitude numbers into a much more human-friendly format. The table below compares sound pressures and sound pressure levels for some everyday sounds.

Sound Pressure amplitude (in μPa)

Sound Pressure Level (in dB)

 Hearing threshold  20 0
 Rice krispies  600 30
 Conversation  20,000 60
 Leaf blower  600,000 90
 Loud indoor arena  20,000,000 120

Decibels take advantage of logarithms, which you’ll tackle in more detail later.

Stop to think 2

While you listen to the stereo, air pressure in your ear might vary between 101,200 Pa and 101,400 Pa. What is the pressure amplitude of this sound? (The abbreviation “Pa” stands for “Pascal”- the metric unit for pressure).

Online resources

At the end of most sections of this book, you will find links to online content pertaining directly to the material you’ve just read. For best results, try to view/explore as many of these resources right after doing the reading. (A fuller list of resources can be found in the “Electronic resources” in the Useful Information section of this book).

View Measuring Frequency (a 2:26 youTube video) [1]. This video leads you through how to measure period (and frequency) for a spring-mass oscillator, including the numerical calculations.

Play with the animation SpringMassPend (a Desmos simulation)[2]. Play with the sliders for frequency and amplitude and see how the motion changes. BTW, this simulation was written using the free online graphing calculator at Desmos.com, developed by Texas Instruments. If you like math even a little bit, learn Desmos yourself. It’s easy and fun!

Stop to Think Answers

Stop to think answers

  1. The eagle’s flapping has the longer period. The bee’s wings flap more frequently.
  2. The pressure amplitude of the sound is one hundred Pascals. (Subtract the lowest pressure from the highest and divide by two). Notice that the variation in air pressure in the sound is pretty small compared to the average air pressure.

Image credits

  1. Accutron Watch: https://commons.wikimedia.org/wiki/File:Accutron.jpg
  2. Amplitude diagram: created by David Abbott

References


  1. Abbott, D. (2018, January 18). Measuring Frequency. Retrieved from https://youtu.be/GIqKJpu1fcU
  2. Abbott, D. (2018, August 10). SpringMassPend. Retrieved from https://www.desmos.com/calculator/glflmhla4m

License

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Understanding Sound by abbottds is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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