Vibrations

3 Time domain graphs

The vibrations that cause sound are (mostly) invisible- too small and too quick for the human eye to see. Sound engineers and scientists use electronics and the graphs they produce to make sense of vibrations and the sounds they cause. Time domain graphs are the most basic of these graphs. This book will also explore frequency domain graphs and the spectrograms in later chapters.

Time domain graphs

A time domain graph shows how some quantity fluctuates over time. Time is plotted along the horizontal axis and the fluctuating quantity is plotted along the vertical axis. Time domain graphs are sometimes called history graphs. Position-time graphs and pressure-time graphs are two common examples of time domain graphs.

To understand a time domain graph is, watch Spray Paint Oscillator (0:50 youTube video by the MIT Physics Demo Group) [1] now. The video shows a spray paint can bobbing up and down on the end of a spring as paper is pulled from right to left in front of spraying can. The paint trace on the paper is a record of where the spray can was (and when it was there). Time is recorded from left to right along the horizontal axis, while the motion of the can is recorded along the vertical axis. Paint near the top of the paper was laid down when the can was near the top of the paper; paint at the bottom of the paper was applied when the can was near the bottom of its motion.

Sine curves, complex vibrations and noise

You can tell a lot about a vibration (or a sound) by looking at its time-domain graph. Most musical sounds are created by periodic vibrations- vibrations with a repeating pattern. If the time domain graph is a sine curve, like the MIT spray paint example, the vibration is called simple harmonic oscillation (SHO for short). SHO are the simplest possible type of vibration- the source is vibrating at a single, unique frequency. The sound caused by an SHO is often called pure tone.

Pure tones are not common in real life. Most musical sounds are caused by complex vibrations– periodic vibrations that are more complex than the SHO. The time domain graph of a complex vibration shows a repeating pattern, but the graph isn’t a sine curve. When a musical instrument like a bassoon or a violin, plays a note, the result is a complex vibration. Complex tones are made up of many distinct frequencies happening simultaneously. 

Three different periodic vibrations, all with the same period

Not all vibrations have a repeating pattern. If there’s no repeating pattern on the time domain graph, the vibration is called aperiodic or noise. Sounds like “shhhh” are created by aperiodic vibrations. Because noise is aperiodic, the concepts of period and frequency don’t apply. Noise is made of a random mix of all possible frequencies..

Non-periodic vibration
Time-domain graph of an aperiodic vibration (aka noise).

Period and amplitude on time domain graphs (SHM)

Period is defined as the time for one full cycle. On a time-domain graph, the period is the horizontal “distance” from the beginning of one cycle to the beginning of the next. (Keep in mind that horizontal “distances” on the time graph actually represent amounts of time). Amplitude is defined as the maximum deviation from equilibrium. On a time-domain graph, amplitude is the vertical “distance” on the graph from the middle of the vibration to either the “top” or the “bottom.” What the vertical “distance” on the time domain graph depends on the graph. If the graph is a pressure-time graph, vertical distances on the graph corresponds to differences in pressure. If the graph is a position-time, vertical “distances” on the graph corresponds to differences in position and so on.

Period and amplitude of a simple harmonic motion on a time domain graph

Measuring period, amplitude and frequency from real world graphs

To get an accurate value for period from real world graphs, it is often useful to measure the total time for a large number of cycles and divide the total time by the number of cycles:

\tau = \frac{amount \: of \: time \: for \: N \: cycles}{N}

Why measure the time for lots of cycles? Have you even tried to measure the thickness of a single sheet of paper with a ruler? Hopeless, right? However, you can easily measure the thickness of a large number of 500 sheets and the total thickness by 500.

You can also find frequency directly from a time domain graph, since frequency and period are fundamentally related:

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

An example

The example below shows how to read period and amplitude from a time-domain graph.

Time domain graph of SHO with amplitude 4 Volts and period of 5 millisecond

There are lots of correct ways to calculate period from a time domain graph. All are based on the equation

\tau = \frac{amount \: of \: time \: for \: N \: cycles}{N}

Simply count off a number of full cycles, and note when the cycles begin and end. Subtract the two clock readings from the horizontal axis and divide by the number of cycles you used. On the graph above, you could notice that one cycle starts at  t=10 \:ms and the next begins at  t=15 \:ms or you could notice that the first four cycles take 20 \:ms . Either way results in the same correct answer-  5 \:ms . Once you have the period, you calculate the frequency of the vibration. (If the vibration is complex, the frequency you calculate from the fundamental period is called the fundamental frequency).

On a time domain graph, amplitude is the vertical “distance” from the equilibrium to one extreme. Keep in mind that vertical “distances” on the graph might not represent distances. To figure out what vertical “distances” mean for your graph, consult the label on the y-axis. In the graph above, the y-axis is in Volts, so the amplitude is in Volts. There are multiple correct ways of finding the amplitude from the graph. All boil down to subtracting y-values. In the example above, you could subtract the equilibrium voltage (3 V) from the maximum voltage (7 V). Alternatively, you could subtract the minimum voltage (-1 V) from the maximum voltage (7 V) and divide the result by two. (Remember that amplitude is from the middle to one extreme- not extreme to extreme). Either way, the result is the same (4 Volts).

Online resources

Watch Spray Paint Oscillator (0:50 youTube video) [2] now! (if you haven’t done it already). It’s just that good/important!

Image credits

  1. Time domain graph of three vibrations: David Abbott; created using Desmos.com.
  2. Time-domain graph of an aperiodic vibration (aka noise): David Abbott; created by saying “shhh” into a microphone using LoggerPro software.
  3. Time domain graph of SHO with amplitude 4 Volts and period of 5 millisecond: David Abbott; created using Desmos.com.

  1. MIT Physics Demo Group. (2009, March 23). Spray Paint Oscillator. Retrieved from https://youtu.be/P-Umre5Np_0
  2. MIT Physics Demo Group. (2009, March 23). Spray Paint Oscillator. Retrieved from https://youtu.be/P-Umre5Np_0

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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