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An introduction to electronics
An introduction to electronics

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4.4 Signals and sine waves

Interactive 2 shows a signal based on a sine wave, sin(2πft), where f is the frequency of the wave and t is time. Here the frequency of the wave is f = 200 Hz. The period of the wave, T, is the time between successive peaks; in this case, T = 5 ms (milliseconds). As would be expected, T = 1/f = 1/200 = 0.005 s. Similarly, f = 1/T = 1/0.005 = 200 Hz.

The signal is written as

 s(t) = A sin(2πft + ϕ)

where A is called the amplitude of the wave, i.e. the largest value of the wave above or below the horizontal axis. Here it is set to A = 0.5.

ϕ is the phase of the wave, which means how far the wave is shifted to the left or the right. Here it is set to 0, since the wave goes through the origin and is not shifted either left or right.

Right-click on the image below to open the interactive in a new tab, then press the loudspeaker button to hear what the signal sounds like.


View interactive version
Interactive 2 Investigating a sine wave
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This is an interactive that may be used to display a visual representation of a sine wave on the screen and also hear it as a pure tone. At the top is the sine wave equation, s(t) = A sin(2πft + ϕ), but A for amplitude, f for frequency and ϕ for phase have been replaced by type-in text boxes into which values can be entered or selected by using the up and down arrows on the right of each box. Below this is a graph showing voltage in volts against time in milliseconds. The sine wave described by the specific values entered into the equation is displayed. To start with, because the default values of the interactive are an amplitude of 0.5, a frequency of 200 and a phase of 0, the sine wave varies between 0.5 and minus 0.5 volts, and completes one cycle in 5 milliseconds. The wave starts at the origin, rises to the maximum, falls to the minimum and then returns to 0. This part of the interactive changes according to the parameters you enter and therefore you may need a sighted helper to describe the waveforms to you. Beside the graph is a loudspeaker button; if you click on this, you will hear the sine wave shown on the graph as a pure tone.

Interactive 2 Investigating a sine wave
Interactive feature not available in single page view (see it in standard view).

SAQ 7

  • a.Change A to 1.0 and play the sound. What do you observe?
  • b.Change f to 400 and play the sound. What do you observe?
  • c.Change ϕ to π/2 and play the sound. (You can enter a phase of π/2 into the interactive as 90 degrees.) What do you observe?

Answer

  • a.The wave is displayed with twice the height and the sound is louder.
  • b.The wave is displayed with the lines much closer vertically, and the sound is at a higher pitch (it is an octave higher than before).
  • c.The wave is moved along by a quarter of its period. Now it has the highest value at t = 0, where previously it was 0 at t = 0. However, it sounds the same.