Waves

What if we want to graph the pressure changes in air made by somebody playing the flute? The graph might look a bit like this:

The vertical axis is pressure and the horizontal axis is time. We can see the pressure increase, decease and increase again. The idealised wave form shown here is a sine wave. This wave has exactly one frequency in it and is the simplest possible wave form.

If you generate a sine wave in your DAW and then zoom way in, you’ll see exactly the same shape, but in that case, the Y axis is how much the speaker cone will offset when we play back the sound. This makes sense. The speaker needs to push the air to make the sound wave. If we were looking at an analogue signal to the speaker via an oscilloscope, the Y axis would be the amount of voltage.

If the wave is taller, the speaker moves more air and the sound is louder. The height of the wave is the amplitude.

The height of the wave is labelled amplitude and the distance peak to peak is the wavelength. — Figure 2: Wave, wavelength and amplitude CC BY 3.0 <https://creativecommons.org/licenses/by/3.0>, via Wikimedia Commons

The distance from one peak to another, λ, is the wavelength. If the wavelength is shorter, the speaker cone moves faster. A faster movement and a shorter wavelength means a higher frequency.

We’ve measured from the peaks, but we could measure from any point along the curve, for instance, from the zero crossings, as long as the wave has been through a complete cycle.

Figure 3: Phase by Baskurtf at Turkish Wikipedia., CC BY-SA 2.5 https://creativecommons.org/licenses/by-sa/2.5, via Wikimedia Commons

If waves start at different points but have the same wavelength, we say they are the same frequency but have different phases. In figure 3, the red line starts at zero and is a sine wave. The blue line starts at 1 and is a cosine wave. They both are the same frequency.

We have the unit circle (with radius = 1) in green, placed at the origin at the bottom right.

In the middle of this circle, in yellow, is represented the angle theta (θ). This angle is the amount of counter-clockwise rotation around the circle starting from the right, on the x-axis, as illustrated. An exact copy of this little angle is shown at the top right, as a visual illustration of the definition of θ.

At this angle, and starting at the origin, a (faint) green line is traced outwards, radially. This line intersects the unit circle at a single point, which is the green point spinning around at a constant rate as the angle θ changes, also at a constant rate.

The vertical position of this point is projected straight (along the faint red line) onto the graph on the left of the circle. This results in the red point. The y-coordinate of this red point (the same as the y-coordinate of the green point) is the value of the sine function evaluated at the angle θ, that is:

y coordinate of green point = sin θ

As the angle θ changes, the red point moves up and down, tracing the red graph. This is the graph for the sine function. The faint vertical lines seen passing to the left are marking every quadrant along the circle, that is, at every angle of 90° or π/2 radians. Notice how the sine curve goes from 1, to zero, to -1, then back to zero, at exactly these lines. This is reflecting the fact sin(0) = 0, sin(π/2) =1, sin(π) = 0 and sin(3π/ 2) -1

A similar process is done with the x-coordinate of the green point. However, since the x-coordinate is tilted from the usual convention to plot graphs (where y = f(x), with y vertical and x horizontal), an “untilt” operation was performed in order to repeat the process again in the same orientation, instead of vertically. This was represented by a “bend”, seen on the top right.

Again, the green point is projected upwards (along the faint blue line) and this “bent” projection ends up in the top graph’s rightmost edge, at the blue point. The y-coordinate of this blue point (which, due to the “bend” in the projection, is the same as the x-coordinate of the green point) is the value of the cosine function evaluated at the angle θ, that is:

x coordinate of green point = cos θ

The blue curve traced by this point, as it moves up and down with changing θ, is the the graph of the cosine function. Notice again how it behaves at it crosses every quadrant, reflecting the fact cos(0) = 1, cos(π/2) = 0, cos(π) = -1 and cos(3π/2) = 0. — Figure 4: Sine and Cosine wave by Lucas Vieira, Public domain, via Wikimedia Commons

In figure 4, we can see an animation of the cosine and sine wave moving at the same frequency and how they are related to each other.

Summary

In the last three posts, we learned that sound is made up of tiny pressure waves which travel at 340 m/s. When these strike our ear drums, this in turn causes our basilar membrane to vibrate. Distinct vibrations on the membrane are heard as distinct frequencies.

We can graph the pressure waves of the sound. This is the same as the waveform graph in our DAW and is the same as the change in voltage of the signal going to our speakers. All signals going to our speakers have an amplitude, where taller is louder. Periodic sounds, like sine waves, also have a frequency, where a shorter wave length is a faster vibration and a higher pitch.

Waves can have the same frequency but be out of phase with each other, so their peaks and troughs do not line up.

Supplementary Reading

Everest, F.A. and Pohlmann, K.C. (2015). Master Handbook of Acoustics. Sixth edition. New York: McGraw-Hill Education. – Chapter 1

Activity

Materials

Audacity
Sonic Visualiser
A Microphone
An audio interface (or other way to get microphone input into your computer.)
A quiet corridor with a wall some meters distant
A tape measure
Optional: a room thermometer

Method

Place your microphone so it points at the wall. Start recording into Audacity. Stand behind the microphone. Clap. Stop recording.

Check your recording. You should have two impulses on the recording. One is the loud clap and the second is the echo of the clap. If these are too close together, move further from the wall.

Once you have a clean recording, export it as a WAV file and open it in Sonic Visualiser. Use the tape measure to measure how far you are from the wall.

Listen for when the first echo appears, and see if you can measure the distance in milliseconds using the display.

You might need to experiment a bit with the zoom controls, and possible other controls in Sonic Visualiser to make it clearer to see where the echo appears.

Also, it won’t necessarily be an exact point, so you may have to use your judgement.

Remember that the sound has to travel to the wall and back, so the total distance is double what you measured.

What was the speed of the sound. Is it what you expected? If you were able to measure the temperature, how much impact did that have on the speed?

Summary

Supplementary Reading

Activity

Materials

Method

Published by

Charles Céleste Hutchins

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