Written by Travis M. Moore
Last edited 20-Sep-2019
What is a 1000-Hz sine wave plus another 1000-Hz sine wave? What about a 1000-Hz wave + a 1200-Hz wave? Combining sine waves isn't difficult, but you will need to keep in mind that sine waves have amplitudes have negative and positive values.
When adding sine waves together, the only equations you need are:
The only wrinkle is minor, and already addressed in the "equations" above. A sine wave starts at 0 deg, peaks at 90 deg, and returns to 0 deg. However, that is only half a cycle. The sine wave continues to -90 deg and back up to 0 deg to complete a full "revolution." This means at times we'll be taking amplitude away when one wave is at a positive part of its cycle and the other wave is at a negative part of its cycle. This reduction in amplitude from summing a positive and negative value is called destructive interference. The opposite is also true. When both waves are in the positive part of their cycles, we will be adding amplitudes. This increase in amplitude when summing two waves in the positive part of their cycles is called constructive interference. These processes make for some interesting waveforms, but stick with the basic arithmetic and you won't go wrong.
Figure 1 shows an animation of the addition process. It shows two waves, one blue and one green. A specific point in time is selected, marked by a blue or green circle. Note the blue and green circles are at different amplitudes, but always at the same time. Next, the simple addition of the blue and green amplitudes is indicated by a red circle. This will repeat for most of a cycle. Try to see the "shape" of the summed wave by connecting the red dots in your head. If you're unimaginative, don't worry! The red dots are connected with a red line to show you the true shape of the resulting wave.
Figure 2 is another way to look at the same thing going on in Figure 1, but in real time. The top panel in Figure 2 shows a blue stationary wave. The middle panel in Figure 2 displays a red wave that is moving to the left. The bottom panel is the most interesting; it plots the sum of the blue and red waveforms across all time points, every time the red wave moves another "step". Single points with vertical lines show the summation at a single point. The numbers next to the vertical lines show the amplitude of the colored dots in real time. In other words, the bottom panel is the sum of the blue and red panels.
Notice what happens in Figure 2 when the waves are exactly opposite each other. Every time a point in the blue wave is positive, the same time point is equally negative in the red wave. The result in a flat line! This is essentially how noise-cancelling headphones work. There is a microphone outside the earcup to collect the ambient sound. That sound is then altered until it is exactly out of phase (180 deg) with the sound leaking in naturally around the earcup. The result is a cancellation of pressures, and without traveling pressure, there is no sound.
There are two especially interesting cases of sine wave addition, and each one has to do with adding together sine waves that are multiples of a given frequency. The frequency used in the multiplication is referred to as the fundamental frequency, abbreviated F0, and represents the lowest frequency in the set. The table below illustrates multiples of 250 Hz.
Table 1. Integer multiples of 250 Hz. |
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F0 (Hz) | Multiple | Result (Hz) |
250 | 1 | 250 |
250 | 2 | 500 |
250 | 3 | 750 |
250 | 4 | 1000 |
The multiples of 250 Hz in Table 1 are also known as harmonics of 250 Hz. Any integer multiple of a frequency is an harmonic. As a random example, $2090$ is a multiple of $418\; (2090\; =\; 418\; *\; 5)$. It turns out we can make waveforms called square waves and sawtooth waves by adjusting the amplitude and adding together certain harmonics of a given frequency. The formulae for these waves are quite simple:
Table 2. Square Wave with Fundamental of 250 Hz. |
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F0 (Hz) | Multiple | Harmonic (Hz) | Amplitude Value | |
250 | 3 | 750 | 1/3 | |
250 | 5 | 1250 | 1/5 | |
250 | 7 | 1750 | 1/7 | |
250 | 9 | 2250 | 1/9 |
The animation below (Figure 3) shows the creation of a square wave adding one odd-numbered harmonic at a time. The result really does have squared sides (even though it technically looks more like a rectangle).
And now for a sawtooth wave. Note that the "N=" in the bottom left-hand corner is a tally of how many harmonics are being added to the fundamental. The wave looks more and more like the tooth of a saw as more harmonics are added.
Table 2. Sawtooth Wave with Fundamental of 250 Hz. |
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F0 (Hz) | Multiple | Harmonic (Hz) | Amplitude Value | |
250 | 2 | 500 | 1/2 | |
250 | 4 | 1000 | 1/4 | |
250 | 6 | 1500 | 1/6 | |
250 | 8 | 2000 | 1/8 |