Superposition of pure tones
A superposition of two waves is just the sum of the two waves. It's easiest to see how to do the sum by looking at a fixed position; we then add pressure differences from the two waves at each instant in time. The best way to do this is by an example. We'll work this one out in gory detail. Other examples of superposition we'll look at in the coming few lectures will follow exactly the same method, but we'll use a computer to do the actual calculations.
We'll superimpose two pure tones corresponding to a fundamental (or first harmonic), having a frequency of 250 Hz, and the second harmonic, having a frequency of 500 Hz. A graph of the pressure oscillations at a fixed position, say a microphone, look like
The pure tones we're adding have different amplitudes; in general, in a superposition, the waves we add can have different amplitudes. The fundamental has an amplitude, p1 = 2 x 10-2 N/m2, and the second harmonic has half that amplitude. The other thing to note is that the second harmonic completes two full cycles in the same time that the fundamental completes one; this is a general feature of the harmonic series. We can represent the two waves shown graphically above as
The mathematical addition of the two waves requires us to evaluate the pressure associated from the fundamental and the second harmonic at a time, t, and then add the two pressures together. We'll do this in a table below, and then graph the results.
There's a couple things to note when you try reproducing the entries in the table. First, the time column is in units of milliseconds (10-3 seconds). Second, going from column 2 to 3 (or 4 to 5) using your calculator, you need to make sure that you see an "R" or "RAD" in the display meaning that angles for trig functions are in radians; otherwise you'll be entering angles in "D" or "DEG", i.e., in degrees.
It's hard to picture what the waveform resulting from the superposition looks like from the table. The only reason to go through this exercise is to show you that the procedure for doing wave superposition is simple! The complexity arises from the tedious task of first computing all of the sines (or cosines) corresponding to different harmonics at each instant in time, and then adding them together. This task is extremely well suited to a computer. For our example, to see the resulting waveform, it's necessary to graph the results from the table.
The points from the table are shown as dots in the graph. A smooth curve is drawn through the points to represent the waveform from the superposition. What's the end result from this exercise? The superposed wave has the period of the fudamental harmonic, yet has a more complex waveform! The procedure of superposition will enable us to produce complex waves!
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Last updated: 30 Oct 1999
Comments: bland@indiana.edu