# What does it mean to multiply two sine waves together

Suppose I have a wave pool with two sources, A and B. The two sources both produce sine waves of the same frequency, and the sources are separated by a distance of the period of the wave.

So they will make an interference pattern on the surface of the pond (or if they are light waves, an interference pattern of light).

Now I want to add another wave source - this one the product of A and B. The puzzle is how to do that. I can mathematically multiply a sine by another sine - it will give a wave of the same period, but the shape is different - however, I don't know that you could generate a wave like this with either water or light.

Is there any solution other than that one? My goal is to find the values of A and B such that a sum such as "A + B - AB" is maximized and I thought of using waves as a way to do this kind of math.

• Not sure what's going on with all the waves and stuff, but is your only goal to just find $A,B$ such that $A+B-AB$ is maximised? If so, what are $A$ and $B$? Can they take negative values? – Displayname Jan 27 at 23:05
• What's the purpose of the "distorted sine wave" $AB$? Physically, multiplying two sine waves together amounts to using one to modulate the amplitude of the other. In this case, you're getting a non-sinusoidal but still periodic wave of the same frequency, so basically adding a load of harmonics to it. – timtfj Jan 28 at 0:25
• Also $where$ is $A+B-AB$ maximised? If the wave sources aren't all in the same place, there'll be different phase differences at different locations (hence the interference pattern). – timtfj Jan 28 at 0:38
• I think for your $AB$ source you'd be best generating the shape electronically, unless there's some non-linear physical effect for filtering the actual wave once it's been turned into light or vibration or whatever. – timtfj Jan 28 at 0:45
• I didn't have room in a comment to say everything relevant, so I posted my replies as an 'answer'. – Mark Springer Jan 28 at 11:57