# 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

## 1 Answer

To answer these comments in order:

Think of a network of nodes (like a neural net or a Hopfield net) which are connected by symmetrical bidirectional connections. If two nodes that have an activation of 1 are connected by a weight of 1, then they reinforce each other. If they are connected by a weight of -1, then they inhibit each other. If two nodes have an activation of -1, then again, a weight of 1 means reinforcement. If one node has an activation of 1, and the other has an activation of -1, then a weight of 1 between them would be inhibitory.

People set up these networks, and let them evolve over time, and try to get the most activations that are consistent with each other. The network doesn't always settle into the best solution, but it can settle into a almost-good solution.

You can express networks like these with equations such as the above (A + B - AB + CD - CA.....

So you need to be able to simulate both sums and products, including products with negative signs. My idea is that somewhere in the interference pattern there is a maximal solution (maybe it would appear darkest on a photo, for instance). The nice thing about an interference pattern is that it tries out all combinations (I think). But can it be done?

An equation such as A + B - C can be expressed as waves - A and B can be the source of waves that are in phase and with the same frequency, and C can be the source of a wave that is 90 degrees out of phase with the others. But how do you express the products (AB, CD, etc)?

On the last comment, are you saying that you generate any arbitrary wave shape electronically?