If you have an oscilloscope you'll see a new wave which is the sum of the two originals. An alternative way is to look at a wave or sum of waves in the frequency domain , which is amplitude vs. frequency , rather than time. To do that you need a spectrum analyzer . The spectrum analyzer won't show you the sum of the waves, as did the 'scope. And the way to do that, mathematically, is to multiply the two waves together, instead of simply adding them.
Multiplication is nonlinear addition.
In radio work this is called heterodyning . Now they are interacting with each other, big time. Here is where we find something unexpected. We find that multiplying two waves together in the time domain is exactly the same as shifting frequencies up and down, simultaneously, in the frequency domain. They appear whenever two or more waves intermodulate one another, and they are how low frequency audio waves get shifted up the spectrum to cluster around the carrier wave, which is way above the limit of audibility.