To expand on the math inherit within waves:
Basic Math
Wave A: amplitude and frequency 3 Wave B: amplitude and frequency 4
Mix both waves = 2 new waves each with amplitude 6 where new wave C has a frequency of 1 and new wave D has a frequency of 7.
What this represents is simply by mixing the waves (no math needed), I have a difference (subtraction) |3-4|=1, addition 3+4=7, multiplication 3 * 4=(amplitude of either wave)*2=12. note: multiplication is a result of when the 2 new waves phase and create a maximum amplitude, in this case 12. We don't need to wait for this to happen though because we know it 'must' happen and we know that the amplitude of the 2 new waves are exactly the same.
Trig
Wave A: amplitude and frequency 3 Wave B: amplitude and frequency 4 (phase 90 degrees)
If no other input is given we will assume 90 degree angle and phase one of the waves by 90 degrees. This quite literally gives us the Pythagorean theorem simply by reading the result of the waves. When mixed, the resulting amplitude is 5 which is exactly a^2+b^2=c^2 (in this case 3^2 + 4^2 = 5^2). So by mixing the 2 waves with one phased 90 degrees, we simply read the resulting amplitude to get the hypotenuse of a triangle.
The wave interference naturally encodes ALL triangle geometry through amplitude + phase relationships. Taking the same waves and if no other impute is give we can assume a 90 degree angle. All we have to do is phase one of the waves by 90 degrees.
Calculus (Rate of Change)
The derivative of a sine wave IS a cosine wave (simply phasing the original wave by 90 degrees) d/dt[sin(wt)]=w*cos(wt) So to get the derivative of a wave - shift phase that wave by 90 degrees and scale amplitude by frequency To get the second derivative of a wave - shift another 90 degrees (now 180 degrees) and scale by w^2. etc...
Integrals is simply the reverse. fsin(wt)dt=-1/w*cos(wt) Simply phase the wave -90 degrees and divide amplitude by frequency.