From the wikipedia article on sine waves:
The sine wave is important in physics because it retains its wave
shape when added to another sine wave of the same frequency and
arbitrary phase and magnitude. It is the only periodic waveform that
has this property. This property leads to its importance in Fourier
analysis and makes it acoustically unique.
I don't follow these statements. If you add two sine waves together with identical phase, isn't the amplitude doubled? Sure, it's still a sine wave, but isn't the same true of a square wave? If I add two square waves with equal phase, it's still a square wave, just with doubled amplitude. It has retained its shape. What am I missing?
Best Answer
This is referring to the sum identity for sines: $\sin(\alpha) + \sin(\beta) = 2\cos(\frac{\alpha-\beta}{2})\sin(\frac{\alpha+\beta}{2})$.
From this we see that we can add sine waves of the same frequency but different phases and still get a sine wave of the original frequency. Specifically, applying the above to $\sin(x)$ and $\sin(x+\phi)$, we get $\sin(x)+\sin(x+\phi) = 2\cos(\frac{\phi}{2})\sin(x+\frac{\phi}{2})$. A little work generalizes this to arbitrary sinusoids of the same frequency.