What does it mean that a sine wave is unchanged when added to another sine wave?

Question

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?

Answer 1

The statement means that $$\alpha_1 \sin(\omega x + \delta_1) + \alpha_2\alpha_3\sin(\omega x + \delta_2)$$ can be expressed as another sine wave of the form $\alpha_3\sin(\omega x + \delta_3)$, e.g. in the case $\alpha_1=\alpha_2 = 1$ $$ 2\cos(\frac{\delta_2}{2}-\frac{\delta_1}{2})\sin(\omega x + \frac{\delta_1}{2} + \frac{\delta_2}{2}) $$

Note that the cosine term out front does not depend on $x$, it is just a constant in terms of $\delta_1$ and $\delta_2$.

Answer 2

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.

Answer 3

Well you already gave the answer that when you add 2 sinusoids you get a sinusoid again.

You thought you gave a counter example by saying when 2 square wave get added it produces a square wave. But what you missed out is that a square wave can be represented by a summation of many sinusoids, which is the Fourier Series.

So in general "any" signal can be represented in terms of sinusoids by using Fourier Series and thus we only talk any theory in terms of sinusoids.

Hope you understood the crux of this discussion!