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

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?

• If you add two sinusoids with the same frequency but different phases, you get another sinusoid with the same frequency. If you add two square waves with the same frequency but different phase/amplitude you won't get a square wave. – Math1000 Jun 7 '15 at 1:57

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.

• But can you explain how this isn't also true of other waves? I get it now for square waves (since they have to alternate between max and min) but....I mean is this just like saying any wave that isn't exactly square or triangular or sawtoothed is sinusoidal? – temporary_user_name Jun 7 '15 at 2:06
• Not quite the answer you wanted, but this shows that the addition identities uniquely determine sine and cosine. math.stackexchange.com/questions/1303044/… – Zach Effman Jun 7 '15 at 2:15
• Presumably the identity $f(kx+\phi_1) + f(kx+\phi_2) = f(kx+\phi_3)$ alone forces $f$ to be a variant of $\sin$; that seems to be what the author claims – Zach Effman Jun 7 '15 at 2:21

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 $\delta_2$.

• But can you explain how this isn't also true of other waves? I get it now for square waves (since they have to alternate between max and min) but....I mean is this just like saying any wave that isn't exactly square or triangular or sawtoothed is sinusoidal? – temporary_user_name Jun 7 '15 at 2:06
• Ahh, the fact that no other waveform has this property. This is a deep fact. Look at the fourier transform of your waveform. If it has at most one nonzero fourier coefficient, then it is a sine wave. If it has at least two nonzero coefficients, then you can show that you can break the the given condition. – nullUser Jun 7 '15 at 2:19
• @nullUser So would this only to apply to sine waves of infinite extent? – AnonSubmitter85 Jun 9 '15 at 4:39
• @AnonSubmitter85 Yes that is correct. This only applies for functions $f:\mathbb{R}\to\mathbb{R}$, otherwise the "change the phase" condition wouldn't make any sense. – nullUser Jun 16 '15 at 0:20
• @AnonSubmitter85 But also we can always extend a function on an interval periodically over all of $\Bbb R$. Or first oddly, then periodic, or first evenly, etc. – GPerez Jun 16 '15 at 15:34

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!