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(Guitar player and programmer here, don't know much about math. So go easy ;) ).

I recently learned that an audio sine wave is called that way because it is of the shape of the graph of a sine function.

So (correct me if I'm wrong), the equation for a sine function is:

$$p = \sin(t)$$

Where: $p$ is the point on the graph, and $t$ is the point in time. Writing this in Wolfram Alpha indeed shows the expected graph.

So my question: why does the Wikipedia article show much more complex formulas? If Wolfram Alpha says that the formula I wrote above is correct, why all the weird stuff in Wikipedia? The Wikipedia article doesn't show the formula $f(t) = \sin(t)$ even once. What am I missing?

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    $\begingroup$ If someone asked me to give the equation of a sine curve, I would write $\sin(x)$. If someone asked me to describe sinusoidal motion, I would give the equation $A \sin(\omega x+\varphi )$. I think this is a pretty standard way of looking at things (the wiki article does clarify by saying "or sinusoid"). By virtue of the angle addition identities, sinusoidal motion can also be described as $A \cos(\omega x+\varphi)$ or $A \sin(\omega x)+B \cos(\omega x)$, with any replacement of $+$ with $-$ that you want. So if you see these equations they're equivalent to the wikipedia article. $\endgroup$
    – user12029
    Commented Sep 23, 2014 at 16:11
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    $\begingroup$ Sine wave - physicskey.com/35/… $\endgroup$
    – Saral
    Commented Dec 29, 2018 at 5:18

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As a guitar player, you know about dynamics. Some notes are LOUD and others are soft. This is related to the amplitude of the signal, $A$.

There are also different pitches. An A and a G are different pitches, which correspond to frequency, $f$ or angular frequency, $\omega$.

Finally, there's the notion of phase, $\varphi$ which how humans can detect whether there are one or two flutes (which emit nearly perfectly sinusoidal acoustic waves), even if they're playing the exact same note.

If we only used $p=\sin(t)$, we would only have one amplitude, one frequency and no phase shift. Instead we use $p = A\sin(\omega t + \varphi)$ or more commonly $p = A\cos(\omega t + \varphi)$ because the only difference between the two is the value of $\varphi$.

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    $\begingroup$ Thanks for answering. I know about frequency and amplitude, but what do you mean about 'angular frequency'? And what exactly is 'phase'? $\endgroup$
    – Aviv Cohn
    Commented Sep 23, 2014 at 23:22
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    $\begingroup$ @AvivCohn: Angular frequency $\omega = 2\pi f$, where $f$ is the frequency. $\endgroup$
    – JamalS
    Commented Sep 25, 2014 at 11:21
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Simply read the wikipedia article.

$y(t)= A \sin(2\pi f t+\varphi)$

Here $A$ is the amplitude of the wave,i.e. the maximum height of the wave; $f$ the frequency, i.e. is the number of oscillations (cycles) that occur each second of time; $\varphi$, the phase, specifies (in radians) where in its cycle the oscillation is at $t = 0$.

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