Why are sine waves denoted as s(t) = Asin(2πft + Φ) 

After what is a probably a long sabbatical from physics and the mathematics of travelling waves, I went back to school and there we began to review an old-new topic i.e.  a topic that I had encountered before but because i hadn't dealt with it for a while, it seemed novice; namely "wave propagation". The course text book that we used began  rather stifly

"A sine wave is represented using the function:

s(t) = Asin(2πft + Φ) 

where A is the amplitude of the wave, f is the frequency of the wave and Φ is its phase"

All physics/engineering students and enthusiasts of waves will tell you that is a core tenet of wave theory. And while I cannot be certain that I have paid a second thought to this mathematical description before, I began wondering what the basis could be for such a representation. Here I was starting a unit, this was a core albeit simple and taken for grant statement and I wanted to get to really understand what this representation meant as a good basis for its obvious use all through the course.

So after a bit of research, here goes. 

The sine function is used to represent the distance from the center of a circle (ideally who's radius is 1 unit) to the extrapolated length of a radius touching its circumference at a specific angle, Φ

Sound cryptic? Here is a diagram illustrating that rather daunting definition:

<diagram here>

And so looking at the diagram above, the sine of angle Φ is the distance in red i.e. the distance from the extrapolated radius line to the center of the circle. Other text book definitions state that the sin is calculated by dividing the opposite side of a right angled triangle with the hypotenuse. A little consideration will show that the two mean the same. 

And so the sine function of a specific angle Φ will be given as 

 s(x) = sin (Φ)

It's easy to see from the diagram that sin (90) = 1, sin (180) = 0 and so forth. 

Therefore, in a way a circle is actually the locus of the sine of angles starting from 0 to 360. Ok, so far? (it was interesting here to view sine as a measure of length :) )

So let's pivot back to the sine wave (as used in wave text books). You will notice that if we find the "sine" of a sine wave (confusing, ha?), the values that we obtain are actually similar to those that we obtain from finding the "sine" of a circle. Thus in essence, a sine wave may similar to a circle be represented by the function

 s(x) = sin (Φ)

However, representing a sine wave this way ignores two very important differences between a sine wave and a circle: 

• Sine waves vary in amplitude 
• Sine waves represent motion/changes in value over time 

Let's consider each of these as we try to figure out why the mathematical representation of sine waves is as it is. Let's start off with looking at the sine wave as varying in magnitude.

Indeed a sine wave varies in magnitude or more accurately amplitude. A stronger sine wave is perceived as one with a larger amplitude; it's a stronger wave. So how can we modify the equation 

to capture the fact that sine waves can have varying amplitudes. It turns out that it is fairly easy to modify this equation to consider amplitude. Simply multiply the value that would be returned from finding the sine of a wave with the maximum amplitude of the wave, A.

So the new form of the equation that incorporates change in the amplitude A is given by 

s(x) = Asin (Φ)

Ok.

The next question that we need to ask ourselves is how could we standardize this formula to allow for sine waves that have varying wavelengths...wavelength that are not the default 1 unit length.  Turns out that this can be done by adding the coefficient 2π before the Φ in the equation. 

s(x) = Asin (2π/λ x Φ)

where λ is the wavelength that can now be varied to our desire. Therefore, if we were to replace λ with say π/2 (i.e. the wave completes one cycle in 180 equivalent), then the sin at Φ for this wave would be 

s(x) = Asin (2π/ (π/2) x Φ)

which would be 

s(x) = Asin (4Φ)

Comparing this with hand calculated values will confirm this. 

So now we have an equation that allows us to vary amplitude of the sine wave as well as its wavelength. Great. So what's remaining. Since the sine wave (unlike the circle represents a travelling sine wave), we need the equation modifiable so that we can set different values for phase i.e. the time at which the wave begins to travel from a theoretical 'zero' start time.

Turns out that the equation can be fairly easily standardized to accomodate this by adding on a phase shift factor. So the equation representing the travelling sine wave is now denoted as 

s(x) = Asin (2π/λ x Φ + ψ) where ψ is the phase. Testing out different values of ψ will confirm this. 

So now our equation s(x) = Asin (2π/λ x Φ + ψ)  is looking almost similar to the equation whose representation we set off to determine s(t) = Asin(2πft + Φ) 

The final bit to transform our equation so far to the desired equation is given by the realization that Φ/λ is equivalent in units to ft. Thus the two equations are equivalent.