Usually we use a sinusoidal wave form to represent a alternating quantity. Why not a cosinusoidal wave or a ramp wave form?
In sine wave forms we can indicate the maximum and minimum amplitude and its variation with respect to time. Ramp waves have the same properties; then why not a ramp wave form?
Best Answer
Sine and cosine waves are, physically, the most common. They are definitely the best description to what comes out of a wall socket, not because we like them mathematically, but because it's what comes out; electromotive force is generated in the power plant as a sinusoidal pattern with frequency 50/60 Hz. In the usual kind of generator, this is because in the generator, the rotating motion of the magnetic rotor leads to sinusoidal variation of EMF in the winding of the stator and consequently in any circuit connected to the socket. (Note that sine and cosine waves are equivalent, and choice between them is merely convention; the neutral word that can be used to describe the shape is harmonic.)
Even better, if we have some more complicated waveform - from slightly deformed all the way to ramp and step waveforms - then we can use Fourier series to decompose them into a sum of sine waves. We can then study the response of the circuit to each sine wave independently, and add up the responses at the end; this is usually much simpler than using the waveform directly.
The reason this works is because most circuits are linear. That is, if we input some voltage $v(t)$ and we measure some property $p(t)$ of the circuit, then adding a voltage $v'(t)$ will result in the property being $p(t)+p'(t)$. Thus if our complicated waveform $v(t)$ can be expressed as a Fourier series, say, as $$v(t)=\sum_n v_n \sin(n\omega t),$$ and we know (because it's easier to study) that a sinusoidal stimulus $v_n\sin(n\omega t)$ will result in a property $p_n(t)$, then the full waveform $v(t)$ will induce a response $$p(t)=\sum_n p_n(t).$$
This can, of course, break down if you have nonlinear elements in your circuit, such as diodes or overdriven vacuum tubes or transformers. These will behave differently and will induce distortion of your waveform, which may or may not be a good thing depending on what the application is. This distortion is, of course, the same distortion as you get with an electric guitar amp.
A final word on square and ramp waveforms. Fourier treatments of these waveforms are sometimes a bit difficult, and convergence near the sharp edges can be very slow (see e.g. the Gibbs phenomenon). This represents some very important physics of square waves: the instantaneous change in voltage is not actually possible in any physical circuit. This is because if the source circuit has any inductance $L$, the instantaneous change in current $i$ will mean an infinite current change $\frac{di}{dt}$ and therefore an infinite inductance back-voltage. Typical sources have very small inductances, but they can never be zero. The comparison of this inductance and the source internal resistance will give the timescale in which the voltage can suddenly change sign.
If you want a good description of an actual physical square wave, then, you have two choices: you can account for a finite "ramp" time, or equivalently you can simply take a finite number of terms in the Fourier series. The latter is, of course, a lot easier!