naturewaves
The oscillation of light waves and sound waves (and many other waves) can be described with sinusoidal functions in time (frequency) and in space (wavenumber). Moreover, oscillation of monochromatic waves is perfectly described by a single sinusoidal (single frequency and wavenumber) function.
But why sine function and not any other function/wavelet? Or, why pressure waves, for example, have anything to do with the Cartesian projections of circular motion?
Here's how I would suggest thinking about it. This is a sinusoidal wave:
Is that described by cosine formula or a sine formula? Well, as it stands, neither. It's just a thing that exists in space, there are no numbers attached.
In order to represent this wave by a formula, you need to find a way to represent points in space and time by numbers. That's what a coordinate system is for. And in order to apply a coordinate system, you need to choose some spacetime point to be labeled $(t, x) = (0, 0)$. For example, you might choose that point like this:
Now that every point is labeled by a number, you can meaningfully talk about what formula describes the wave. To do so, take a look at one specific "slice" of time, and at one specific point, for example the time and location that we chose to be $(t, x) = (0, 0)$.
You'll notice that at $(0,0)$, the wave has an value of zero, and its derivative is at a maximum. That uniquely matches the formula $y = \sin(kx - \omega t)$.
But what if you take the same wave and put a different coordinate system on it - that is, choose a different location to be position zero? You might get something like this:
It's the same wave, we've just chosen different labels for the points in the space the wave is traveling through. Suppose we call these new labels $(t',x')$. So let's look at the snapshot at $t'=0$ (when I write something with a prime like this, it means "$t$ in the primed coordinate system") with this new coordinate system.
Now, at the point we've chosen to call $(0,0)$, the wave is at its maximum. So with this new coordinate system, it doesn't match $y = \sin(kx' - \omega t')$; instead, the formula that matches the wave is $y = \cos(kx' - \omega t')$.
Of course, there are an infinite number of other ways you could choose the coordinate system. For example, suppose you choose a double-primed coordinate system $(t'',x'')$ so that the same time is labeled zero, but instead choose the position coordinates such that the $t'' = 0$ snapshot looks like this:
With this labeling, the wave doesn't match either the sine formula or the cosine formula. So what can you do? Use a shifted coordinate. For example, you can see that the wave crosses zero with a positive derivative at some negative value around $-2$. (The value is actually $-8\pi/11$.) Since you already know how to write the formula for a wave which crosses zero with positive derivative at position zero, you can do that and just use $x'' + 8\pi/11$ as your position coordinate. You can think of this as shifting the coordinate system back so that the zero point falls where the wave crosses zero.
So the formula would be
$$y = \sin\biggl[k\biggl(x'' + \frac{8\pi}{11}\biggr) - \omega t\biggr]$$
Alternatively, since you already know how to write the formula for a wave which hits its maximum at position zero, you could do that instead: shift the coordinate system back only enough so that the zero point falls where the wave hits its maximum.
You would need to shift by $5\pi/22$, which is equivalent to using $x'' + 5\pi/22$ as your position coordinate. So in this case the formula would be
$$y = \cos\biggl[k\biggl(x'' + \frac{5\pi}{22}\biggr) - \omega t\biggr]$$
You may notice that all these different equations for the wave take the general form
$$y = \sin(kx - \omega t + \phi_s)$$
or
$$y = \cos(kx - \omega t + \phi_c)$$
You can use either of these; they both describe the same kind of wave, as long as you pick the value of $\phi$ correctly for your coordinate system. For example, in the latest of the three cases I presented (the double-primed coordinates), you could choose $\phi_s'' = 8\pi/11$ and use the sine formula, or you could choose $\phi_c'' = 5\pi/22$ and use the cosine formula. It's the same thing either way (and you can show this mathematically, too). In the second case, the single-primed coordinates, the cosine formula worked with $\phi_c' = 0$, but you could just as well have used the sine formula with $\phi_s' = \pi/2$. And in the first case, the sine formula worked with $\phi_s = 0$, but you could also have used the cosine formula with $\phi_c = -\pi/2$.
In most resources you will see at your level, the author will choose to use one or the other of these formulas consistently, i.e. always cosine, or always sine. In fact, they might always choose the coordinate system that makes $\phi$ be zero. But it's useful to remember that, in case you ever have to use a different coordinate system (which you usually will, in higher-level physics), you can always use either of the general formulas, the sine or the cosine, and just pick the value of $\phi$ that makes things work out.
See for example: http://www.acs.psu.edu/drussell/Demos/reflect/reflect.html
Reflection from hard vs. soft boundary. Closed tube is like a hard boundary, open tube a soft boundary. In either case there is an impedance discontinuity and hence reflection, however the phase of the reflected wave differs depending on the nature of the discontinuity.
Best Answer
The solutions to wave equations are sinusoids, or seem to be, because sinusoids always represent solutions to a wave equation. But in reality the general solutions are linear sums of sinusoids, with different frequencies.
When you sum a lot of sinusoids you can can get any wave packet you want. So the solution then depends on what other conditions you impose.
For one pure sinusoid the condition is a fixed constant energy. Constant energy fixes the frequency. For electromagnetism (EM) a single frequency sinusoid it is a single color (if light, a very finely defined freq. channel if radio, etc. If you have a sum of many of those with frequents spanning the spectrum (of light) you get white light. If you changes frequencies Ina radio signals to do frequency modulation you have a bunch of freqs within that band. Something similar is true for sound, where freq represents pitch.
So, yes, you can form packets or wavelets or whatever you want, by summing with different amplitudes a bunch of those.
So, it is not that sinusoids are the only solutions, it is that any solution can be represented (and calculated) as the sum of sinusoids. If you look at the signal then so you can see all its frequencies, you are looking at its spectrum. Mathematically it is simply a Fourier transform to go from the time waveform to the spectrum.. A sinusoid spectrum is simply one line in the spectrum (two, one a positive freq and one at negative freq, if you want to be exact).
It is also said that sinusoids, or complex exponentially, are eigenvectors of the wave equation represented as an operator on the eigenvectors, with fixed eigenvalues, which are the energy.
Bets to understand this about waves well before doing any real physics.