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.