Like @sansuiso said, compressed sensing is a way of acquiring signals that happens to be efficient if the signals are sparse or compressible.
Compressed Sensing is efficient because signals are multiplexed, hence the number of multiplexed samples (called measurements) is smaller than the number of samples required by Shannon-Nyquist where there are no strong assumptions on the signal.
In the noiseless case, it can be shown that compressive sensing reconstruction solver can recover an exact solution.
In the compressible case, as opposed to the strictly sparse case, it can be shown that the reconstruction error is bounded.
And yes most signals, including ultrasounds are somehow either sparse or compressible. It generally comes down to figuring out the dictionary where the signal is sparse. Domain experts generally know these things.
The interesting question you have is: Imagine you have a non sparse signal and then add zeros to make it sparse and then use compressed sensing to sample that signal, wouldn't it better than directly sampling the full signal?
The answer is no.
It turns out that the sampling requirements for which CS work require more infromation than just performing a full sampling of the original (full/non-zero) signal. In other words, the number of CS measurements required would be higher than the number of non-zero elements in the signals. By sparsifying the signal, you are "losing" on purpose the information about where the signal is supported (i.e. non-zero). The hard part of Compressive Sensing and attendant reconstruction solvers is to find that location where those non zero elements of the signal live: If you know beforehand the locations of those non zero elements, then there is no need to go to a less efficient method of sampling that signal. Indeed, finding the location of the non-zero elements of a signal is the reason we talk about compressive sensing being NP-Hard, BPP and so forth..... Up until 2004, we thought it was hard to do.
Let me put it some other way: Let us assume a signal has K non zero components. If you know the location of these K elements, then you need only K information to know your signal. If you add zeros anywhere in the signal and make that signal of size N, you now need to sample the signal N times through traditional sampling or O(Klog(K/N)) times with a compressive sensing approach. Since O(Klog(K/N) > K, losing the information about the location of the non zeros elements has yielded a larger set of samples/measurements.
You might be interested in reading my small blog on the subject:
And the following resource: