Actually, I disagree on some points.
Here's an easy example. Think of the Beats phenomenon. You have 2 waves that are similar in frequency(f) and are equal in amplitude(A). At points they will cancel out completely, and at other points they will have a cumulative amplitude of 2A, with a peak to peak beat frequency of |f1-f2|. This can be applied to any two frequencies, and many more actually, but it isn't as easy. This operation is the basic principal of AM transmission.
Regardless of the frequency, it will shift in and out of phase with another signal.
Wave and signal can be used synonymously as long as both of them are sinusoidal.