Now that we have reason to believe light is a wave, of whatever kind, it becomes necessary to determine wether that description is consistent with everything about light that can be observed. There is a particular experiment that will concern us when we get to quantum mechanics, involving light inside a red-hot cavity. In fact, it was this experiment that first indicated that whatever field light `lived on', it was not a classical one. To understand, first, why the experiment presented problems with the classical ways of thinking, and second, how those should be corrected, we need a different and more refined way of describing waves than the intuitive pictures of pond ripples that we have employed so far.
The particular description that is useful is actually already familiar in experience. Suppose you are in a roomful of sound, like a concert hall. Your eardrum responds to oscillations in the value of the pressure field, and you hear. The ear happens to be a device to which pure tones are simple sounds. When acted upon by a complex sound, it decomposes it and recognizes it as some collection of pure tones, present in different amounts. To some lesser degree, the ear is able to identify from which direction a sound came, so there must be something in the sound field, in addition to how much of each pitch it contains, that identifies that directionality.
It turns out that these two quantities, pitch and directionality, are enough to uniquely identify every simple component of a sound wave (`simple' as regarded by the ear). A specification of how much of each of the simple waves occurs is then enough to determine the whole content of the sound wave. Moreover, this kind of description can be generalized to apply to any wave of any kind. It will also be the one in which the problem with light in a red-hot cavity can be understood simply.
The full identification of all the properties of pitch and directionality in a sound wave, though, along with its relation to the intuitive picture of pond ripples, is somewhat complex. Therefore we will start with a simpler problem, and address one characteristic at a time.
The simpler system to consider is that of waves on a string. Analogous to asking why two orchestras sound different in a concert hall, or why two kinds of light emanating from a cavity might look different, we can ask how two kinds of waves on a string can be different. For example we can take a piano or harpsichord string. Why does a piano sound different from a harpsichord? They both have similar strings and similar sound boxes. In fact, in some crude way, a piano can be made to sound like a harpsichord by putting thumbtacks in the hammers where they strike the strings. Thus, we understand most of the question if we can understand why the same piano string, struck differently, sounds different. Obviously, even sound, and our ability to hear sound, are not issues at the heart of this question, because it is the differences in the behavior of the struck string itself, all other things being kept the same, that create the differences in the sounds we hear. Our hearing them only serves to make the difference familiar. Thus we may ignore all the complexity of sound in air, and simply address what kinds of waves can exist on the string itself, and how those can be described and can differ.