Why is “wavelength” one word when “wave height” isn't?
As another example, wave speed is two words. But wavelength is only one word. What is the reason for this?
In Swedish and other contructs, both words are only one word:
The reason that wavelength is only one word is that humans commonly form new words by combining existing words together. This is called compounding, and is observed over and over by linguists studying the evolution of languages.
The reason that the other terms you mention, “wave height” and “wave speed”, have not compounded is that they are different from “wave length” in a significant way.
Online Etymology Dictionary tells us that the compound wavelength was coined in the 1840s when scientists began to explore the properties of light spectra (rainbow colors). As Kaz’s answer points out, they needed a technical term for the “distance over which the wave’s shape repeats”. (Wikipedia)
However, a new word was not coined for light’s “wave height” or intensity. Instead, the much older existing term amplitude was used. Nor was “wave speed” used for light’s velocity: instead the term speed of light was adopted.
Those choices are just an accident of history. But the consequence is that “wave height” and ”wave speed” did not gain currency in the 19th century explosion of literature on the subject of electromagnetic waves.
We can actually see compounding occurring in the chart below. Three similar terms – wave length (purple), wave-length (green), and wavelength (blue) – rapidly gained currency as scientific inquiry progressed. We see that before 1920 the term was almost always spelled using a hyphen or space. During the next three decades, the compound spelling wavelength gradually displaced the other spellings. By 1950 the majority of authors used the compound spelling.
Wavelength has been adopted as a term for a dimension (of measurement).
The other phrases have their respective terms (wave height: amplitude), as such they are separate individual words.
Note that wavelength is a noncompositional compound. It does not refer to the length of a wave, but rather to the length of a single oscillation period of the wave: the reciprocal of the frequency. It is technical jargon which does not mean wave length.
Generally we write two words as one when they are used so often and in such a specific way, that the meaning is no longer just the combination of the meanings of the two words. For example, a blackboard is not a black board. It is not just any board which is black but a tablet for writing with chalk, which may in fact be produced in colors other than black, such as green. Moreover, this is not just an orthographic (written) difference. The word blackboard is phonologically different from black board. It places accent on black and removes it from board. There can be an accent on black in black board, but it is not integrated into the word. It is used for emphasis, as in "I said bring me the black board, not the white one!". The accent could also be on board: *"I said bring me the black board, not the black cord.". In blackboard, there is no freedom to play with the relative stress. An example which shows that this has nothing to do with orthography is (The) White House, which is is written as two capitalized words, yet it is phonetically a unit like blackboard.
Waveheight has not been coined as a technical term probably because English-speaking scientists and engineers adopted the term amplitude instead, which is as precise as wavelength. Amplitude is the degree of displacement of an oscillation from the system's rest position.
If waveheight referred to amplitude, it would be different from wave height, which does not have a clear meaning. Wave height could refer to amplitude, or it could be a peak-to-peak height, which is twice the amplitude. Or it could be something else: the height of shoaling as water waves hit shallow water.
The only full answer is “because it is”. There are rarely complete answers to “why” questions in linguistics.
But a clearly relevant factor is that there is a need to talk about wavelength much more often than wave height, because wavelength is a property of all kinds of wave (including sound and electromagnetic radiation), whereas height is a property only of transverse waves in a physical medium, such as water.
The real answer to this question, in my opinion, is that to use the word height to describe a wave displays both a misunderstanding of the actual physical form of many waves and an incomplete understanding of the word height as it relates to mathematics.
First, most people think of waves in terms of the surface of water, or the readout of an oscilloscope, however many waves are not physically shaped like that.
Second, mathematically, height is the third dimension of measurement and it is perpendicular to the plane created by length and width. While a wave travels three-dimensionally, it doesn't actually have that kind of dimensionality, a wave as we measure it is shown to have dimensionality, because we prefer to perceive things as we are, not as they are, but a wave as it is, is actually just energy travelling.
As a quick aside, when I try to think of a wave as it is, I always come back to the word impulse, I kind of feel like that it the closest word we have to describe waves-as-energy-transfer that relates to an identifiable feeling we experience.
