What are harmonic waves? I am studying waves for my examination. Harmonic waves is also in my syllabus and I don't know what it is. I searched for it on google and got two possibly different answers. One answer is from en.wikipidea.org, and the other from britannica.com.
Quoting from Wikipedia

A harmonic is a wave with a frequency that is a positive integer multiple of the fundamental frequency, the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the 1st harmonic, the other harmonics are known as higher harmonics.

Quoting from Britannica

The properties of harmonic waves are illustrated in the mathematical expression for the displacement in both space and time. For a harmonic wave traveling in the x-direction, the spatial and time dependence of the displacement ϕ is
$$ϕ(x, t) = Acos\Big(\frac {2\pi x} {\lambda} - 2\pi ft\Big)$$

From Wikipedia definition, it looks as it is standing wave while from Britannica, it looks as if standing wave is same as sinusoidal wave. To add to the confusion, my classmates have told me that it is same as periodic wave.
So, to conclude, my question is:
What is harmonic wave? Is it same as a sinusoidal wave or it is a standing wave or something else?
Please help me out. Thank you.
 A: In the world of music...
In many different kinds of resonator, the system is capable of vibrating at multiple, distinct modes all at the same time.
Most tonal musical instruments are based around some kind of resonator; an vibrating air column, a vibrating string, a vibrating bar of metal,... Musicians call the lowest frequency of any of the vibration modes the "fundamental," and they call all of the others "harmonics."
For a strictly linear resonator (e.g., in a stringed instrument, or in a woodwind instrument with a cylindrical bore) the harmonics are integer multiples of the fundamental. For instruments that vibrate in more complex ways—metallophones, brass horns, and I don't know what else—some of the harmonics can be other, fixed ratios of the fundamental.
The ratios of the amplitides of the fundamental and its various harmonics are what give each instrument its own distinct timbre (i.e., characteristic sound).
A: Even though it's better to check the definition in the context, I'd use the term "harmonic" for a periodic sinusoidal signal (you get cosine with phase shift), thus a signal that has 1 frequency only in its spectrum.
Once the definition of the term is shared among the participants to a discussion, I'll be fine to define an harmonic travelling wave as
$f^t(x,t) = A \cos(kx - \omega t)$,
traveling with velocity $c = \omega/k$, as well as harmonic standing waves, those coming from the transformation of a traveling wave as the superposition of two standing waves (using properties of addition of cosines)
$f^t(x,t) = A \left[ \cos(kx)\cos(\omega t) + \sin(kx)\sin(\omega t) \right] = \\ \qquad \ \ \ 
= A \cos(kx)\cos(\omega t) + A \sin(kx)\sin(\omega t) = \\ \qquad \ \ \ =f^{s,1}(x,t) + f^{s,2}(x,t) .$
