# Do traveling sound waves have harmonics?

I know that standing waves produce harmonics but what about traveling waves that dont reflect back on itself?

Do traveling sound waves have partials that are harmonic(integer multiple of the fundamental frequency)? Why?

I am trying to get a better understanding of Acoustics.

• Instruments that produce sound waves are generating the harmonics. The actual traveling wave itself is a given frequency, each harmonic has its own traveling wave. – K7PEH Apr 3 '18 at 17:03
• A purely linear medium will not induce any harmonics. A non-linear medium will, but the extent to which they are introduced will vary with any number of factors. – Jon Custer Apr 3 '18 at 17:08

Waves don't have harmonics. When you say, "standing waves produce harmonics," the underlying phenomenon that you're trying to describe is that a resonator (e.g., the pipe of a wind instrument, a vibrating string, or the head of a drum) can have many vibrational modes.

Each of the different modes of a resonator has a particular frequency, and all of the different frequencies are ratios of some fundamental frequency. Musicians refer to the different frequencies as harmonics.

• In general, acoustical resonators have overtones that are not integer multiples and thus harmonics of the fundamental frequency. See Wikipedia: en.wikipedia.org/wiki/Overtone. – freecharly Apr 3 '18 at 19:01

The harmonics of a wave are defined by boundary conditions of the medium the wave is living in. In the case of the vibrating string, the allowed vibrations are defined by whether the string is fixed at both ends, fixed at one end and open at the other, or open at both ends. The lowest frequency wave the string allows is called the fundamental mode, and the higher frequency waves are called harmonics to that fundamental.

The same thing happens with tubes and cavities containing air, drum heads, metal bars, etc.

Now, traveling waves in open air are only described as harmonics if they were (originally) produced by a process like the ones above. You can, however, get traveling waves that show this sort of harmonic structure due to boundary conditions associated with a ring. Say, for example, you have a tube of air that forms a donut (torus). Then the waves inside of the tube are perfectly happy to travel in circles around the tube, and the allowed waves have a harmonic structure. The same thing can also happen on drum heads, with the waves moving around the head one direction or another.

Wait, I mentioned drum heads and pipes in both situations, so what's going on here? Well, standing waves are just two traveling waves of equal amplitude and frequency going in opposite directions, right? That relationship can be flipped: traveling waves are just two standing waves of equal amplitude but offset in phase and position added together. In trig functions (but this is more general than them) it's \begin{align} \mathrm{standing\ waves} &\hphantom{=}\ \ \mathrm{traveling\ waves} \\ 2\cos(kx)\cos(\omega t) &= \cos(kx-\omega t)- \cos(kx+\omega t)\ \mathrm{and} \\ \cos(kx)\cos(\omega t)- \sin(kx)\sin(\omega t) &=\cos(kx-\omega t). \end{align}

When the sound wave is produced by a time periodic signal $$f(t)=f(t+T) \tag 1$$ with frequency $f=\frac {1}{T}$, it produces a sound wave $$f(t-\frac {z}{v}) \tag 2$$ propagating in positive z-direction with the sound velocity $v$ and a corresponding time and distance periodicity. Thus the time and position dependence of the wave can be decomposed into a Fourier series $$f(t-\frac {z}{v})=\sum_{n=-\infty}^{+\infty} c_n \exp i\frac {2\pi n}{T}(t-\frac {z}{v})= \sum_{n=-\infty}^{+\infty} c_n \exp 2\pi i(f_n t-\frac {z}{\lambda_n})\tag3$$ where the terms for $|n|\gt 1$ are the higher harmonics with frequencies $$f_n=\frac {n}{T}=nf \tag 4$$ and wavelengths $$\lambda_n= \frac {Tv}{n}=\frac {\lambda}{n}\tag 5$$ The amplitudes of the partial waves are given by the coefficients $$c_n=\frac {1}{T}\int_{t_0}^{t_0+T} f(t)\exp(-i \frac {2\pi nt}{T}) \tag 6$$ Thus any periodic sound wave can, in general, be decomposed into an infinite series of harmonic waves. This happens in a linear medium, non-linear effects are not necessary.