Do all periodic sound waves have harmonics?

Question

I 've read that all periodic sound waves have a fundamental frequency and I was wondering if all periodic sound waves also have harmonics (integer multiples of a fundamental frequency). If that question rings true then how come its like that and how can I apply this information in composing a song?

I am aware that a simple sine wave doesn't have harmonics but I was thinking more in the "real world" not computer simulations.

Answer 1

I 've read that all periodic sound waves have a fundamental frequency

That's true - if a wave is periodic, it has a fundamental frequency.

(Oddly, just because a wave has a fundamental frequency, it doesn't necessarily mean that there's any energy at that frequency though - it may be that all the energy is in the harmonics!)

I was wondering if all periodic sound waves also have harmonics. (I am aware that a simple sine wave doesn't have harmonics but I was thinking more in the "real world", not computer simulations.)

Yes, all periodic waves except (as you say) sinusoidal waves have harmonics. This is a result of the Fourier theorem, which states that any PERIODIC function f(x) may be expressed as the sum of a series of sinusoidal functions (possibly with different amplitudes and phase offsets).

While that might sound a bit abstract, it's an excellent way to analyse sound, as it's exactly what our ear does - pick out energy at different frequencies.

If you're interested in "real-world" rather than computer-generated scenarios, it's worth noting that few sounds - certainly no acoustic instrument sounds - are exactly periodic - they change over time. Having said that, even when a sound is only roughly periodic, It's still a valid idea to think of it as consisting of a set of harmonics that themselves change in amplitude (and even frequency) over time.

There may also be other partials in a sound that aren't close to being integer multiples of the fundamental - these are often called 'inharmonic partials'. Some energy in a sound often isn't easily modelled as a 'partial' of a steady frequency at all - this energy is often just called 'noise'.

Answer 2

Talking about periodic signals does not require any knowledge of sines / Fourier analysis, nor reference to any physical systems. The no-nonense definition is just this:

A periodic signal with period 𝑇 is a function 𝑓 : ℝ → ℝ such that 𝑇 : ℝ⁺ is the unique smallest positive number for which
               𝑓(𝑡 + 𝑇) = 𝑓(𝑡) ∀ 𝑡∈ℝ
(there is no 𝜏 < 𝑇 that would also always give 𝑓(𝑡 + 𝜏) = 𝑓(𝑡)).

(We need the restriction forbidding 𝜏 < 𝑇 because else any signal with period 𝑇 would also qualify as periodic with period 2·𝑇 or 3·𝑇 and so on.)

That readily allows us to also define the fundamental frequency as simply the reciprocal: 𝜈_f = ¹⁄_𝑇.

As I promised, this does not yet make any reference whatsoever to sinusoidal components – in fact, it also applies to functions that cannot be represented by a Fourier series, such as

       ⎧ 0                  if 𝑡 is integral
𝑓(𝑡) := ⎨
       ⎩ sin (1⁄1−cos(2·π·𝑡))   else

In this signal, you have in essence locally (close to 0, 1, 2 etc.) the ripple-frequency go up arbitrarily high, without decreasing its amplitude). This can't be constructed as just a superposition of constant-frequency sine parts. Thus it's also not really possible to talk of harmonics.

Now, such a “signal” fortunately can never actually turn up in the physical world – you'd basically have an ultraviolet catastrophe at each of the points where the frequency goes infinite. Actual physical signals, if they are periodic, can be decomposed into a superposition of sinusoidals, with the slowest-oscillating sharing the fundamental frequency of the signal. (That includes, of course, the audio example above, which is effectively bandlimited through Nyquist-sampling. Mind, this is for this signal not possible without hefty aliasing artifacts.)

And then, the answer to your question is yes, as topo morto explained: if a signal has no harmonics, then by definition of “harmonic”, the signal is actually just a single sinusoidal.

Answer 3

The more mathematical way to state it is: if a signal is periodic with period ‘T’ then it’s power spectrum can have non-zero values only at frequencies of ‘1/T, 2/T, 3/T...’. that is, integer multiples of the fundamental frequency ‘f0=1/T’.

Answer 4

According to http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/funhar.html:

The lowest resonant frequency of a vibrating object is called its fundamental frequency. [Emphasis added.]

And according to http://hep.physics.indiana.edu/~rickv/Standing_Waves_on_String.html:

The lowest standing wave frequency is called the fundamental or first harmonic.

So to your question, "[Do] all periodic sound waves also have harmonics (integer multiples of a fundamental frequency)" the answer is no. These two citations show that, in physics, harmonics refer somewhat narrowly to natural/resonant modes of vibration. But many sounds we hear are not the result of resonance--they are instead generated through forced vibration, which is a different category from natural/resonant vibration.

The passage below (about bees' wingbeat frequency) does a good job of illustrating the difference. It's comparing the fundamental frequency of a bee's wings with the actual frequency at which bees vibrate their wings:

In support of the stiff element hypothesis, the first fundamental frequency (602±145 Hz) was substantially higher than the wingbeat frequency of our individual bees (234±13.9 Hz; paired t-test, P=9.2×10−8, n=6; Fig. 3).

^{http://jeb.biologists.org/content/220/15/2697}

Just because bees can beat their wings at a periodic frequency (which produces a sound that we hear) does not mean that the wingbeat frequency automatically qualifies as a fundamental frequency or as a harmonic. Only those special resonant modes qualify as harmonics.

I've read that all periodic sound waves have a fundamental frequency

As you'll appreciate now, this is also false. There's an important difference between forced vibration/oscillation and natural vibration/oscillation. According to the definitions above, the fundamental frequency only refers to the lowest resonant vibrational mode.

Note about Terminology

The term "harmonic" can describe: - component frequencies of a complex waveform (detached from any physical origin), and - vibrational modes of an oscillating physical system (which is the source of sound waves)

If we were to see a wave oscillating like this picture shows and asked "what are we hearing?" I would reply "the third harmonic frequency [of the string's vibration]." I agree that, if we let the string vibrate this way, then the waveform of the resulting sound wouldn't have a third harmonic. But despite this, I still think we should say that we are hearing the third harmonic frequency [of the oscillating system]. If we are attaching the sound wave itself to its physical origins--the real-world oscillating system--and ignoring computer-generated noises (and this is what I see the OP asking), then we should speak in terms of the oscillatory frequencies that produce the sounds. For these oscillatory frequencies, the term harmonic refers more narrowly to resonant frequencies.