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I'm a Software Developer by profession and my physics knowledge is limited what I had learned at high school level. Please excuse me if the question is trivial.

Question:

From what I know, a sound wave is set of different amplitudes spread across a time line. The amplitudes vary greatly. Hence, sound wave is largely aperiodic (amplitudes don't repeat often). Now, where does frequency come into picture here? If the wave is periodic like a sine wave or a deterministic mathematical function of time, then frequency can be measured as the number of cycles (wave reaching same amplitude) in one second. How can we define frequency of highly aperiodic sounds like human speech. If all that is recorded on a gramophone disc is varying amplitudes across timeline, where is frequency accounted for? Does the frequency remain constant in a typical human speech?

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The sound that reaches your ear is just air pressure fluctuating over time. You can use a transducer of some sort to convert the value of air pressure to some other form - for example:

  • to the depth of a groove being cut into a helical track on a layer of wax on a rotating drum

  • to the depth of a groove being cut into a spiral track on a circular disc of metal from which other plastic disks are pressed.

  • to a strength of magnetisation of a magnetic layer on a plastic tape being wound onto a spool

  • to a series of numbers representing the pressure at regular tiny intervals of time.

The idea that the variations in pressure over time are due to, or consist of, a collection of frequencies is just a mathematically equivalent description but it does not represent extra information, it's just a different way of describing the same information.

Here's some diagrams from a synthesizer manual

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Above are three very different sounds with apparently the same frequency (say 440 Hz)

enter image description here

Above is shown how you can add sine waves of two frequencies to produce a more complex waveform

enter image description here

Above is shown how you can continue adding sine waves of differing frequencies to construct an arbitrary waveform (a sawtooth).

The sawtooth waverform can be recorded directly as depths in a groove on a record. But you could "record" the same thing as a set of numbers that represent the frequencies of a dozen sine waves you could add together to produce a single pressure wave that varies over time in the same way.

See Fast Fourier Transform

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  • $\begingroup$ @RedGrittyBrick..thanks for the answer! The first part of the answer pretty much explains what I want. Is it like at every point in time (say at every ms) we record all the harmonics that contribute to the sound wave at that instant of time? Also, I can understand that recording this information in a digital form is easy. How is stored on mechanical device like Gramophone? Do we just record grooves with varying depths over time? $\endgroup$ – Gopal Jan 22 '13 at 10:21
  • $\begingroup$ @Gopal: at every interval in time we record the single value of air pressure that results from the sum of all the harmonics (+ other sounds/noise). Yes, this information can be stored as the depth of a groove cut in some surface. The depth varies over distance along the groove, a needle moving along the groove experiences changes in depth over time. $\endgroup$ – RedGrittyBrick Jan 22 '13 at 10:36
  • $\begingroup$ @RedGrittyBrick..as I accept the answer I would like to ask you a closing question, Essentially a gramophone doesn't actually do any fourier transforms. All it records is the displacement of microphone at every instant of time and reproduces same using a diaphragm. $\endgroup$ – Gopal Jan 22 '13 at 14:33
  • $\begingroup$ @Gopal: That is true. $\endgroup$ – RedGrittyBrick Jan 22 '13 at 15:10
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You can make an arbitrary sound (or any waveform) by adding together a bunch of pure tones at different frequencies. So a sound, unless it happens to be a pure tone, does not contain a single frequency component, rather a range of frequencies. The mathematics behind this is called Fourier analysis and you can see many examples on Wikipedia or by searching the web.

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  • $\begingroup$ ..>>You can make an arbitrary sound (or any waveform) by adding together a bunch of pure tones at different frequencies << bang on! Liked this lucid explanation. $\endgroup$ – Gopal Jan 23 '13 at 7:18
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Your question is specifically about how the concept of frequency can be applied to aperiodic signals. Simplest example is a finite width rectangular pulse - the signal is zero outside the pulse. This is certainly aperiodic. Now for periodic signals f(t), you can, for each frequency which is a multiple of the period, compute the n'th Fourier coefficient as $$ c_n = \int^{\pi}_{-\pi}f(t)e^{-int}dt$$ This represents "how much" of frquency n is present in the periodic signal, as explained in the other answers.

Correspondingly, for our aperiodic isolated rectangular pulse f(t), we can choose an interval of length T which is bigger than the pulse width, and again calculate the Fourier coefficients over this interval$$ c_n=\int^{\frac{T}{2}}_{-\frac{T}{2}}f(t)e^{-2\pi i (\frac{n}{T})t}dt $$ Now if we let $T\rightarrow \infty$, the discrete values $\frac{n}{T}$ can be replaced by a continuous variable $\xi$, and the set of Fourier coefficients is replaced by a function $\hat{f}(\xi)$: $$ \hat{f}(\xi) = \int^{\infty}_{-\infty}f(t)e^{2\pi i \xi t}dt $$ This is the Fourier transform of f.

So for aperiodic signals (or truncated periodic signals), this is what you want. You can calculate the Fourier transform of anything from a rectangular pulse to the latest David Bowie track.

The "frequencies" that are present in the signal are just the Fourier transformed variables $\xi$.

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  • $\begingroup$ @twistor..thanks for detailed explanation. I'll try to understand fourier transform (had a bit of it in my Engineering course) $\endgroup$ – Gopal Jan 23 '13 at 7:17

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