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Sine Waves and Degrees of Phase

Sine Waves and Degrees of Phase

Video 1 - Sine Waves and Degrees of Phase

Sine Function

Fig. 1 - Unit Circle mathematical relation to a Sine Wave.

Sine wave is defined as 'a curve representing periodic oscillations of constant amplitude as given by a sine function.'

Sine waves are sometimes described as "pure tones" because they represent a consistent, single oscillation. Oscillators in a synthesizer (or otherwise) produce these Alternating Current signals along with various other types of waveform (Square waves, triangle waves, sawtooth waves) to be employed as musical devices.

All sounds in nature are fundamentally constructed of sine waves. More complex sounds simply contain more oscillations at different frequencies, stacked one upon another. Higher-frequency, oscillations which are tonally related to the fundamental frequency (the base note or tone) are known as harmonics. All sounds in nature produce harmonics simply because the physics of air and other materials create simultaneous tones in higher octaves, the process of which is described in the standing waves wiki page.

In electronics, an unvarying electrical signal in an AC circuit can be represented graphically by a sine wave, as the voltage varies over time. This, along with some commentary on phase is illustrated in Video 1, and is the basis for modern electrical analogue recording technology.

Sine Function

The shape of a sine wave is relative to a vector moving around a regular circle, as defined by the Sine Function. This method of plotting a sine wave using sine function is illustrated in Video 1 and demonstrated further in terms of [pi] in Fig. 1.

A full understanding of the mathematics of the sine function is not necessary to understand the basics of the nature of a sine wave - it simply aids description of the features which a sine wave exhibits: that of mathematically predictable and consistent oscillation governed by a set of trigonometic rules.

Use in Audio

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Fig. 2 - Analogue audio signal as it relates to a sound pressure wave in air.

Sine waves are useful in audio recording because of the way an electrical sine wave signal relates to (is analogous to) the way sound moves in air, as seen in Fig. 2. Understanding the sine wave is a good gateway to understanding the factors involved in analogue recording.
Frequencies

Fig. 3 - Audible Frequency Bands

There are three main features which describe any kind of wave, and are particularly relevant to analogue audio. Those features are frequency, wavelength and amplitude. Frequency is measured in Hertz (Hz) which means times per second. Amplitude (in analogue electrical circuitry) is measured in Volts, and Wavelength is measured in small divisions of distance (E.G. Micrometres and nanometres.)

Frequency is important to audio because frequencies determine the pitch or note of the sound we hear. As illustrated in Fig. 3, a higher frequency is perceived (due to human psychoacoustics) as a 'higher' note in the scale. Jumping up from the C note (on a piano or any other instrument) to the next C in the scale is known in music as a jump of one octave, and the actual physical change here is that the frequency of the sound has been doubled. Going down an octave cuts the frequency in half. The fact that this is not a linear scale has implications on human hearing which are described in more detail on the psychoacoustics wiki page.

Wavelength is important to audio especially in acoustics scenarios where a room used in music production needs to be treated for ideal listening conditions. The wavelength of a sound wave affects the sort of material which is requried to stop it in its tracks. For example, the long-wavelength, low-frequency bass sound shown at the bottom of Fig. 3 would require a deeper and more substantial piece of absorbent material to stop it from reflecting back into the room and creating unwanted interference than the high-frequency (10-18kHz), short-wavelength treble waves at the top. Sound waves in that sort of high-end frequency range can be absorbed by even thin materials like carpet, because the waves can be effectively trapped between two strands of the material due to its very short wavelength. The bass waves can travel around small objects like these, and indeed bass waves frequently travel through the floor and walls of a building for this very reason. The measurement of Wavelength and Amplitude are shown in Fig. 4.

The other important factor to bear in mind when calculating features of a wave is the speed of sound, which in air (under normal circumstances, on Earth, at sea level) is 343 m/s. This is a rounded figure because variations in altitude, air pressure, temperature, humidity and gravitational forces affect the actual speed of sound. However, for the sake of most theoretical calculations, the speed of sound can indeed be considered to be 343 m/s, or rounded even further to 340m/s. The conventions on which number should be used vary considerably.
Wavelength

Fig. 4 - Wavelength and amplitude.

The relationship between frequency and wavelength can be easily calculated with a simple equation:

c = f * λ

When dealing with equations:

  • The speed of sound is represented by the lower-case Latin letter c (celeritas).
  • Frequency is represented by the lower-case Latin letter f.
  • Wavelength is represented by the Greek letter lambda: λ.

For example, if a wave is known to have a frequency of 100Hz (a low-frequency range bass tone), and the speed of sound in air is known to be 343m/s, then its wavelength can be calculated by substituting c for λ in the above equation:

Speed=Frequency * Wavelength

Wavelength = Speed / Frequency

Wavelength = 343 / 100

Wavelength = 3.43m

Imagining that the wavelength of a 100Hz bass tone is 3.43m puts the problem of acoustic treatment into perspective. A single cycle of that bass tone spreads out 3.43m, from peak-to-peak (as shown in Fig. 4).

Wavelength is, however, less important when dealing with electrical analogue signals moving through wires and cables. Upon recording a sound, the frequency content and its amplitude are more important. As the frequency defines the pitch, amplitude defines the volume or perceived loudness of the signal. These factors must be taken into account when using microphones and amplifiers in combination for analogue recording.

Sine Wave Summary

  • Sine waves represent repeating, consistent oscillations in any given parameter (such as SPL [Sound Pressure Level, in air] or Voltage [in an electrical circuit])
  • Sine waves exhibit three main features: frequency, wavelength and amplitude
  • The relationship between frequency and wavelength is given by the equation c = f * λ (where c=the speed of sound, f=frequency and λ=wavelength)
  • The shape and pattern of a sine wave is subject to the rules of the sine function, hence its name
  • Sine waves in electrical circuits operate in the same way as sound pressure waves operate in air, giving rise to electrical analogue audio recording methods
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