In Physical, the superposition of two Wave S of Frequency, of Number of wave or different Phase gives place to the phenomenon of beat . One observes it in fact in all the fields or intervene of the waves, in particular the Acoustique and the Optique.

Musical acoustics

In Acoustics musical, the beat is a sound perception due to the mixture of two sounds, close fundamental frequencies, or containing close harmonic frequencies. It is the sound equivalent of the fringes of moire that one can observe in Optique.

When two sounds are frequencies ƒ₁ and ƒ₂ very close - thus close heights - the ear perceives a kind of slow pulsation of which the frequency is the difference ƒ₁ - ƒ₂ in absolute value.

For example, a the with 440 Hz played at the same time as a the with 443 Hz will jointly produce a pulsation of 1,5 beats a second:

A beat is also perceived between sounds of frequency ƒ₂ and ƒ₃ if the latter is a simple harmonic frequency of ƒ₁. For example, if ƒ₃ is the Quinte of ƒ₁, then:

$f\_3\; =\; f\_1\; \backslash \; times\; \backslash \; frac\; \{3\}\; \{2\}$

The beats occur in fact in great number between all the involved frequencies, but the majority are not audible. Indeed, it is possible that these beats or the resulting sounds correspond to existing frequencies, which are then reinforced, or although they are of too low intensity or that they vary too slowly (a beat every 5 seconds or slower). It will not be heard them either if the beats are too fast: beyond 20 beats per seconds, they are not discernible any more, and one enters the field of the audio frequencies: 20 Hz (extreme low register) being regarded as the auditive threshold of an ordinary ear. The phenomenon remains the same one, but one speaks then, because of the change of perception, His resulting.

The phenomenon of beat gets along very well when a person grants a string instrument (for example a Guitare): one hears a vibration of the sound, due to the mixture of the sounds emitted by the two cords pinches together. It is this phenomenon which makes it possible to carry out, simply with the ear, the agreement of the musical instruments: an interval is pure when one hears any more no beat.

But conversely, the existence of a recognized beat also makes it possible to carry out the agreement or the intonation. For example, the third major is never used pure (except in old Musique), the quality of its beat makes it possible to the instrumentalists to make sure that they play just.

Methods of Accordage also employ like tool infallible, but not always practical, the counting of the beats: their speed indicates with a high degree of accuracy the state as of the moderate intervals essential at the agreement of an instrument to fixed sounds.

Intervals less consonants, although “right”, such as the second or minor Sixth, can also generate beats which are constitutive of their nature. It is there the reason of their weak Consonance. Even intervals enough consonants contain some: to see the interesting case of the third .

General case

The Onde S can be represented by goniometrical functions: indeed, the theorem of Fourier guarantees that one can break up any periodic function like summons goniometrical functions.

Let us suppose linear waves, solutions of the equation of Alembert, being propagated transversely on a dimension: for example a vibrating Cord. Then displacement in a point of X-coordinate X , to one moment T compared to the home position is given by the formula:

$A\; \backslash \; left\; (X,\; T\; \backslash \; right)\; =\; A\_0\; \backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; \backslash \; cdot\; T\; +\; K\; \backslash \; cdot\; X\; +\; \backslash \; alpha\; \backslash \; right)$

with $\backslash \; omega$ the Pulsation (in rad·s^-1), $k$ the Number of wave (in rad·m^-1), $\backslash \; alpha$ the phase in the beginning (in rad) and A₀ the Amplitude of the wave.

One can connect the Pulsation to the Fréquence by this equation:

$\backslash \; Omega\; =\; 2\; \backslash \; pi\; f$

Beats in time

To simplify, one places oneself in a fixed point, of X-coordinate $x\_0$, and one studies the beats which occur in this point, in time.

$x\_0$ is chosen so that:

$k\; \backslash \; cdot\; x\_0\; +\; \backslash \; alpha\; =\; 0$

One has then, at any moment T :

$A\; \backslash \; left\; (x\_0,\; T\; \backslash \; right)\; =\; A\_0\; \backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; \backslash \; cdot\; T\; \backslash \; right)$

Since one has Interférence S between two waves, two goniometrical functions should be summoned. That can be made by using the formulas of addition:

$\backslash \; cos\; \backslash \; left\; (\backslash \; right\; has)\; +\; \backslash \; cos\; \backslash \; left\; (B\; \backslash \; right)\; =\; 2\; \backslash \; cdot\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{has\; +\; B\}\; \{2\}\; \backslash \; right)\; \backslash \; cdot\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{has\; -\; B\}\; \{2\}\; \backslash \; right)$.

Let us suppose that two of the same waves are propagated amplitude, but of different pulsations. Then displacement in a point and the sum of the contributions of the two waves:

$A\_1\; \backslash \; left\; (x\_0,\; T\; \backslash \; right)\; +\; A\_2\; \backslash \; left\; (x\_0,\; T\; \backslash \; right)\; =\; 2\; A\_0\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; omega\_1\; +\; \backslash \; omega\_2\}\; \{2\}\; \backslash \; cdot\; T\; \backslash \; right)\; \backslash \; cdot\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; omega\_1\; -\; \backslash \; omega\_2\}\; \{2\}\; \backslash \; cdot\; T\; \backslash \; right)$

It appears that the total wave can be broken up into a “basic” wave, of fast pulsation $(\backslash \; omega\_1\; +\; \backslash \; omega\_2)\; /2$, and in a wave of slow pulsation $(\backslash \; omega\_1\; -\; \backslash \; omega\_2)\; /2$ which varies the amplitude of the first.

Beats in space

It is possible to make a complementary study: one fixes one moment t₀ and one looks at the wave in space.

One chooses T ₀ so that:

$\backslash \; Omega\; \backslash \; cdot\; t\_0\; +\; \backslash \; alpha\; =\; 0\; \backslash ,$

Just as previously, the total wave is the sum of the two waves of different numbers of wave:

$A\_1\; \backslash \; left\; (X,\; t\_0\; \backslash \; right)\; +\; A\_2\; \backslash \; left\; (X,\; t\_0\; \backslash \; right)\; =\; 2\; \backslash \; cdot\; A\_0\; \backslash \; cdot\; \backslash \; cos\; \backslash \; left\; (X\; \backslash \; cdot\; \backslash \; frac\; \{k\_1+k2\}\; \{2\}\; \backslash \; right)\; \backslash \; cdot\; \backslash \; cos\; \backslash \; left\; (X\; \backslash \; cdot\; \backslash \; frac\; \{k\_1\; -\; k\_2\}\; \{2\}\; \backslash \; right)\; \backslash ,$

one obtains a space figure of interference, having also a variation of small Wavelength ( K ₁ + K ₂) /2 and one variation big wavelength ( K ₁ - K ₂) /2

Difference in phase

If one considers of the same waves now amplitude has , of the same pulsation ω (thus of the same number of wave K ) but of different phase α, one a:

$A\_1\; \backslash \; left\; (X,\; T\; \backslash \; right)\; +A\_2\; \backslash \; left\; (X,\; T\; \backslash \; right)\; =\; 2\; \backslash \; cdot\; A\_0\; \backslash \; cdot\; \backslash \; cos\; \backslash \; left\; (\backslash \; Omega\; \backslash \; cdot\; T\; +\; K\; \backslash \; cdot\; X\; +\; \backslash \; frac\; \{\backslash \; alpha\_1\; +\; \backslash \; alpha\_2\}\; \{2\}\; \backslash \; right)\; \backslash \; cdot\; \backslash \; cos\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; alpha\_1\; -\; \backslash \; alpha\_2\}\; \{2\}\; \backslash \; right)$

The resulting wave thus has the same pulsation, but its phase in the beginning and its amplitude depend on the phases of the interfering waves.

If α₁ = α₂, the factor cos ((α₁ - α₂) /2) is worth cos (0) = 1, one thus has a double wave of amplitude; one speaks about constructive interferences and one says that the waves are “in phase”.

If on the other hand α₁ = α₂ + π, the factor cos ((α₁ - α₂) /2) is worth cos (π/2) = 0, the waves are cancelled; one speaks about destructive interferences and one says that the waves are “in opposition of phase”.

Between these situations, the amplitude passes from 2· has ₀ to 0 according to the factor cos ((α₁ - α₂) /2). The places where one has an extinction of the sound for two loudspeakers connected in opposition of phase correspond to the places for which the waves are always in opposition of phase.

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