Beat (acoustics)

Beat is referred to as the resultant of the additive superposition ( superposition ) of two oscillations which differ from each other only slightly in frequency, so that there is interference. Beats occur among others in the field of signal processing, when the two signal frequencies and close to each other. Next it occurs in waves, for which the superposition principle applies, so for example when sound waves and electromagnetic waves.

When the beat no new frequencies are, in contrast to the method, as you will find mixers application generates, and also there are no frequency shifts.

Derivation of the equations for beat frequency and period

Two harmonic oscillations with slightly different frequencies

For simplicity, it is assumed that the two oscillations have the same amplitude.

Then the sum of vibration can be represented as follows:

This expression can be formed by use of the trigonometric addition theorem:

The sum of the two frequencies can be expressed as a product of a local oscillation and a frequency of the envelope:

The amount of the envelope is the so-called beat frequency

Their value is much smaller than. The resulting beat period

Is the time interval between two points of minimum amplitude (nodes) of the beat function. The closer the two output frequencies and lie together, the greater is the beat period.

If the amplitudes and frequencies of the two are not equal, then one speaks of the so-called unclean beat. In this general form, there is a fluctuation of the beat period.

Acoustic beats

In acoustics, the beat can be heard clearly: sounding of two tones whose frequencies differ only slightly, so you will hear a tone whose frequency corresponds to the average of the frequencies of the two superimposed tones. This tone is modulated, its volume varies with the so-called beat frequency corresponding to the difference of the frequencies of the two tones.

Increases the frequency difference, so the ear is able to increasingly rapid changes in volume not to follow and one hears a sound harsh timbre, which splits upon further increasing the frequency difference in two single tones. Exceed the beat frequency, the hearing threshold of approximately 20 Hz, it will be audible as a difference frequency.

This phenomenon demonstrates the following sound example / i: A sine wave with a constant frequency of 440 Hertz is a second sine wave superimposed whose frequency increases from 440 hertz to 490 hertz.

As the beats of an interval ( a semitone here ) are perceived depends very strongly on the altitude, as will be seen in the following example:

For example, the ( sinusoidal ) tones e and f of the large to the three-line octave first individually, then together Played. The frequency of f in each octave higher than 6.6% of E.

Sound Samples

Beats in the superposition of two tones at 440 Hz and 440.5 Hz

Two chromatic semitones ( frequency difference 4%) in harmony

Listen? / I

Listen? / I

Sound samples with special waveforms

To facilitate an understanding of the acoustic beating something here found four exemplary different vibrations. All have the same start frequency of 110 Hz, but they differ in their waveform: triangle, square, sawtooth, sine

• Triangle wave? / I
• Square wave? / I
• Sawtooth wave? / I
• Sine wave? / I

In all four examples, two sound vibrations were superimposed, which initially have the same frequency. From 4 seconds, the frequency of a vibration is gradually increased ( in 8 seconds to 50 cents), then it remains equal to 6 seconds, then rapidly decreased ( by 100 cents) and after a further stable phase ( -50 cents) it is changed back to the output frequency. An exact course provides the following chart shows:

Beats with impure intervals

For unclean intoned intervals, one can calculate the beats of the harmonics as follows:

Example of this in the mean tone Mood: meantone fifths / i?

The usually lying outside the critical area intervals you hear a beat, when two clearly audible colorations or an overtone and a fundamental frequency are close together.

As can be seen from the following wave images with pure sine tones scarcely a beat is perceived ( The amplitudes does not change much ), but at a high overtone it is clearly audible.

Beats at intervals play a major role in the pure, the Mean, the well-tempered and of equal temperament tuning. For example, one hears at a pure third no beat, at the same stage a significant - perceived as friction - beat. The beats of the meantone fifths are tuned so low that they are not perceived as discord.

Acoustic deception?

The auditory perception of beats not generally rely on an acoustic illusion, but to physically real processes. This is different for the so-called binaural beats, where the ears via headphones ever one of two differing frequencies is supplied and the perception of beats is constructed by the signal processing in the brain.

Applications

The phenomenon of the beat can be applied versatile. In the musical practice as it is

• Invigorating sound effect used for example as so-called switchable tremolo effect or as a special register in pipe organs in musical instruments.
• The Leslie speaker cabinet uses the Doppler effect to produce the beat. Here, the constant sound is superimposed with a vibrating in pitch sound.
• The tremolo harmonica ( Viennese tuning ) and most Handzuginstrumenten the tone with two punch tongues, which are tuned to a beat is occurring.
• The tuning a musical instrument by ear (without tuner with LED display ), so the actual attunement to concert pitch as the reference frequency, is performed until no more beat is heard: the sound is " beatless zero - it 's true ."
• The Tonharmonie of bamboo instrument Angklung is based on the principle of two to four in- beat ensemble (bass, melodic instruments and chords ) that are shaken at the same time.

Unpleasant disturbing the beat is, however, when two instruments with approximately sinusoidal tones ( flute ) play closely spaced tones - they say the sounds rub. In unison interplay of two recorder beginner it may even happen that in the depths of an extremely penetrating difference tone is audible at extreme impropriety.