Growl

With growl (growl),

• A voice long low to come out of a throat when I strain and suffer.
• A low voice to utter unintentionally or consciously when I admired. →「I let audience groan」
• The battle cry that I sing at a low tone as if turning down a voice in a Noh song, naniwabushi recitation, bright lapis lazuli intentionally and talk about.
• A phenomenon to occur because of the interference of the wave in physics. I explain it in detail in the following.

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Two waves which are slightly different from growl (English: beat) in the physics in number of vibrations (or frequency) interfere it and say a phenomenon to produce the composition wave that the amplitude turns into periodically slowly [¹].

For example, the strength and weakness of the sound are asked in a period equivalent to a difference of the frequency of each fundamental tone when two sound waves that slightly vary in pace sound. I hear two sounds then to be one sound, but hear both as a two sound of the distinction when pace leaves it to some extent.

Table of contents

Mathematical explanation

As the simplest case, I think about the composition of two signatures wave that are equal to both strength and phase. When only width 2 α composes two sound sin (ω-α)t and sin (ω+α)t (in t time) where were out of an angular frequency back and forth, mainly on angular frequency ω [^{note 1}], the composition sound is as follows (the transformation of the expression refers to a trigonometric function).

${\displaystyle {\begin{aligned}&\sin(\omega -\alpha) t+\sin(\omega +\alpha) t\\&=(\sin \omega t\cos \alpha t-\cos \omega t\sin \alpha t)+(\sin \omega t\cos \alpha t+\cos \omega t\sin \alpha t)\\&=(2\cos \alpha t)\cdot \sin \omega t\end{aligned}}}$ 

As a result, the synthetic sound becomes the wave pattern that the amplitude of the sound of angular frequency ω fluctuates in angular frequency 2 α [^{note 2}].

Example

	Growl of 438Hz and 442Hz At first I oscillate a sine wave of 442Hz in a sine wave of 438Hz next from the right side from the left side. I let you oscillate both at the same time and produce a growl afterwards.
When cannot watch this sound and picture well; Help: Please see the reproduction of a sound, the animation.

For example, frequency cannot understand a conjugation of the sound pressure of one one between the people for a sound of 440Hz. However, frequency of the growls of 438Hz and 442Hz is 4Hz (four times a second); can be reasonable easily.

 

Green is the wave which a wave, blue of frequency 104Hz put on top of one another a wave of frequency 110Hz red, and a growl of 6Hz is seen in the upper figure.

Footnote

1. I used angular frequency ω to become easy to see a numerical formula ^ here, but am the same because frequency f has a relationship of angular frequency and ω = 2πf even if I think at frequency.
2. A factor of ^ cos is plus, but even minus number is careful about showing the same effect as the amplitude.

References

1. For ^ Zensuke Tokuoka edition "physics outline bottom" arts and sciences book publishing company, 1,988 years, it is page 31. ISBN 4-87361-022-2. 

• Serway, R. A., Vuille, C. College Physics. Volume 1, Ninth Edition, Cengage Learning, 2011, p. 499.

Allied item

• Moire
• Frequency-modulate
• Tuning
• Different tone

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