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en/3.9.19.md
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| + | ### Statement | ||
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| + | $3.9.19.$ [Insert the problem statement] | ||
| + | |||
| + | ### Solution | ||
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| + | ### Statement | ||
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| + | $3.9.19.$ Why is the hollow body of a violin and cello made figured? How does the tone of the sound depend on its dimensions? | ||
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| + | ### Solution | ||
| + | |||
| + | <b>1. Figured shape of the body:</b> | ||
| + | If the body had a simple, geometrically symmetrical shape (e.g., a parallelepiped, cylinder, or sphere), it would produce a small number of highly pronounced natural frequencies (standing waves). This would cause the instrument to act as a narrow-band filter: it would loudly resonate only on specific notes, while others would sound muffled. | ||
| + | The complex (figured) shape with non-parallel soundboards and curves breaks spatial symmetry. This creates a vast number of closely spaced resonant frequencies (vibration modes). As a result, the body acts as a broadband resonator that responds evenly to sounds of almost any frequency, making the tone of the instrument smooth and rich in overtones. | ||
| + | |||
| + | <b>2. Dependence of tone on dimensions:</b> | ||
| + | The body of the instrument acts as a volume acoustic resonator (Helmholtz resonator) and simultaneously as a system of vibrating plates. To form an effective standing wave, the linear dimension of the resonator $L$ must be comparable to the acoustic wavelength $\lambda$. | ||
| + | Since the frequency $\nu$ and wavelength are related by $\nu = \frac{v}{\lambda}$ (where $v$ is the speed of sound), increasing the dimensions of the body allows it to support longer standing waves, which correspond to a **lower frequency**. Therefore, the larger the instrument and its internal volume, the more efficiently it radiates low-frequency acoustic waves, resulting in a lower (deeper) fundamental tone (compare a violin, a cello, and a double bass). | ||
| + | |||
| + | #### Answer | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
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| ### Statement | |||
| $3.9.19.$ [Insert the problem statement] | |||
| ### Solution | |||
| ### Statement | |||
| $3.9.19.$ Why is the hollow body of a violin and cello made figured? How does the tone of the sound depend on its dimensions? | |||
| ### Solution | |||
| <b>1. Figured shape of the body:</b> | |||
| If the body had a simple, geometrically symmetrical shape (e.g., a parallelepiped, cylinder, or sphere), it would produce a small number of highly pronounced natural frequencies (standing waves). This would cause the instrument to act as a narrow-band filter: it would loudly resonate only on specific notes, while others would sound muffled. | |||
| The complex (figured) shape with non-parallel soundboards and curves breaks spatial symmetry. This creates a vast number of closely spaced resonant frequencies (vibration modes). As a result, the body acts as a broadband resonator that responds evenly to sounds of almost any frequency, making the tone of the instrument smooth and rich in overtones. | |||
| <b>2. Dependence of tone on dimensions:</b> | |||
| The body of the instrument acts as a volume acoustic resonator (Helmholtz resonator) and simultaneously as a system of vibrating plates. To form an effective standing wave, the linear dimension of the resonator $L$ must be comparable to the acoustic wavelength $\lambda$. | |||
| Since the frequency $\nu$ and wavelength are related by $\nu = \frac{v}{\lambda}$ (where $v$ is the speed of sound), increasing the dimensions of the body allows it to support longer standing waves, which correspond to a **lower frequency**. Therefore, the larger the instrument and its internal volume, the more efficiently it radiates low-frequency acoustic waves, resulting in a lower (deeper) fundamental tone (compare a violin, a cello, and a double bass). | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||