# 亥姆霍兹共振

## 定性解释

${\displaystyle \omega _{H}={\sqrt {\gamma {\frac {A^{2}}{m}}{\frac {P_{0}}{V_{0}}}}}}$ (rad/s),

• ${\displaystyle \gamma }$ (gamma) 指的是绝热指数 或某种比热容。对于空气和其他双原子分子气体来说，这个值通常是1.4。
• ${\displaystyle A}$ 是开口的横截面积；
• ${\displaystyle m}$ 是开口的质量；
• ${\displaystyle P_{0}}$ 是共振腔的静态压力；
• ${\displaystyle V_{0}}$ 是共振腔的静态容积。

${\displaystyle A={\frac {V_{n}}{L_{eq}}}}$,

• ${\displaystyle L_{eq}}$ 是校正后瓶口的等效长度，可以由：${\displaystyle L_{eq}=L_{n}+0.3D}$计算出来， 其中${\displaystyle L_{n}}$是瓶口的实际长度，${\displaystyle D}$ 是瓶口的水力直径[4]
• ${\displaystyle V_{n}}$ 是开口处的空气体积，

${\displaystyle \omega _{H}={\sqrt {\gamma {\frac {A}{m}}{\frac {V_{n}}{L_{eq}}}{\frac {P_{0}}{V_{0}}}}}}$.

${\displaystyle \omega _{H}={\sqrt {\gamma {\frac {P_{0}}{\rho }}{\frac {A}{V_{0}L_{eq}}}}}}$ ,

${\displaystyle v={\sqrt {\gamma {\frac {P_{0}}{\rho }}}}}$ ,

${\displaystyle f_{H}={\frac {v}{2\pi }}{\sqrt {\frac {A}{V_{0}L_{eq}}}}}$.

## 应用

• 一个单片金属（或其它材料）上面有很多小孔以规则或不规则的形式分布，这就是所谓的阻力片；
• 一系列的“蜂窝腔”（但实际上起作用的是容积）。

## 参考文献

1. ^ Helmholtz, Hermann von (1885), On the sensations of tone as a physiological basis for the theory of music页面存档备份，存于互联网档案馆）, Second English Edition, translated by Alexander J. Ellis. London: Longmans, Green, and Co., p. 44. Retrieved 2010-10-12.
2. ^ Helmholtz resonator at Case Western Reserve University. Helmholtz Resonator. [16 February 2016]. （原始内容存档于2016-04-15）.
3. ^ Derivation of the equation for the resonant frequency of an Helmholtz resonator.. [2016-06-07]. （原始内容存档于2017-02-28）.
4. ^ End Correction at a Flue Pipe Mouth. [2016-06-07]. （原始内容存档于2020-02-19）.
5. ^ Ocarina Physics - How Ocarinas Work. ocarinaforest.com. [2012-12-31]. （原始内容存档于2013-03-14）.
6. ^ Wings that waggle could cut aircraft emissions by 20%. [2016-06-07]. （原始内容存档于2020-11-09）.
7. ^ Why Do Slightly Opened Car Windows Make That Awful Sound?. [2016-06-07]. （原始内容存档于2020-11-11）.