Acoustic waves: Difference between revisions
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{{NumEqn|<math> | {{NumEqn|<math> | ||
a=-\rho\frac{da}{d\rho} | a=-\rho\frac{da}{d\rho} | ||
</math>}} | </math>}|label=eq-speed-of-sound-a}} | ||
The one-dimensional momentum equation between station 1 and station 2 gives | The one-dimensional momentum equation between station 1 and station 2 gives | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{da}{d\rho}=-\frac{1}{2a\rho}\left(\frac{dp}{d\rho}+a^2\right) | \frac{da}{d\rho}=-\frac{1}{2a\rho}\left(\frac{dp}{d\rho}+a^2\right) | ||
</math>}} | </math>|label=eq-speed-of-sound-b}} | ||
{{EquationNote|label=eq-speed-of-sound-b|nopar=1}} in {{EquationNote|label=eq-speed-of-sound-a|nopar=1}} gives | |||
{{NumEqn|<math> | {{NumEqn|<math> | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
a^2=\frac{dp}{d\rho} | a^2=\frac{dp}{d\rho} | ||
</math>}} | </math>|label=eq-speed-of-sound-c}} | ||
Sound wave: | Sound wave: | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
a^2=\left(\frac{dp}{d\rho}\right)_s | a^2=\left(\frac{dp}{d\rho}\right)_s | ||
</math>}} | </math>|label=eq-speed-of-sound-d}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
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</math>}} | </math>}} | ||
where <math>\tau_s</math> is the compressibility of the gas. | where <math>\tau_s</math> is the compressibility of the gas. {{EquationNote|label=eq-speed-of-sound-d|nopar=1}} is valid for all gases. It can be seen from the equation, that truly incompressible flow (<math>\tau_s=0</math>) would imply infinite speed of sound. | ||
Since the process is isentropic, we can use the isentropic relations if we also assume the gas to be calorically perfect | Since the process is isentropic, we can use the isentropic relations if we also assume the gas to be calorically perfect | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
