38 2 Derivation of Acoustic Wave Equation
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mavca2 |
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PoVa ¼ |
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So, the change in translation energy will be: |
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dðPoVaÞ ¼ d 3 mavca2 |
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2.3.3Derivation of Equation of State
Equating two equations derived from the previous two sections by d mav2ca yields:
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PoVa |
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α |
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2 PodVa |
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3 d |
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! 2Po ∙ dVa ¼ 3αðdP ∙ Va þ Po ∙ dVaÞ |
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! 3α ∙ dP ∙ Va ¼ ð3α þ 2ÞPo ∙ dVa |
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dP |
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3α þ 2 |
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dVa |
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Po |
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3α |
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Va |
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the system remains |
constant, that is, ρ |
V |
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constant, |
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Because the mass dVa |
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dρ |
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o |
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ρodVa + dρVa ¼ 0, or |
Va |
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¼ ρo , the equation above changes to: |
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Po |
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3α |
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ρo |
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dP |
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3α þ 2 |
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dρ |
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In the above equation, dP is the pressure difference between the absolute pressure P and the constant average pressure Po (Po¼ 1 atm).
Similarly, in the above equation, dρ is the density difference between the absolute density ρ and the constant average density ρo.
Therefore, we can replace dP with P Po and dρ with ρ ρo to get:
Po |
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3α |
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ρo |
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P Po |
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3α þ 2 |
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ρ |
ρo |
Finally, the ratio between the normalized pressure difference and the normalized
ð3αþ2Þ
density difference 3α is called the ratio of specific heats and is defined as: