Acoustic waves - Revision history

Nian at 15:37, 2 April 2026

2026-04-02T15:37:49Z

← Older revision		Revision as of 15:37, 2 April 2026
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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}}

	~~Eqn. \ref~~{eq~~:speedofsound:~~b} in ~~\ref~~{eq~~:speedofsound:~~a} gives		{{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. ~~Eqn. \ref~~{eq~~:speedofsound:~~d} 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.		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>

Nian at 07:56, 1 April 2026

2026-04-01T07:56:27Z

← Older revision		Revision as of 07:56, 1 April 2026
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	--><noinclude><!--		--><noinclude><!--
	-->{{#vardefine:secno\|3}}<!--		-->{{#vardefine:secno\|3}}<!--
	-->{{#vardefine:eqno\|0}}<!--		-->{{#vardefine:eqno\|1}}<!--
	--></noinclude><!--		--></noinclude><!--

Nian at 07:44, 1 April 2026

2026-04-01T07:44:16Z

Show changes

Nian at 19:40, 20 March 2026

2026-03-20T19:40:30Z

← Older revision		Revision as of 19:40, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	~~[[Category:Topic]]~~
	[[Category:One-dimensional flow]]		[[Category:One-dimensional flow]]
	[[Category:Inviscid flow]]		[[Category:Inviscid flow]]
	[[Category:Wave solution]]		[[Category:Wave solution]]
			<noinclude>
			[[Category:Compressible flow:Topic]]
			</noinclude>

	__TOC__		__TOC__

Nian at 17:14, 20 March 2026

2026-03-20T17:14:02Z

← Older revision		Revision as of 17:14, 20 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
			[[Category:Topic]]
	[[Category:One-dimensional flow]]		[[Category:One-dimensional flow]]
	[[Category:Inviscid flow]]		[[Category:Inviscid flow]]

Nian: /* Acoustic Waves */

2026-03-17T13:31:37Z

Acoustic Waves

← Older revision		Revision as of 13:31, 17 March 2026
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	__TOC__		__TOC__

	~~==Acoustic Waves==~~

	<!--		<!--

Nian: Nian moved page Acoustic Waves to Acoustic waves

2026-03-16T10:08:17Z

Nian moved page Acoustic waves to Acoustic waves

← Older revision	Revision as of 10:08, 16 March 2026
(No difference)

Nian at 10:05, 16 March 2026

2026-03-16T10:05:21Z

← Older revision		Revision as of 10:05, 16 March 2026
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	[[Category:Compressible flow]]		[[Category:Compressible flow]]
	[[Category:One-dimensional flow]]		[[Category:One-dimensional flow]]
	[[Category:~~inviscid~~ flow]]		[[Category:Inviscid flow]]
			[[Category:Wave solution]]

	__TOC__		__TOC__

Nian: Created page with "Category:Compressible flow Category:One-dimensional flow Category:inviscid flow TOC ==Acoustic Waves== In Fig. \ref{fig:soundwave}, station 1 represents the flow state ahead of the sound wave and station 2 the flow state behind the sound wave. Set up the continui..."

2026-03-16T10:04:26Z

Created page with "Category:Compressible flow Category:One-dimensional flow Category:inviscid flow __TOC__ ==Acoustic Waves==  In Fig. \ref{fig:soundwave}, station 1 represents the flow state ahead of the sound wave and station 2 the flow state behind the sound wave. Set up the continui..."

New page

[[Category:Compressible flow]]
[[Category:One-dimensional flow]]
[[Category:inviscid flow]]

__TOC__

==Acoustic Waves==



In Fig. \ref{fig:soundwave}, station 1 represents the flow state ahead of the sound wave and station 2 the flow state behind the sound wave. Set up the continuity equation for one-dimensional flows between 1 and 2. If we could change frame of reference and follow the sound wave, we would see fluid approaching the wave with the propagation speed of the wave, <math>a</math>, and behind the wave, the fluid would have a slightly modified speed, <math>a+da</math>. There would also be a slight in all other flow properties. Let's apply the one-dimensional continuity equation between station 1 and station 2.

<math display="block"> \rho_1 u_1=\rho_2 u_2</math>

<math display="block">\rho a=(\rho+d\rho)(a+da)</math>

<math display="block">\cancel{\rho a}=\cancel{\rho a} + \rho da + ad\rho +\underbrace{d\rho da}_{\sim 0} \Rightarrow</math>

<math display="block">
a=-\rho\frac{da}{d\rho}
</math>

The one-dimensional momentum equation between station 1 and station 2 gives

<math display="block">\rho_1 u_1^2+p_1=\rho_2 u_2^2+p_2</math>

<math display="block">\rho a^2+p=(\rho+d\rho)(a+da)^2+(p+dp)</math>

<math display="block">\cancel{\rho a^2}+\cancel{p}=\cancel{\rho a^2}+2\rho ada+\underbrace{\rho da^2}_{\sim 0}+d\rho a^2+\underbrace{2d\rho ada}_{\sim 0}+\underbrace{d\rho da^2}_{\sim 0}+\cancel{p}+dp \Rightarrow</math>

<math display="block">dp=-2\rho ada-d\rho a^2 \Rightarrow</math>

<math display="block">da=-\frac{dp+d\rho a^2}{2\rho a}=-\frac{d\rho}{2a\rho}\left(\frac{dp}{d\rho}+a^2\right) \Rightarrow</math>

<math display="block">
\frac{da}{d\rho}=-\frac{1}{2a\rho}\left(\frac{dp}{d\rho}+a^2\right)
</math>

Eqn. \ref{eq:speedofsound:b} in \ref{eq:speedofsound:a} gives

<math display="block">a=\frac{1}{2a}\left(\frac{dp}{d\rho}+a^2\right) \Rightarrow</math>

<math display="block">
a^2=\frac{dp}{d\rho}
</math>

Sound wave:

* gradients are small
* irreversible (dissipative effects are negligible)
* no heat addition

Thus, the change of flow properties as the sound wave passes can be assumed to be an isentropic process

<math display="block">
a^2=\left(\frac{dp}{d\rho}\right)_s
</math>

<math display="block">
a=\sqrt{\left(\frac{dp}{d\rho}\right)_s}=\sqrt{\frac{1}{\rho \tau_s}}
</math>

where <math>\tau_s</math> is the compressibility of the gas. Eqn. \ref{eq:speedofsound:d} 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\\

<math display="block">\frac{p_2}{p_1}=\left(\frac{\rho_2}{\rho_1}\right)^\gamma \Rightarrow p=C\rho^\gamma</math>

<math display="block">a^2=\left(\frac{dp}{d\rho}\right)_s=\gamma C\rho^{\gamma-1}=\gamma \underbrace{\left[C\rho^\gamma\right]}_{=p}\rho^{-1}=\frac{\gamma p}{\rho}\Rightarrow</math>

<math display="block">
a=\sqrt{\frac{\gamma p}{\rho}}
</math>

or

<math display="block">
a=\sqrt{\gamma RT}
</math>

From the relation above, it is obvious that the local speed of sound is related to the temperature of the flow, which in turn is a measure of the motion of elementary particles (atoms and/or molecules) of the fluid at a specific location. This stems from the fact that sound waves are propagated via interaction of these elementary particles. Since information in a flow is propagated via molecular interaction the relation between the speed at which this information is conveyed and the speed of the flow has important physical implications. Figure~\ref{fig:speed:of:sound} compares three sound wave patterns generated by a a beacon. In the left picture, the sound transmitter is stationary and thus the acoustic waves are centered around the transmitter. In the middle image, the transmitter is moving to the left at a speed less than the speed of sound and thus the transmitter will always be within all sound wave circles but it will be off-centered with a bias in the direction that the transmitter is moving. In the right image the transmitter is moving faster than the speed of sound and thus it will always be located outside of all acoustic waves. In a supersonic flow, no information can travel upstream and therefore there is no way for the flow to adjust to downstream obstacles. This is compensated for by the introduction of shocks in the flow. Over a shock flow properties changes discontinuity. An example is given in Figure~\ref{fig:supersonic:flow}.