Acoustic theory
Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.
For sound waves of any magnitude of a disturbance in velocity, pressure, and density we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} + \nabla\cdot(\rho'\mathbf{v}) & = 0 \qquad \text{(Conservation of Mass)} \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \qquad \text{(Equation of Motion)} \end{align} }
In the case that the fluctuations in velocity, density, and pressure are small, we can approximate these as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p'& = 0 \end{align} }
Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}(\mathbf{x},t)} is the perturbed velocity of the fluid, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_0} is the pressure of the fluid at rest, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p'(\mathbf{x},t)} is the perturbed pressure of the system as a function of space and time, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_0} is the density of the fluid at rest, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho'(\mathbf{x}, t)} is the variance in the density of the fluid over space and time.
In the case that the velocity is irrotational (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\times \mathbf{v} = 0} ), we then have the acoustic wave equation that describes the system:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi = 0 }
Where we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{v} & = -\nabla \phi \\ c^2 & = (\frac{\partial p}{\partial \rho})_s\\ p' & = \rho_0\frac{\partial \phi}{\partial t}\\ \rho' & = \frac{\rho_0}{c^2}\frac{\partial \phi}{\partial t} \end{align} }
Derivation for a medium at rest
Starting with the Continuity Equation and the Euler Equation:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho}{\partial t} +\nabla\cdot \rho\mathbf{v} & = 0 \\ \rho\frac{\partial \mathbf{v}}{\partial t} + \rho(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p & = 0 \end{align} }
If we take small perturbations of a constant pressure and density:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \rho & = \rho_0+\rho' \\ p & = p_0 + p' \end{align} }
Then the equations of the system are
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial}{\partial t}(\rho_0+\rho') +\nabla\cdot (\rho_0+\rho')\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla (p_0+p') & = 0 \end{align} }
Noting that the equilibrium pressures and densities are constant, this simplifies to
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} }
A Moving Medium
Starting with
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{w}+\nabla\cdot \rho'\mathbf{w} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{w}}{\partial t} + (\rho_0+\rho')(\mathbf{w}\cdot\nabla)\mathbf{w} + \nabla p' & = 0 \end{align} }
We can have these equations work for a moving medium by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{w} = \mathbf{u} + \mathbf{v}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} is the constant velocity that the whole fluid is moving at before being disturbed (equivalent to a moving observer) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v}} is the fluid velocity.
In this case the equations look very similar:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' + \nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{u}\cdot\nabla)\mathbf{v} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} }
Note that setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u} = 0} returns the equations at rest.
Linearized Waves
Starting with the above given equations of motion for a medium at rest:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} }
Let us now take Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v},\rho',p'} to all be small quantities.
In the case that we keep terms to first order, for the continuity equation, we have the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho'\mathbf{v}} term going to 0. This similarly applies for the density perturbation times the time derivative of the velocity. Moreover, the spatial components of the material derivative go to 0. We thus have, upon rearranging the equilibrium density:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} }
Next, given that our sound wave occurs in an ideal fluid, the motion is adiabatic, and then we can relate the small change in the pressure to the small change in the density by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p' = \left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}\rho' }
Under this condition, we see that we now have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial p'}{\partial t} +\rho_{0}\left(\frac{\partial p}{\partial \rho_0}\right)_{s}\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} }
Defining the speed of sound of the system:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \equiv \sqrt{\left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}} }
Everything becomes
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial p'}{\partial t} +\rho_0c^2\nabla\cdot \mathbf{v} & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} }
For Irrotational Fluids
In the case that the fluid is irrotational, that is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\times\mathbf{v} = 0} , we can then write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} = -\nabla\phi} and thus write our equations of motion as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial p'}{\partial t} -\rho_0c^2\nabla^2\phi & = 0 \\ -\nabla\frac{\partial\phi}{\partial t} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} }
The second equation tells us that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p' = \rho_0 \frac{\partial \phi}{\partial t} }
And the use of this equation in the continuity equation tells us that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho_0\frac{\partial^2 \phi}{\partial t} -\rho_0c^2\nabla^2\phi = 0 }
This simplifies to
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} -\nabla^2\phi = 0 }
Thus the velocity potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} obeys the wave equation in the limit of small disturbances. The boundary conditions required to solve for the potential come from the fact that the velocity of the fluid must be 0 normal to the fixed surfaces of the system.
Taking the time derivative of this wave equation and multiplying all sides by the unperturbed density, and then using the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p' = \rho_0 \frac{\partial \phi}{\partial t}} tells us that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{c^2}\frac{\partial^2 p'}{\partial t^2} -\nabla^2p' = 0 }
Similarly, we saw that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p' = \left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}\rho' = c^{2}\rho'} . Thus we can multiply the above equation appropriately and see that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{c^2}\frac{\partial^2 \rho'}{\partial t^2} -\nabla^2\rho' = 0 }
Thus, the velocity potential, pressure, and density all obey the wave equation. Moreover, we only need to solve one such equation to determine all other three. In particular, we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{v} & = -\nabla \phi \\ p' & = \rho_0 \frac{\partial \phi}{\partial t}\\ \rho' & = \frac{\rho_0}{c^2}\frac{\partial\phi}{\partial t} \end{align} }
For a moving medium
Again, we can derive the small-disturbance limit for sound waves in a moving medium. Again, starting with
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' + \nabla\cdot \rho'\mathbf{v} & = 0 \\ (\rho_0+\rho')\frac{\partial \mathbf{v}}{\partial t} + (\rho_0+\rho')(\mathbf{u}\cdot\nabla)\mathbf{v} + (\rho_0+\rho')(\mathbf{v}\cdot\nabla)\mathbf{v} + \nabla p' & = 0 \end{align} }
We can linearize these into
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{v} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} }
For Irrotational Fluids in a Moving Medium
Given that we saw that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial \rho'}{\partial t} +\rho_0\nabla\cdot\mathbf{v}+\mathbf{u}\cdot\nabla\rho' & = 0 \\ \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{v} + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} }
If we make the previous assumptions of the fluid being ideal and the velocity being irrotational, then we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} p' & = \left(\frac{\partial p}{\partial \rho_{0}}\right)_{s}\rho' = c^{2}\rho' \\ \mathbf{v} & = -\nabla\phi \end{align} }
Under these assumptions, our linearized sound equations become
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{1}{c^2}\frac{\partial p'}{\partial t} -\rho_0\nabla^2\phi+\frac{1}{c^2}\mathbf{u}\cdot\nabla p' & = 0 \\ -\frac{\partial}{\partial t}(\nabla\phi) - (\mathbf{u}\cdot\nabla)[\nabla\phi] + \frac{1}{\rho_0}\nabla p' & = 0 \end{align} }
Importantly, since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}} is a constant, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbf{u}\cdot\nabla)[\nabla\phi] = \nabla[(\mathbf{u}\cdot\nabla)\phi]} , and then the second equation tells us that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\rho_0} \nabla p' = \nabla\left[\frac{\partial\phi}{\partial t} + (\mathbf{u}\cdot\nabla)\phi\right] }
Or just that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p' = \rho_{0}\left[\frac{\partial\phi}{\partial t} + (\mathbf{u}\cdot\nabla)\phi\right] }
Now, when we use this relation with the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{c^2}\frac{\partial p'}{\partial t} -\rho_0\nabla^2\phi+\frac{1}{c^2}\mathbf{u}\cdot\nabla p' = 0} , alongside cancelling and rearranging terms, we arrive at
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} - \nabla^2\phi + \frac{1}{c^2}\frac{\partial}{\partial t}[(\mathbf{u}\cdot\nabla)\phi] + \frac{1}{c^2}\frac{\partial}{\partial t}(\mathbf{u}\cdot\nabla\phi) + \frac{1}{c^2}\mathbf{u}\cdot\nabla[(\mathbf{u}\cdot\nabla)\phi] = 0 }
We can write this in a familiar form as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\frac{1}{c^2}\left(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla\right)^{2} - \nabla^{2}\right]\phi = 0 }
This differential equation must be solved with the appropriate boundary conditions. Note that setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u}=0} returns us the wave equation. Regardless, upon solving this equation for a moving medium, we then have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{v} & = -\nabla \phi \\ p' & = \rho_{0}\left(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla\right)\phi\\ \rho' & = \frac{\rho_{0}}{c^{2}}\left(\frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla\right)\phi \end{align} }
See also
References
- Landau, L.D.; Lifshitz, E.M. (1984). Fluid Mechanics (2nd ed.). Butterworth-Heinenann. ISBN 0-7506-2767-0.
- Fetter, Alexander; Walecka, John (2003). Fluid Mechanics (1st ed.). Courier Corporation. ISBN 0-486-43261-0.