Acoustic theory

Starting with the above given equations of motion for a medium at rest:

${\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} +\nabla \cdot \rho '\mathbf {v} &=0\\(\rho _{0}+\rho '){\frac {\partial \mathbf {v} }{\partial t}}+(\rho _{0}+\rho ')(\mathbf {v} \cdot \nabla )\mathbf {v} +\nabla p'&=0\end{aligned}}}$

Let us now take ${\displaystyle \mathbf {v} ,\rho ',p'}$ to all be small quantities.

In the case that we keep terms to first order, for the continuity equation, we have the ${\displaystyle \rho '\mathbf {v} }$ 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:

${\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} &=0\\{\frac {\partial \mathbf {v} }{\partial t}}+{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

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

${\displaystyle p'=\left({\frac {\partial p}{\partial \rho _{0}}}\right)_{s}\rho '}$

Under this condition, we see that we now have

${\displaystyle {\begin{aligned}{\frac {\partial p'}{\partial t}}+\rho _{0}\left({\frac {\partial p}{\partial \rho _{0}}}\right)_{s}\nabla \cdot \mathbf {v} &=0\\{\frac {\partial \mathbf {v} }{\partial t}}+{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

Defining the speed of sound of the system:

${\displaystyle c\equiv {\sqrt {\left({\frac {\partial p}{\partial \rho _{0}}}\right)_{s}}}}$

Everything becomes

${\displaystyle {\begin{aligned}{\frac {\partial p'}{\partial t}}+\rho _{0}c^{2}\nabla \cdot \mathbf {v} &=0\\{\frac {\partial \mathbf {v} }{\partial t}}+{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

In the case that the fluid is irrotational, that is ${\displaystyle \nabla \times \mathbf {v} =0}$, we can then write ${\displaystyle \mathbf {v} =-\nabla \phi }$ and thus write our equations of motion as

${\displaystyle {\begin{aligned}{\frac {\partial p'}{\partial t}}-\rho _{0}c^{2}\nabla ^{2}\phi &=0\\-\nabla {\frac {\partial \phi }{\partial t}}+{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

The second equation tells us that

${\displaystyle p'=\rho _{0}{\frac {\partial \phi }{\partial t}}}$

And the use of this equation in the continuity equation tells us that

${\displaystyle \rho _{0}{\frac {\partial ^{2}\phi }{\partial t}}-\rho _{0}c^{2}\nabla ^{2}\phi =0}$

This simplifies to

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}-\nabla ^{2}\phi =0}$

Thus the velocity potential ${\displaystyle \phi }$ 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 ${\displaystyle p'=\rho _{0}{\frac {\partial \phi }{\partial t}}}$ tells us that

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}p'}{\partial t^{2}}}-\nabla ^{2}p'=0}$

Similarly, we saw that ${\displaystyle p'=\left({\frac {\partial p}{\partial \rho _{0}}}\right)_{s}\rho '=c^{2}\rho '}$. Thus we can multiply the above equation appropriately and see that

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\rho '}{\partial t^{2}}}-\nabla ^{2}\rho '=0}$

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

${\displaystyle {\begin{aligned}\mathbf {v} &=-\nabla \phi \\p'&=\rho _{0}{\frac {\partial \phi }{\partial t}}\\\rho '&={\frac {\rho _{0}}{c^{2}}}{\frac {\partial \phi }{\partial t}}\end{aligned}}}$

Again, we can derive the small-disturbance limit for sound waves in a moving medium. Again, starting with

${\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} +\mathbf {u} \cdot \nabla \rho '+\nabla \cdot \rho '\mathbf {v} &=0\\(\rho _{0}+\rho '){\frac {\partial \mathbf {v} }{\partial t}}+(\rho _{0}+\rho ')(\mathbf {u} \cdot \nabla )\mathbf {v} +(\rho _{0}+\rho ')(\mathbf {v} \cdot \nabla )\mathbf {v} +\nabla p'&=0\end{aligned}}}$

We can linearize these into

${\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} +\mathbf {u} \cdot \nabla \rho '&=0\\{\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {v} +{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

Given that we saw that

${\displaystyle {\begin{aligned}{\frac {\partial \rho '}{\partial t}}+\rho _{0}\nabla \cdot \mathbf {v} +\mathbf {u} \cdot \nabla \rho '&=0\\{\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {v} +{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

If we make the previous assumptions of the fluid being ideal and the velocity being irrotational, then we have

${\displaystyle {\begin{aligned}p'&=\left({\frac {\partial p}{\partial \rho _{0}}}\right)_{s}\rho '=c^{2}\rho '\\\mathbf {v} &=-\nabla \phi \end{aligned}}}$

Under these assumptions, our linearized sound equations become

${\displaystyle {\begin{aligned}{\frac {1}{c^{2}}}{\frac {\partial p'}{\partial t}}-\rho _{0}\nabla ^{2}\phi +{\frac {1}{c^{2}}}\mathbf {u} \cdot \nabla p'&=0\\-{\frac {\partial }{\partial t}}(\nabla \phi )-(\mathbf {u} \cdot \nabla )[\nabla \phi ]+{\frac {1}{\rho _{0}}}\nabla p'&=0\end{aligned}}}$

Importantly, since ${\displaystyle \mathbf {u} }$ is a constant, we have ${\displaystyle (\mathbf {u} \cdot \nabla )[\nabla \phi ]=\nabla [(\mathbf {u} \cdot \nabla )\phi ]}$, and then the second equation tells us that

${\displaystyle {\frac {1}{\rho _{0}}}\nabla p'=\nabla \left[{\frac {\partial \phi }{\partial t}}+(\mathbf {u} \cdot \nabla )\phi \right]}$

Or just that

${\displaystyle p'=\rho _{0}\left[{\frac {\partial \phi }{\partial t}}+(\mathbf {u} \cdot \nabla )\phi \right]}$

Now, when we use this relation with the fact that ${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial p'}{\partial t}}-\rho _{0}\nabla ^{2}\phi +{\frac {1}{c^{2}}}\mathbf {u} \cdot \nabla p'=0}$, alongside cancelling and rearranging terms, we arrive at

${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}\phi }{\partial t^{2}}}-\nabla ^{2}\phi +{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}[(\mathbf {u} \cdot \nabla )\phi ]+{\frac {1}{c^{2}}}{\frac {\partial }{\partial t}}(\mathbf {u} \cdot \nabla \phi )+{\frac {1}{c^{2}}}\mathbf {u} \cdot \nabla [(\mathbf {u} \cdot \nabla )\phi ]=0}$

We can write this in a familiar form as

${\displaystyle \left[{\frac {1}{c^{2}}}\left({\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla \right)^{2}-\nabla ^{2}\right]\phi =0}$

This differential equation must be solved with the appropriate boundary conditions. Note that setting ${\displaystyle \mathbf {u} =0}$ returns us the wave equation. Regardless, upon solving this equation for a moving medium, we then have

${\displaystyle {\begin{aligned}\mathbf {v} &=-\nabla \phi \\p'&=\rho _{0}\left({\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla \right)\phi \\\rho '&={\frac {\rho _{0}}{c^{2}}}\left({\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla \right)\phi \end{aligned}}}$

• Acoustic attenuation
• Sound
• Fourier analysis

• 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.