1.1. Acoustic waves

We consider the propagation of sound waves in a homogeneous isotropic medium viewed as an inviscid fluid.

• Geometry: A given open domain \(\Omega\subset\mathbb R^3\), and time interval \([0,T], T>0\)

• Constituents: The (vectorial) velocity field (unit \(m/s\)), the pressure (\(kg/(ms^2)\)) and the density (\(kg/m^3\)) of the fluid given as functions \(v:\Omega\times [0,T]\to\mathbb R^3\), \(p,\rho:\Omega\times [0,T]\to\mathbb R\).

• Balance relations: We impose the Euler equation of momentum

(1.1)\[\rho\left(\partial_t v+v\cdot\nabla v\right)+\nabla p = 0,\]

and the equation of continuity

(1.2)\[\partial_t\rho+\nabla\cdot (\rho v)=0,\]

where \(\partial_t\) denotes the time-derivative and \(\nabla\cdot\) the divergence. Note that these balance relations ad-hoc hold only in integral form (integration over each volume and time) and can be stated pointwise only for sufficiently regular functions.

For our model we choose \(p,v\) to be the primal quantities.

To close our system of equations we need another equation which is provided by the

• Material law, an equation of state, relating \(p\) and \(\rho\):

(1.3)\[p = h(\rho)\]

for some scalar function \(h\), which is usually assumed to be sufficiently regular and strictly increasing.

Next we assume that the quantities \(p,\rho,v\) are small perturbations of an equilibrium state, i.e.,

\[\begin{aligned} \rho&=\rho_0+\rho_1, &p&=p_0+p_1, &v&=v_0+v_1, \end{aligned}\]

thus to derive a linear model we may neglect the quadratic quantities of \(\rho_1,p_1,v_1\). We impose that the medium is in a static state, i.e. \(p_0,\rho_0\) are independent of \(t\) and \(v_0=0\) and obtain the linearized equation of momentum

(1.4)\[\rho_0\partial_t v_1+\nabla p_0+\nabla p_1 = 0.\]

Since the static state \(p_1,\rho_1,v_1=0\) also satisfies the (linearized) balance of momentum we immediately obtain that \(p_0(x)=const.\) From the constitutive relation (1.3) we also obtain that \(\rho_0(x)=const.\) Thus the linearized conservation of mass and momentum become

(1.5)\[\begin{split}\begin{aligned} \partial_t\rho_1+\rho_0\nabla\cdot v_1&=0,\\ \rho_0\partial_t v_1+\nabla p_1 &= 0. \end{aligned}\end{split}\]

It remains to use material law (1.3). Taking the derivative with respect to time and linearizing we obtain

\[\partial_t \rho_1= h'(\rho_0)\partial_t\rho_1=c^2\partial_t\rho_1,\]

where the parameter \(c^2:=h'(\rho_0)\) is the speed of sound in the respective medium (unit \(m/s\)). As another parameter we introduce the bulk modulus \(\kappa_0\) of the material as \(\kappa_0=\rho_0 c^2\), (unit \(kg/(ms^2)\)).

Combining the balance relations and material laws above, and ommiting the index \(\cdot_1\) yields the first order (mixed) form of the acoustic wave equation

(1.6)\[\begin{split}\begin{aligned} \partial_tv+\frac{1}{\rho_0}\nabla p &= f,&\text{in }&[0,T]\times \Omega,\\ \partial_t p+\kappa_0\nabla\cdot v &=0,&\text{in }&[0,T]\times \Omega,\\ p(0,\cdot)&=0,&\text{in }&\Omega,\\ p&=0,&\text{in }&[0,T]\times \partial\Omega,\\ \end{aligned}\end{split}\]

where we also add an external force as the right-hand-side term (acceleration, unit \(m/s^2\)) \(f\), in the vectorial equation and added homogeneous Dirichlet boundary conditions, and initial data to close the system.

Remark 1.1 (Variable materials)

For non-constant material-parameters \(\rho_0,\kappa_0\) the constitutive relation (1.3) needs to be replaced by

\[p = f(\rho,s)\]

where \(s\) is the entropy (unit \(kg\,m^2/(s^2\,K)\)) which satisfies the conservation law (adiabatic hypothesis)

\[\partial_t s +v\cdot\nabla s = 0.\]

A similar reasoning as above then leads to (4.10) with variable in space \(\rho_0,\kappa_0\) (but still uniform initial pressure \(p_0\)).

Remark 1.2 (Energy conservation)

Multiplying the first and second equation of (4.10) with \(f=0\) by \(\rho_0 v\) and \(\frac{p}{\kappa_0}\) respectively yields

\[\frac{1}{2}\left(\rho_0\partial_t|v|^2+\frac{1}{\kappa_0}\partial_t |p|^2\right)+\nabla\cdot(pv)=0.\]

Integration over \(\Omega\) and applying the Gauss theorem yields

\[\partial_t\frac{1}{2}\int_\Omega \rho_0|v|^2+\frac{1}{\kappa_0}|p|^2 = \int_{\partial\Omega}pv\cdot n = 0\]

due to the homogeneous boundary conditions. Thus the energy

\[E(t):=\frac{1}{2}\int_\Omega \rho_0|v|^2+\frac{1}{\kappa_0}|p|^2,\]

is conserved over time. The two terms can be interpreted as kinetiv and potential energy respectively.

The well-known second order form of the acoustic wave equation is then obtained by taking the time-derivative of the scalar equation and inserting the vectorial equation to obtain

(1.7)\[\begin{split}\begin{aligned} \partial_t^2 p-\kappa_0\nabla\cdot\frac{1}{\rho_0}\nabla p &= -\kappa_0 \nabla\cdot f,&\text{in }&[0,t]\times \Omega,\\ p(0,\cdot)&=p_0,&\text{in }&\Omega,\\ \partial_t p(0,\cdot)&=-\kappa_0\nabla\cdot v_0,&\text{in }&\Omega,\\ p&=0,&\text{on }&[0,t]\times \partial\Omega,\\ \end{aligned}\end{split}\]

Defining a velocity potential \(\phi\) we obtain

\[\begin{align*} \partial_t \phi &= -\frac{1}{\rho_0}p\\ v &= \nabla \phi \end{align*}\]

and thus the scalar wave equation for \(\phi\)

(1.8)\[\begin{align} \frac{1}{c^2} \partial_t^2\phi&=\Delta\phi-f\\ &+\text{i.c., b.c.} \end{align}\]

In the mathematical treatment of the acoustic wave equation the physical meaning of the modeled quantities is often neglected and the letter \(u\) is used for the scalar unknown.

Boundary conditions

The homogeneous Dirichlet boundary conditions used above corresponds to a sound-soft boundary, the pressure of the total wave vanishes at the boundary. Alternatively one could use sound-hard boundary conditions which are in the acoustic case modeled by the Neumann boundary condition

\[\begin{align*} \partial_n p &= 0,&\text{on }\partial\Omega. \end{align*}\]

Also a mix of boundary conditions is possible.

Another important type of boundary conditions for acoustic problems are so-called impedance boundary conditions given by

\[\begin{align*} \partial_n p &= \lambda \partial_t p,&\text{on }\partial\Omega. \end{align*}\]

for a suitable function \(\lambda\).

Example 1.1 (First order absorbing boundary condition)

An important example of impedance boundary conditions are first order absorbing boundary conditions which can be motivated as follows: D’Alembert’s solution of the one-dimensional (homogeneous) wave-equation with initial data \(p(0)=p_0, v(0)=v_0\) is given by

(1.9)\[\begin{aligned} p(t,x) = \frac{1}{2}\left(p_0(x-ct)+p_0(x+ct)+\frac{1}{c}\int_{x-ct}^{x+ct} v_0(\xi)d\xi\right). \end{aligned}\]

The first two terms constitute waves travelling to the right and left respectively. Thus, if we assume compactly supported initial data on a finite interval \((a,b)\) and want to impose boundary conditions at \(a,b\) that do not change the solution we may use

(1.10)\[\begin{split}\begin{aligned} c\partial_x p(t,a)&=\,\partial_t p(t,a),\\ c\partial_x p(t,b)&=-\,\partial_t p(t,b), \end{aligned}\end{split}\]

for all times \(t>0\).

In higher dimensions it is not so straightforward any more to differentiate incoming from outgoing waves. Still the higher dimensional counterpart of (1.10) can be stated as

(1.11)\[c\nabla p(t,x)\cdot n =\partial_t p(t,x)\]

for all \(x\in\partial\Omega\) and times \(t>0\), where \(n\) is the outward normal vector of the boundary \(\partial \Omega\). The condition (1.11) can be interpreted as a prescribed impedance at the interface \(\partial\Omega\) or, alternativly as a rough approximation to the exact absorbing boundary conditions.