2. Benchmark solution

2.1. Calculation method

The problem is spherically symmetric. We therefore place ourselves in the \((r,\mathrm{\theta },\mathrm{\varphi })\) frame of reference. We consider the case where \({p}_{\mathit{imp}}\) does not depend on \(\mathrm{\theta }\), nor on \(\mathrm{\varphi }\). This hypothesis allows us to express the analytical solution for the problem in a harmonic regime and semi-analytic for the transient problem by exploiting the symmetry of the problem. In fact, by the symmetry in the problem, the structural displacement is written as:

(2.1)\[ {u} _ {s} (r,\ mathrm {\ theta},\ mathrm {\ varphi}, t) =u (r, t) {e} _ {r}\]

And the sound pressure in the fluid:

(2.2)\[ {p} _ {f} (r,\ mathrm {\ theta},\ mathrm {\ varphi}, t) =p (r, t) =p (r, t)\]

The linearized Green-Lagrange strain tensor is therefore given by:

(2.3)\[ \ mathrm {\ epsilon} (r, t) =\ frac {\ partial u (r, t)} {\ partial r} (r, t) {e} _ {r}\ otimes {e} _ {r} _ {r} +\ frac {u (r, t)} +\ frac {u (r, t)} {\ frac {u, t)} {r, t)} {\ frac {u (r, t)} {r} _ {\ mathrm {\ theta}} + {e}} _ {\ mathrm {\ varphi}}\ otimes {e} _ {\ mathrm {\ varphi}}}\ right)\]

With an isotropic elastic material, the Cauchy stress is then equal to:

\[\]

: label: eq-4

mathrm {sigma} (r, t) =left ((mathrm {lambda} +2mathrm {mu})frac {partial u (r, t)} {partial r}} {partial r} +2mathrm {lambda} +2mathrm {lambda}lambda}frac {lambda}frac {u (r, t)} {r}otimes {e} _ {r} +left (mathrm {lambda}frac {partial u (r, t)} {partial r} + (mathrm {lambda} +2mathrm {mu})frac {mu})frac {u (r, t)} {r, t)} {r, t)} {r}right)left ({e} _ {mathrm {theta}}}otimes {e} _ {mathrm {theta}} + {e}} _ {mathrm {varphi}}otimes {e} _ {mathrm {varphi}}right)

\(\mathrm{\lambda }\) and \(\mathrm{\mu }\) being the Lamé coefficients.

The expression for the conservation of momentum allows us to write:

\[\]

: label: eq-5

frac {{r} ^ {2}} {{c}} {{c} _ {s} ^ {2}}frac {{partial} ^ {2}} u (r, t)} {partial {t} ^ {2}}} - {r} {2}} - {r}}} - {r} {2}}} -rfrac {partial u (r, t)}} {partial r} +2u (r, t) =0

With the speed of sound propagation in structure \({c}_{s}\) which is equal to:

(2.4)\[ {c} _ {s} ^ {2} =\ frac {\ mathrm {\ lambda} +2\ mathrm {\ mu}} {{\ mathrm {\ rho}}} _ {s}}\]

\(\mathrm{\lambda }\) and \(\mathrm{\mu }\) being the Lamé coefficients.

(2.5)\[ \ frac {1} {{c} _ {f} ^ {2}}}\ frac {{\ partial} ^ {2} p} {\ partial {t} ^ {2}} (t, x) -\ mathrm {\ Delta} p (t, x) =0\]

In spherical coordinates, the sound pressure propagation equation is written as:

(2.6)\[ \ frac {1} {{c} _ {f} ^ {2}}}\ frac {{\ partial} ^ {2} p} {\ partial {t} ^ {2}} (r, t) -\ frac {{\ partial}} ^ {2}}}\ frac {2}} {\ partial} ^ {2}}} (r, t) -\ frac {2} {r} {r} {r}\ frac {\ partial} ^ {2}}} (r, t) -\ frac {2} {r} {r}}\ frac {\ partial} {\ partial}} partial p} {\ partial r} (r, t) =0\]

The condition of continuity of the normal stress on the surface \({\mathrm{\Gamma }}_{\mathit{ext}}\) is written in the form of a scalar equation:

:math:`(mathrm{lambda }+2mathrm{mu }){frac{partial u(t)}{partial r}

}_{r={R}_{mathit{ext}}}+2mathrm{lambda }{frac{u(t)}{{R}_{mathit{ext}}}

}_{r={R}_{mathit{ext}}}=-{p(t)

}_{r={R}_{mathit{ext}}}`

The condition for the continuity of normal acceleration on surface \({\mathrm{\Gamma }}_{\mathit{ext}}\) is written as:

:math:`{frac{{partial }^{2}u(t)}{partial {t}^{2}}

}_{{R}_{mathit{ext}}}=-{frac{1}{{mathrm{rho }}_{f}}frac{partial p(t)}{partial r}

}_{{R}_{mathit{ext}}}`

The pressure is imposed on surface \({\mathrm{\Gamma }}_{\text{int}}\) and is written as:

:math:`(mathrm{lambda }+2mathrm{mu }){frac{partial u(t)}{partial r}

}_{r={R}_{text{int}}}+2mathrm{lambda }{frac{u(t)}{{R}_{mathit{ext}}}

}_{r={R}_{text{int}}}=-{p}_{mathit{imp}}(t)`

Finally, we apply the Sommerfeld condition (non-reflection of waves) on the outer surface \({\mathrm{\Gamma }}_{\mathit{bgt}}\):

(2.7)\[ \ underset {r\ to\ mathrm {\ infty}} {\ mathrm {lim}}}\ left (\ frac {1} {{c} _ {f}}\ frac {\ partial p} {\ partial p} {\ partial p} {\ partial p} {\ partial r}}\ frac {\ partial p} {\ partial r}} (r, t)\ right) =0\]

2.2. Harmonic resolution

We are interested here in soliciting the form \({p}_{\mathit{imp}}(t)={P}_{\mathit{imp}}{e}^{i\mathrm{\omega }t}\) and we are looking for the solution of the form \(u(r,t)=U(r){e}^{i\mathrm{\omega }t}\) and \(p(r,t)=P(r){e}^{i\mathrm{\omega }t}\). In this form, partial time derivatives are trivial. For example:

(2.8)\[ u (r, t) =U (r) {e} ^ {i\ mathrm {\ omega} t}\ to\ frac {\ partial u} {\ partial t} =i\ mathrm {\ omega} U (r) {\ omega} U (r) {e} ^ {e} ^ {e} ^ {e} ^ {e} ^ {e} ^ {i\ mathrm {\ omega} t}\ to\ frac {{\ partial} ^ {2} u} {\ partial t} u} {\ partial t} ^ {2}} =-i {\ mathrm {\ omega}}} ^ {2} U (r) {e} ^ {i\ mathrm {\ omega} t}\]

The problem comes down to finding \(U\) and \(P\) which are solutions to the following problem:

(2.9)\[ {r} ^ {2}\ frac {{d}} ^ {2} U} {{\ mathit {dr}} ^ {2}} +2r\ frac {dU} {\ mathit {dr}}} +\ left (\ left (\ frac {{\ mathrm {\ omega}}} {{\ mathrm {\ omega}}} {\ mathrm {\ omega}}} ^ {2}}} {r} ^ {2}}} +\ left (\ frac {{\ mathrm {\ omega}}} ^ {2}}} {r} ^ {2}}} {r} ^ {2}} {r} ^ {2}}} +\ left (\ frac {{d}}}} +\ left (\ frac {{d}}\ right) U=0\]

And:

(2.10)\[ \ frac {{d} ^ {2} P} {{\ mathit {dr}} {{\ mathit {dr}}}} ^ {2}} +\ frac {\ mathit {dP}} {\ mathit {dr}}} {\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {{\ mathit {dr}}}} +\ frac {\ mathit {dr}}} +\ frac {{\ mathit {dr}\]

The solution to the problem () is written as:

(2.11)\[ U (r) =A (\ mathrm {\ omega}) {j} _ {1}\ left (\ frac {\ mathrm {\ omega}} {{c} _ {s}} r\ right) +B (\ mathrm {\ omega}) {\ mathrm {\ omega}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}) +B (\ mathrm {\ omega}) +B (\ mathrm {\ omega}}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}}) {\ mathrm {\ omega}) +B (\ mathrm {\ omega}}) {\} r\ right)\]

Where \({j}_{1}\) and \({y}_{1}\) are, respectively, the first-order spherical Bessel function of the first type and the first-order spherical Bessel function of the second type.

The solution to the problem () is written as:

(2.12)\[ P (r) =C (\ mathrm {\ omega})\ frac {{e} ^ {-i\ mathrm {\ omega} r/ {c} _ {f}}} {r}} {r} +D (\ mathrm {\ omega})\ frac {\ omega})\ frac {{e} ^ {e} ^ {i\ mathrm {\ omega} r/ {c}} _ {f}}} {f}} {r}} {r}\]

To find the values \(A(\mathrm{\omega })\), \(B(\mathrm{\omega })\), \(C(\mathrm{\omega })\), and \(D(\mathrm{\omega })\). Boundary conditions are used.

In equation (), the term in \(\frac{{e}^{-i\mathrm{\omega }r/{c}_{f}}}{r}\) corresponds to the wave propagating radially to infinity while the other term \(\frac{{e}^{i\mathrm{\omega }r/{c}_{f}}}{r}\) is the reflection of the wave (towards the source). As we impose a non-reflection condition, on the outer edge, this second contribution is zero and we therefore have \(D(\mathrm{\omega })=0\).

2.3. Reference quantities and results

To calculate the solution, you need to have an estimate of the Bessel functions. For that we use scipy, (special functions). The displacement is recovered at two points: Uint on the surface of the structure \({\mathrm{\Gamma }}_{\text{int}}\), Uext on the surface IFS \({\mathrm{\Gamma }}_{\text{ext}}\). The pressure is recorded Pext on the surface IFS \({\mathrm{\Gamma }}_{\text{ext}}\) and Pbgt on the outer surface \({\mathrm{\Gamma }}_{\text{bgt}}\) .

Real part	Imaginary part

2.4. Uncertainties about the solution

None, the solution is analytical.