= Solution
Use the <Fourier transform> pair $\widehat q(k)=\int q(x)e^{ikx}dx$, $q(x)=(2\pi)^{-1}\int\widehat q(k)e^{-ikx}dk$. The point <force> transforms to $F$. With the <dispersion relation> $D$ defined above, the sheet equation becomes
$$
D(k,\omega)\widehat\eta(k)=F,\qquad
\widehat p_+(k,y)=-\frac{F\rho_0\omega^2e^{-\gamma y}}{\gamma(Tk^2-m\omega^2)-2\rho_0\omega^2}.
$$
The <point-force radiation from a fluid-loaded sheet> is therefore represented exactly by
$$
\rho'(x,y,t)=-\frac{F\rho_0\omega^2}{2\pi c_0^2}e^{i\omega t}
\int_C\frac{e^{-ikx-\gamma y}}{\gamma(Tk^2-m\omega^2)-2\rho_0\omega^2}\,dk.
$$
The causal contour and <outgoing acoustic square-root branch> are fixed first with $\operatorname{Im}\omega<0$, then continued to the desired real frequency. This prescription fixes how poles and the branch points are passed.
For the <acoustic far field> $x=r\cos\theta$, $y=r\sin\theta$, take $0<\theta<\pi$ bounded away from grazing and $k_0r\gg1$. The <method of steepest descent> <saddle point> is $k_s=k_0\cos\theta$, with $\gamma_s=ik_0\sin\theta$. The supplied <saddle point> rule, including its $\sin\theta$ factor, gives, provided the <contour deformation> crosses no poles,
$$
\rho'\sim-\sqrt{\frac{k_0}{2\pi r}}\,
\frac{F\rho_0\omega^2\sin\theta\,e^{i\omega(t-r/c_0)+i\pi/4}}
{c_0^2\left[ik_0\sin\theta\,(Tk_0^2\cos^2\theta-m\omega^2)-2\rho_0\omega^2\right]}.
$$
A convenient simplification, free of division by $T$, is
$$
\boxed{\rho'\sim-\sqrt{\frac{\omega}{2\pi c_0r}}\,
\frac{F\rho_0\sin\theta\,e^{i\omega(t-r/c_0)+i\pi/4}}
{c_0^2\left[ik_0\sin\theta\,(T\cos^2\theta/c_0^2-m)-2\rho_0\right]}.}
$$
Equivalently, when $T\ne0$,
$$
\rho'\sim-\sqrt{\frac{\omega}{2\pi r}}\,
\frac{F\rho_0c_0^{-3/2}\sin\theta\,e^{i\omega(t-r/c_0)+i\pi/4}}
{(\cos^2\theta-mc_0^2/T)i\omega(T/c_0^2)\sin\theta-2\rho_0c_0}.
$$
\b[The expression printed in the PDF is missing sound-speed factors for general dimensional $c_0$.] It agrees with this result if $c_0=1$ in fully normalized units; when $c_0$ is retained as an arbitrary <sound speed>, the numerator needs $c_0^{-3/2}$ and the structural term needs $T/c_0^2$ in the last form. These factors arise respectively from cylindrical spreading, the pressure-density relation, and $k_s=\omega\cos\theta/c_0$.
A direct countercheck is the transparent-sheet limit $m=T=0$. The sheet jump condition then gives $\widehat p_+(k,0)=F/2$, so the <saddle point> rule requires
$$
\boxed{\rho'\sim\frac{F}{2c_0^2}\sqrt{\frac{\omega}{2\pi c_0r}}\sin\theta\,
e^{i\omega(t-r/c_0)+i\pi/4}.}
$$
The printed expression, interpreted continuously after multiplying out its structural factor, instead gives $F(2c_0)^{-1}\sqrt{\omega/(2\pi r)}\sin\theta$ times the same phase. It differs by a factor $c_0^{3/2}$; for example it is eight times too large when $c_0=4$. This limit also verifies the normalization of the corrected density field independently of the sheet's <elastic-sheet tension>.
To decide about poles, track the roots of $\Delta(k,\omega)$ on the chosen square-root sheet and deform the original causal contour to the <steepest descent contour>. A root contributes a residue exactly when it lies in the region swept out by that deformation; its sign is fixed by the contour orientation. Branch cuts must be retained throughout this comparison. Which roots are crossed can depend on observation angle, producing a change of the modal contribution when a pole meets the deformation boundary. A <saddle point> approaching a pole or a grazing endpoint requires an approximation uniform in that limit, rather than the isolated <saddle point> formula above.
The crossed poles are the free fluid-sheet modes of the preceding solution. Real subsonic roots represent <evanescent acoustic surface waves> carrying energy along the sheet, with normal decay; complex continuations represent leaky or radiating modes. Their residues must be added to the <saddle point> sound when the causal contour selects them. The specification “no poles contribute” is therefore a substantive condition on the contour, not permission to ignore zeros of the <dispersion relation>.
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