I'm trying to solve the equation

$$ frac {d ^ 2u} {dt ^ 2} – frac {d ^ 2} {dx ^ 2} left (c_s ^ 2u + nu frac {of} {dt} right) = 0 $ $

with initial conditions

$$ u (x, 0) = 0 $$

$$ frac {du} {dt} | _ {t = 0} = 0 $$

and boundary conditions

$$ frac {d ^ 2u} {dt ^ 2} – frac {d ^ 2} {dx ^ 2} left (c_s ^ 2u + nu frac {of} {dt} right) = 0 $ $

My code is

```
ClearAll((ScriptC), (Nu), (Rho), (ScriptL), (ScriptN), (Kappa), (Omega), (ScriptCapitalV), A, u, v);
(ScriptC) = 1489;
(Nu) = 1.004;
(Rho) = 998.21;
(ScriptL) = 2*10^-2;
(ScriptN) = 8;
(Kappa) = 2*Pi*(ScriptN)/(ScriptL);
(Omega) = (ScriptC)*(Kappa);
(ScriptCapitalV) = (Omega)*10^-8;
A = ((Rho)*(Omega)*(ScriptCapitalV)/((ScriptC)^4+(Nu)^4*(Omega)^2));
(Tau) = 10^-9;
f(t_) := (1-Exp(-t/(Tau)));
(*f(t_) := If(t < 10^-9, 0, 1);*)
pde = {D(v(t, x), t) - ((ScriptC)^2)*D(u(t, x), x, x) - (Nu)*D(v(t, x), x, x) == 0,
v(t, x) == D(u(t, x), t)};
ics = {u(0, x) == 0, v(0, x) == 0};
bcs = {(D(u(t, x), x) /. x -> 0) == A*(((ScriptC)^2)*Sin((Omega)*t) - ((Nu)*(Omega))*Cos((Omega)*t)) f(t),
(D(u(t, x), x) /. x -> 1) == A*(((ScriptC)^2)*Sin((Omega)*t) - ((Nu)*(Omega))*Cos((Omega)*t)) f(t)};
bcs1 = {(D(v(t, x), x) /. x -> 0) == (Omega)*A*(((ScriptC)^2)*Cos((Omega)*t) + ((Nu)*(Omega))*Sin((Omega)*t)) f(t),
(D(v(t, x), x) /. x -> 1) == (Omega)*A*(((ScriptC)^2)*Cos((Omega)*t) + ((Nu)*(Omega))*Sin((Omega)*t)) f(t)};
{U, V} = NDSolveValue({pde, ics, bcs, bcs1}, {u, v}, {x, 0, (ScriptL)}, {t, 0,1},
Method -> {"IndexReduction" -> Automatic, "EquationSimplification" -> "Residual", "PDEDiscretization" -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 500, "MaxPoints" -> 500,"DifferenceOrder" -> 2}}});
{DensityPlot(U(t, x), {x, 0, 1}, {t, 0, 10}, ColorFunction -> "Rainbow", PlotLegends -> Automatic, FrameLabel -> Automatic, PlotLabel -> "u"),
DensityPlot(V(t, x), {x, 0, 1}, {t, 0, 10}, ColorFunction -> "Rainbow", PlotLegends -> Automatic, FrameLabel -> Automatic, PlotLabel -> "v")}
Manipulate(Plot(U(t, x), {x, 0, 1}), {t, 0, 10})
```

where I had to multiply at boundary conditions a sigmoid function, otherwise these are inconsistent with the initial conditions. A step function does not work well because it makes the system rigid (or singular).

With the code above, I always get the following errors and I do not know how to get rid of it:

NDSolveValue :: nderr: failure of the error test at t == 0.`; unable to continue.

InterpolatingFunction :: dmval: The input value {0.000714286.0.0000714286} is outside the data range of the interpolation function. The extrapolation will be used.

**Note**: This is the continuation of my previous question Solving an attenuated wave equation. I am now trying to use parameter values with physical meaning.