  We are interested in a time harmonic acoustic problem in a waveguide with
locally perturbed sound hard walls. We consider a setting where an observer
generates incident plane waves at $-\infty$ and probes the resulting scattered
field at $-\infty$ and $+\infty$. Practically, this is equivalent to measure
the reflection and transmission coefficients respectively denoted $R$ and $T$.
In [9], a technique has been proposed to construct waveguides with smooth walls
such that $R=0$ and $|T|=1$ (non reflection). However the approach fails to
ensure $T=1$ (perfect transmission without phase shift). In this work, first we
establish a result explaining this observation. More precisely, we prove that
for wavenumbers smaller than a given bound $k_{\star}$ depending on the
geometry, we cannot have $T=1$ so that the observer can detect the presence of
the defect if he/she is able to measure the phase at $+\infty$. In particular,
if the perturbation is smooth and small (in amplitude and in width),
$k_{\star}$ is very close to the threshold wavenumber. Then, in a second step,
we change the point of view and, for a given wavenumber, working with singular
perturbations of the domain, we show how to obtain $T=1$. In this case, the
scattered field is exponentially decaying both at $-\infty$ and $+\infty$. We
implement numerically the method to provide examples of such undetectable
defects.
