  We are interested in the lower part of the spectrum of the Dirichlet
Laplacian $A^\varepsilon$ in a thin waveguide $\Pi^\varepsilon$ obtained by
repeating periodically a pattern, itself constructed by scaling an inner field
geometry $\Omega$ by a small factor $\varepsilon>0$. The Floquet-Bloch theory
ensures that the spectrum of $A^\varepsilon$ has a band-gap structure. Due to
the Dirichlet boundary conditions, these bands all move to $+\infty$ as
$O(\varepsilon^{-2})$ when $\varepsilon\to0^+$. Concerning their widths,
applying techniques of dimension reduction, we show that the results depend on
the dimension of the so-called space of almost standing waves in $\Omega$ that
we denote by $\mathrm{X}_\dagger$. Generically, i.e. for most $\Omega$, there
holds $\mathrm{X}_\dagger=\{0\}$ and the lower part of the spectrum of
$A^\varepsilon$ is very sparse, made of bands of length at most
$O(\varepsilon)$ as $\varepsilon\to0^+$. For certain $\Omega$ however, we have
$\mathrm{dim}\,\mathrm{X}_\dagger=1$ and then there are bands of length $O(1)$
which allow for wave propagation in $\Pi^\varepsilon$. The main originality of
this work lies in the study of the behaviour of the spectral bands when
perturbing $\Omega$ around a particular $\Omega_\star$ where
$\mathrm{dim}\,\mathrm{X}_\dagger=1$. We show a breathing phenomenon for the
spectrum of $A^\varepsilon$: when inflating $\Omega$ around $\Omega_\star$, the
spectral bands rapidly expand before shrinking. In the process, a band dives
below the normalized threshold $\pi^2/\varepsilon^2$, stops breathing and
becomes extremely short as $\Omega$ continues to inflate.
