  We use multiple-scale analysis to study a 5-wave system (5WS) composed of two
different internal gravity wave triads. Each of these triads consists of a
parent wave and two daughter waves, with one daughter wave common between the
two triads. The parent waves are assumed to have the same frequency and
wavevector norm co-existing in a region of constant background stratification.
Such 5-wave systems may emerge in oceans, for example, via tide-topography
interactions, generating multiple parent internal waves that overlap. Two 2D
cases are considered: Case 1(2) has parent waves with the same horizontal
(vertical) wavenumber but with different vertical (horizontal) wavenumber. For
both cases, the 5WS is more unstable than triads for $f/\omega_1\gtrapprox0.3$,
where $\omega_1$ and $f$ are the parent wave and the local Coriolis frequency,
respectively. For $f/\omega_1\gtrapprox0.3$, the common daughter wave's
frequency is $\approx \omega_1-f $ and $f$ respectively for Cases 1 and 2. For
3D cases, 5WSs become more unstable as the angle ($\theta$) between the
horizontal wavevectors of the parent waves is decreased. Moreover, for any
$\theta$, 5WSs have higher growth rates than triads for
$f/\omega_1\gtrapprox0.3$. Numerical simulations match the theoretical growth
rates of 5WSs for a wide range of latitudes, except when $f/\omega_1\approx0.5$
(critical latitude). More than three daughter waves are forced by the two
parent waves when $f/\omega_1\approx0.5$. We formulate a reduced order model
which shows that for any $\theta$, the maximum growth rate near the critical
latitude is approximately twice the maximum growth rate of all triads.
