Hamiltonian of Acoustic Turbulence

Let us expand the Hamiltonian (2.11) (expressed in terms of $b,\ b^*$) in power series:

$${\cal H}= {\cal H}_0 +{\cal H}_{\rm int}\ .$$ (39)

Here ${\cal H}_0$ is quadratic in $b$ and $b^*$, giving the Hamiltonian of non-interacting waves:

$${\cal H}_0=\int \, c \, k \, b ({{\bf k}}) b^*({{\bf k}}) d{\bf k},$$ (40)

with linear dispersion relation $\omega_0({\bf k}) = c k$. In the Hamiltonian of interaction ${\cal H}_{\rm int}$ we take into account only three-wave processes:

$${\cal H}_{\rm int} = {1\over2}\int ( V({\bf k},{\bf k}_1,{\bf k}_2) b^*_1b_2b_3+{\rm c.c.})$$

$$\times\delta({\bf k}_1 -{\bf k}_2- {\bf k}_3)\,d{\bf k}_1 d{\bf k}_2d {\bf k}_3 \ .$$ (41)

We neglected here $0\leftrightarrow3$ processes (processes described by $b_1^*b_2^*b_3^*$ and $b_1b_2b_3$ terms), because they are nonresonant. It means, that if we take into account $0\leftrightarrow3$ term, it is not going to change our final results, thus we can neglect it from the very beginning. We also neglected contributions from 4-wave and higher terms, because three-wave interaction is the dominant one.

The coupling coefficient of the 3-wave interaction a given by [1]

$$V({\bf k}_0,{\bf k}_1,{\bf k}_2)$$ (42)

$$= \sqrt {c k k_1 k_2\over 4\pi^3\rho_0} (3g+\cos\theta_{01}+\cos\theta_{02}+\cos\theta_{12})\, ,$$

where $g$ is some dimensionless constant of the order of unity and $\theta _{ij}$ is the angle between ${\bf k}_i$ and ${\bf k}_j$. Since we have almost linear dispersion relation, only almost parallel wavevectors can interact, therefore $\cos \theta_{ij}$ with the high accuracy can be replaced by $1$ and (2.20) reduces to

$$V({\bf k}_0,{\bf k}_1,{\bf k}_2) =\sqrt {c k k_1 k_2\over 4\pi^3\rho_0} 3(g+1)\, , \nonumber$$