Straightforward Approach

Consider the Euler equations for a compressible fluid:

$$\partial \rho/\partial t+{\bbox\nabla}\cdot\,(\rho{\bf v})=0\,,$$

$$\partial{\bf v}/\partial t+({\bf v}\cdot {\bbox{\nabla}}){\bf v}= -{\bbox\nabla} p(\rho)/\rho\ .$$ (23)

Here $v({\bf x},t)$ is the Euler fluid velocity, $\rho({\bf x},t)$ the density, and $p({\bf r},t)$ is the pressure which, in the general case, is a function of fluid density and specific entropy $s$ [$p=p(\rho,s)$]. In ideal fluids where there is no viscosity and heat exchange, the entropy per unit volume is carried by the fluid, i.e. obeys the equation $\partial s/\partial t+({\bf v}\cdot{\bbox\nabla})s=0$. A fluid in which the specific entropy is constant throughout the volume is called barotropic; the pressure in such fluid is a single-valued function of density $p=p(\rho)$. In this case, ${\bbox\nabla} p/\rho$ may be expressed via the gradient of specific enthalpy of unit mass $w=E+p V$ and $d w=V d p=d p/\rho$. Thus, ${\bbox\nabla} p/\rho={\bbox\nabla} w$.

Writing the fluid density $\rho({\bf x},t)$ as $\rho_0(1+\eta({\bf x},t))$, the velocity field as $v({\bf x},t)$, the pressure field as $p=p_0(1+\eta)^\mu$ and the enthalpy as

$$w=\int \frac{d p}{\rho} = \frac{c_0^2}{\mu-1}(1+(\mu-1)\eta + \frac{(\mu-1)(\mu-2)}{2}\eta^2 + \dots )$$

one can write (2.1) to third order in amplitude in the following form

$$\frac{\partial \eta }{\partial t} + \frac{\partial v_i}{\partial x_i} = - \frac{\partial}{\partial x_i} \eta v_i,$$ (24)

$$\frac{\partial v_j}{\partial t} + c^2 \frac{\partial \eta}{\partial x} = - v_m \frac{\partial v_i}{\partial x_m} - \frac{c^2(\mu-2)}{2} \frac{\partial}{\partial x_j}\eta^2$$

$$- \frac{c^2(\mu-2)(\mu-3)}{6} \frac{\partial }{ \partial x_j}\eta^3.$$ (25)

Let us introduce new variables as

$$\eta({\bf x},t) = \int \sum_s \epsilon a^s({\bf k},t)e^{i {\bf k} {\bf x} + i s \omega({\bf k} ) t } d {\bf k}\, ,$$ (26)

$$v_j({\bf x},t) = \int \sum_s \frac{- c^2 k_j}{s \omega({\bf k} ) } \epsilon a^s({\bf k},t) e^{i {\bf k} {\bf x} + i s \omega({\bf k} ) t } d {\bf k}\,,$$ (27)

where $0< \epsilon \ll 1$, $\omega(\vec k ) = c \vert{\bf k} \vert$ and $\sum_s$ connotes summation over $s=\pm 1$. From (2.2) and (2.3),

$$\frac{\partial a^s({\bf k}, t)}{\partial t} = \epsilon \sum_{s_p ... ...,s_q}_{{\bf k},{\bf k}_p,{\bf k}_q} a^{s_p} ({\bf k}_p, t) a^{s_q}({\bf k}_q,t)$$

$$\times \delta({\bf k}_p+{\bf k}_q - {\bf k} ) \exp\{i[ s_p\omega({\bf k}_p)+s_q\omega({\bf k}_q)- s \omega({\bf k} )]t\}$$

$$+ \epsilon ^2 \sum _{s_p ,s_q, s_r} \int d {\bf k}_p d {\bf k}_q d ... ...^{s,s_p,s_q,s_r}_{{\bf k},{\bf k}_p,{\bf k}_q, {\bf k}_r} a^{s_p} ({\bf k}_p,t)$$

$$\times a^{s_q}({\bf k}_q,t)a^{s_r}({\bf k}_r,t) \delta({\bf k}_p + {\bf k}_q + {\bf k}_r - {\bf k})$$

$$\times \exp\{i[s_p\omega({\bf k}_p)+s_q\omega({\bf k}_q)+s_r\omega({\bf k}_r)- s \omega({\bf k})]t\}$$ (28)

where the summation is done over all signs of $s_p,\ s_q,\ s_r$ and we used the shorthand notation $\omega_p=\omega({\bf k}_p)$. The coupling coefficients are,

$$L^{s,s_p,s_q}_{{\bf k},{\bf k}_p,{\bf k}_q} = \frac{i c^2}{4} \Big(\frac{{\bf k} {\bf k}_p}{s_p\omega_p}+\frac{... ..._q\omega_q} + \frac{s\omega}{s_p\omega_p s_q \omega_q}{\bf k}_p {\bf k}_q \Big)$$

$$+\frac{i}{4}(\mu-2)s\omega$$ (29)

$$L^{s,s_p,s_q,s_r}_{{\bf k},{\bf k}_p,{\bf k}_q,{\bf k}_r} = \frac{i\omega}{12}(\mu-2)(\mu-3) \ .$$ (30)

These coefficients have the following important properties:

• $L^{s,s_p,s_q}_{{\bf k},{\bf k}_p,{\bf k}_q}$ is symmetric under the interchange of $p$ and $q$
• $L^{s_p,s,-s_q}_{{\bf k}_p,{\bf k},-{\bf k}_q}= (s_p \omega_p /s\omega) L^{s,s_p,s_q}_{{\bf k},{\bf k}_p,{\bf k}_q}$
• On the resonant manifold $M$, given by
  

  $$\frac{1}{c}h = s_p \vert{\bf k}_p\vert + s_q\vert{\bf k} - {\bf k}_p\vert- s \vert{\bf k}\vert=0\,,$$ (31)

  $$L^{s,s_p,s_q}_{{\bf k},{\bf k}_p,{\bf k}_q} = \frac{i c s}{4}(\mu+1)K\,,$$ (32)

  
  

where $\vert{\bf k}\vert=K$. Note that if ${\bf k} = (K,0,0)$, the resonant manifold is not of codimension one but degenerates to $K_y=K_z=0$, where ${\bf k}_p = (K_x,K_y,K_z)$, ${\bf k}_q=(K-K_x,-K_y,-K_z)$. There are three cases.

1. For $K_x<0<K$, $\vert{\bf k}_p\vert= - K_x$, $\vert{\bf k}_q\vert= K + K_x$, $s_p=-s$, $s_q= s$.
2. For $0<K_x<K$, $\vert{\bf k}_p\vert=K_x$, $\vert{\bf k}_q\vert=K-K_x$ $s_p=s_q=s$.
3. For $0<K<K_x$, $\vert{\bf k}_p\vert=K_x$, $\vert{\bf k}_q\vert=K_x-K$, $s_p=s,s_q=-s$.