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The following notation is used in a paper regarding potential theory of water waves. There are two potentials, incident wave $\phi_I$ and diffraction $\phi_D$. They must satisfy the free surface condition $$\left[\left(i \omega+U\frac{\partial}{\partial x}\right)^2+g\frac{\partial}{\partial z}\right](\phi_I, \phi_D) = 0$$ I guess that this is a short-hand notation for employing the operation in the brackets on both potentials. Furthermore it is written... $$F_j = \rho \iint_S n_j\left(i\omega + U\frac{\partial}{\partial x}\right)(\phi_I+\phi_D)\mathop{ds}$$ and even $$G_j = \rho \iint_S n_j \left(i \omega+U\frac{\partial}{\partial x}\right)\sum_{k=1}^6 \phi_k\eta_k\mathop{ds}$$ Never mind the parameters and what they represent. I just wish to know if I understand the notation. I think it is strange to put a differential-operator in parantheses, like this. For example, for the middle term. Can this be written ? $$F_j = \rho \iint_S n_j \left(i \omega(\phi_I + \phi_D) + U \frac{\partial}{\partial x}(\phi_I + \phi_D)\right)\mathop{ds}$$ Further, can I write the first equation like this ? $$\left(i \omega \phi_j + U \frac{\partial \phi_j}{\partial x}\right)^2+g \frac{\partial \phi_j}{\partial z}, \text{ where }\ \ j=\{I,D\}$$

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It's a handy abbreviation. But your last rewrite is wrong: $$\left(i \omega+U\frac{\partial}{\partial x}\right)^2$$ is the operator in the parentheses applied twice, so $$\left(i \omega+U\frac{\partial}{\partial x}\right)^2\phi =i \omega\psi+U\frac{\partial\psi}{\partial x} \quad\text{where } \psi=i \omega\phi+U\frac{\partial\phi}{\partial x}. $$

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