Eigenmodes

A general perturbation is given by Eq. (31), which is composed of infinit single frequency perturbation of the form $\hat{h} (\omega, \mathbf{r}) e^{- i \omega t}$. It is obvious that a general perturbation can also be considered to be composed of single frequency perturbation of the form

$$\hat{h} (\omega, \mathbf{r}) e^{- i \omega t} + \hat{h} (- \omega, \mathbf{r}) e^{i \omega t},$$ (42)

(it is also obvious that the range of $\omega$ can be limited in $[0, + \infty]$ for this case). Because a physical quanty is always a real function, $h (t, \mathbf{r})$ in the above should be a real function. It is ready to prove that the Fourier transformation of $h (t, \mathbf{r})$ has the following symmetry in frequency domain:

$$\hat{h} (- \omega, \mathbf{r}) = \hat{h}^{\star} (\omega, \mathbf{r}) .$$ (43)

Writing

$$\hat{h} (\omega, \mathbf{r}) = A e^{i \alpha},$$ (44)

where $A$ and $\alpha$ are real numbers, then the expression (42) is written

$$\hat{h} (\omega, \mathbf{r}) e^{- i \omega t} + \hat{h} (- \omega, \mathbf{r}) e^{i \omega t}$$

$$= \hat{h} (\omega, \mathbf{r}) e^{- i \omega t} + \hat{h}^{\star} (\omega, \mathbf{r}) e^{i \omega t}$$

$$= \hat{h} (\omega, \mathbf{r}) e^{- i \omega t} + [\hat{h} (\omega, \mathbf{r}) e^{- i \omega t}]^{\star}$$

$$= 2 A \cos [\alpha (\mathbf{r}) - \omega t]$$ (45)

Using Eq. (45), the Fourier transformation is written

$$h (t, \mathbf{r}) = \int_0^{\infty} A (\omega, \mathbf{r}) \cos [\alpha (\omega, \mathbf{r}) - \omega t] d \omega .$$ (46)

If $\hat{h} (\omega, \mathbf{r})$ satisfies the eigenmode equations (35)-(37), then it is ready to verify that $\hat{h}^{\star} (\omega, \mathbf{r})$ is also a solution to the equations. Since of Eq. (43), $\hat{h} (- \omega, \mathbf{r})$ is also a slution to the equations. Therefore $\hat{h} (\omega, \mathbf{r}) e^{- i \omega t} + \hat{h} (- \omega, \mathbf{r}) e^{i \omega t}$, i.e., $2 A \cos [\alpha (\mathbf{r}) - \omega t]$, is a solution to the linear equations (28), (29), and (30). This tell us how to construct a real (physical) eigenmode from the complex funtion $\hat{h} (\omega, \mathbf{r})$, i.e., the real part of $\hat{h} (\omega, \mathbf{r}) e^{- i \omega t}$ is a physical eigenmode. Note that it is the real part of $\hat{h} (\omega, \mathbf{r}) e^{- i \omega t}$, instead of the real part of $\hat{h} (\omega, \mathbf{r})$, that is a physical eigenmode.

Note that $\omega$ is a real number by the definition of Fourier transformation. However, strictly speaking, the above Fourier transformation should be replaced by Laplace transformation. In this case, $\omega$ is a complex number. In the following, we will assume that we are using Laplace transformation, instead of Fourier transformation. In the following section, we will prove that the eigenvalue $\omega^2$ of the ideal MHD system must be a real number.