91影视

Write the expression for the linear combination of a sinusoidal wave.

$\stackrel{\mathbf{~}}{\mathbf{f}}\mathbf{(}\mathbf{z}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{鈭�}}\stackrel{\mathbf{~}}{\mathbf{A}}\mathbf{\left(}\mathbf{k}\mathbf{\right)}{\mathbf{e}}^{\mathbf{i}\mathbf{(}\mathbf{kz}\mathbf{-}\mathbf{蝇}\mathbf{t}\mathbf{)}}\mathbf{dk}$ 鈥︹�� (1)

Here, $k$is the propagation vector and $蝇$is the angular frequency.

Substitute $t=0$in equation (1).

$\tilde{f}(z,0)={鈭�}_{-鈭�}^{鈭�}\tilde{A}\left(k\right)e^{i(kz-蝇(0\left)\right)}dk$

$={鈭�}_{-鈭�}^{鈭�}\tilde{A}\left(k\right)e^{ikz}dk$

Write the conjugate of the above expression.

$$\begin{gathered} \tilde{f}(z,0{)}^{鈭�}={鈭�}_{-鈭�}^{鈭�}\tilde{A}(-k{)}^{鈭�}{e}^{-ikz}dk \\ \tilde{f}(z,0{)}^{鈭�}={鈭�}_{-鈭�}^{鈭�}\tilde{A}(k{)}^{鈭�}{e}^{-ikz}(-dk) \end{gathered}$$

It is known that,

$f(z,0)=Re\left[\tilde{f}\right(z,0\left)\right]$

$f(z,0)=\frac{1}{2}\left[\tilde{f}(z,0)+\tilde{f}(z,0{)}^{*}\right]$ 鈥︹�� (2)

Substitute $\frac{1}{2}\tilde{A}\left(k\right)e^{ikz}dk\text{for}\tilde{f}(z,0)\text{and}\frac{1}{2}\tilde{A}(-k{)}^{*}{e}^{ikz}dk\text{for}\tilde{f}(z,0{)}^{*}$in equation

(2).

$f(z,0)={鈭�}_{-鈭�}^{鈭�}\frac{1}{2}\left[\tilde{A}\left(k\right)+\tilde{A}(-k{)}^{*}\right]e^{ikz}dk$

Substitute $f(z,0)={鈭�}_{-鈭�}^{鈭�}\frac{1}{2}\left[\tilde{A}\left(k\right)+\tilde{A}(-k{)}^{*}\right]e^{ikz}dk$in equation (2).

$\frac{1}{2}\left[\tilde{A}\left(k\right)+\tilde{A}(-k{)}^{*}\right]=\frac{1}{2蟺}{鈭�}_{-鈭�}^{鈭�}f(z,0)e^{-ikz}dz$鈥︹�� (3)

Solve the conjugate of $f虄(z,0)$

$$\begin{gathered} \tilde{f}(z,t)={鈭�}_{-鈭�}^{鈭�}\tilde{A}\left(k\right)(-i蝇)e^{i(kz-蝇t)}dk \\ \tilde{f}(z,t)={鈭�}_{-鈭�}^{鈭�}[-i蝇\tilde{A}(k\left)\right]e^{ikz}dk \\ \tilde{f}(z,0{)}^{*}={鈭�}_{-鈭�}^{鈭�}\left[-i蝇\tilde{A}(k{)}^{*}\right]e^{-ikz}dk \\ \tilde{f}(z,0{)}^{*}={鈭�}_{-鈭�}^{鈭�}\left[i蝇\tilde{A}(k{)}^{*}\right]e^{-ikz}(-dk) \end{gathered}$$

On further solving, the above equation becomes,

$$\begin{gathered} \tilde{f}(z,0{)}^{*}={鈭�}_{-鈭�}^{鈭�}\left[i蝇\tilde{A}(-k{)}^{*}\right]e^{-ikz}(dk\left) \\ f\right(z,0)=Re[\tilde{f}(z,0)] \end{gathered}$$

$f(z,0)=\frac{1}{2}\left[\tilde{f}(z,0)+\tilde{f}(z,0{)}^{*}\right]$ 鈥︹�� (4)

Substitute$\frac{1}{2}-i蝇\tilde{A}\left(k\right)e^{ikz}dk\text{for}\tilde{f}(z,0)\text{and}\frac{1}{2}-i蝇\tilde{A}\left(k{)}^{*}{e}^{ikz}dk\text{for}\tilde{f}\right(z,0{)}^{*}$for and for in equation (4).

$$\begin{gathered} f(z,0)={鈭�}_{-鈭�}^{鈭�}\frac{1}{2}[-i蝇\tilde{A}(k)+i蝇\tilde{A}(-k\left)\right]e^{ikz}dk \\ f(z,0)=\frac{-i蝇}{2}\left[\tilde{A}\left(k\right)-\tilde{A}(-k{)}^{*}\right] \end{gathered}$$

The inversion theorem for Fourier transformation states that,

$f(z,0)=\frac{1}{2蟺}{鈭�}_{-鈭�}^{鈭�}f(z,0)e^{-ikz}dz$

Equate both the values of $f(z,0)$

$\frac{1}{2}\left[\tilde{A}\left(k\right)-\tilde{A}(-k{)}^{*}\right]=\frac{1}{2蟺}{鈭�}_{-鈭�}^{鈭�}\left[\frac{i}{蝇}f(z,0)\right]e^{-ikz}dz$ 鈥︹�� (5)

Add equations (3) and (5).

$\frac{1}{2}\left[\tilde{A}\left(k\right)+\tilde{A}(-k{)}^{*}\right]+\frac{1}{2}\left[\tilde{A}\left(k\right)-\tilde{A}(-k{)}^{*}\right]=\frac{1}{2蟺}{鈭�}_{-鈭�}^{鈭�}f(z,0)e^{-ikz}dz+\frac{1}{2蟺}{鈭�}_{-鈭�}^{鈭�}\left[\frac{i}{蝇}f(z,0)\right]e^{-ikz}dz$

role="math" localid="1657466135705" $\tilde{A}\left(k\right)=\frac{1}{2蟺}{鈭�}_{-鈭�}^{鈭�}\left[f(z,0)+\frac{i}{蝇}f(z,0)\right]e^{-ikz}dz$

Therefore, the required expression is$\tilde{A}\left(k\right)=\frac{1}{2蟺}{鈭�}_{-鈭�}^{鈭�}\left[f(z,0)+\frac{i}{蝇}f(z,0)\right]e^{-ikz}dz$