9.33P The "inversion theorem" for Four... [FREE SOLUTION]

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Introduction to Electrodynamics

David J. Griffiths

Physics

4th edition Edition 路 613 pages

ISBN: 9780321856562

Chapter 9: 9.33P (page 432)

The "inversion theorem" for Fourier transforms states that
$蠒 (Z) = {鈭�}_{- 鈭�}^{鈭�} 蠒 (k) e^{ikz} dk 鈬� 蠒 (k) = \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} 蠒 (z) e^{- ikz} dz$
Use this to determine $A (k)$ , in Eq. 9.20, in terms of $f (z, 0)$ and $\overset{*}{f} (z, 0)$

Short Answer

Expert verified

The expression for $A (k)$ is $\frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} [f (z, 0) + \frac{i}{蝇} \overset{g}{f} (z, 0)] e^{- ikz} dz$

Step by step solution

Expression for the linear combination of sinusoidal wave:

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

$\overset{0}{f} (z, t) = {鈭�}_{- 鈭�}^{鈭�} \overset{0}{A} (k) e^{i (kz - 蝇t)} dk$ .............(1)

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

Determine A0(k) in term of data-custom-editor="chemistry" f (z,0) and data-custom-editor="chemistry" f *(z,0)

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

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

Write the conjugate of the above expression.

$\overset{0}{f} {(z, 0)}^{*} = {鈭�}_{- 鈭�}^{鈭�} \overset{0}{A} {(- k)}^{*} e^{- ikz} dk \overset{0}{f} {(z, 0)}^{*} = {鈭�}_{- 鈭�}^{鈭�} \overset{0}{A} {(k)}^{*} e^{- ikz} (- dk)$

It is known that,

$f (z, 0) = Re [\overset{0}{f} (z, 0)] f (z, 0) = \frac{1}{2} [\overset{0}{f} (z, 0) + \overset{0}{f} {(z, 0)}^{*}] . . . . (2)$

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

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

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

$\frac{1}{2} [\overset{0}{A} (k) + \overset{0}{A} {(- k)}^{*}] = \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} f (z, 0) e^{- ikz} dk$ ......(3)

Solve the conjugate for $\overset{0}{f} (z, 0)$ .

$\overset{0}{f} (z, t) = {鈭�}_{- 鈭�}^{鈭�} \overset{0}{A} (k) (- i蝇) e^{i (kz - 蝇t)} dk \overset{0}{f} (z, t) = {鈭�}_{- 鈭�}^{鈭�} [- i蝇 \overset{0}{A} (k)] e^{ikz} dk \overset{0}{f} (z, t) = {鈭�}_{- 鈭�}^{鈭�} [i蝇 \overset{0}{A} (k)] e^{ikz} (- dk)$

On further solving, the above equation becomes,

$\overset{g}{f} {(z, 0)}^{*} = {鈭�}_{- 鈭�}^{鈭�} [i蝇 \overset{g}{A} {(- k)}^{*}] e^{- ikz} (dk) \overset{g}{f} (z, 0) = Re [\overset{g}{f} (z, 0)] \overset{g}{f} (z, 0) = \frac{1}{2} [\overset{g}{f} (z, 0) + \overset{g}{f} {(z, 0)}^{*}] . . . . . . . (4)$

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

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

The inversion theorem for Fourier transformation states that,

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

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91影视

The "inversion theorem" for Fourier transforms states that
$蠒 (Z) = {鈭�}_{- 鈭�}^{鈭�} 蠒 (k) e^{ikz} dk 鈬� 蠒 (k) = \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} 蠒 (z) e^{- ikz} dz$
Use this to determine $A (k)$ , in Eq. 9.20, in terms of $f (z, 0)$ and $\overset{*}{f} (z, 0)$

Short Answer

Step by step solution

Expression for the linear combination of sinusoidal wave:

Determine A0(k) in term of data-custom-editor="chemistry" f (z,0) and data-custom-editor="chemistry" f *(z,0)

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

The "inversion theorem" for Fourier transforms states that蠒(Z)=鈭�-鈭�鈭�蠒(k)eikzdk鈬�蠒(k)=12蟺鈭�-鈭�鈭�蠒(z)e-ikzdzUse this to determine A(k), in Eq. 9.20, in terms of f(z,0)andf*(z,0)

Short Answer

Step by step solution

Expression for the linear combination of sinusoidal wave:

Determine A0(k) in term of data-custom-editor="chemistry" f (z,0) and data-custom-editor="chemistry" f *(z,0)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Famous Physicists

Astrophysics

Fields in Physics

Electric Charge Field and Potential

Electrostatics

Study anywhere. Anytime. Across all devices.

The "inversion theorem" for Fourier transforms states that
$蠒 (Z) = {鈭�}_{- 鈭�}^{鈭�} 蠒 (k) e^{ikz} dk 鈬� 蠒 (k) = \frac{1}{2 蟺} {鈭�}_{- 鈭�}^{鈭�} 蠒 (z) e^{- ikz} dz$
Use this to determine $A (k)$ , in Eq. 9.20, in terms of $f (z, 0)$ and $\overset{*}{f} (z, 0)$