Chapter 9: 9.33P (page 432)

The "inversion theorem" for Fourier transforms states that
$ϕ (Z) = \int_{- \infty}^{\infty} ϕ (k) e^{ikz} dk \Leftrightarrow ϕ (k) = \frac{1}{2 π} \int_{- \infty}^{\infty} ϕ (z) e^{- ikz} dz$
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 $\frac{1}{2 π} \int_{- \infty}^{\infty} [f (z, 0) + \frac{i}{ω} \overset{g}{f} (z, 0)] e^{- ikz} dz$

Step by step solution

Expression for the linear combination of sinusoidal wave:

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

$\overset{0}{f} (z, t) = \int_{- \infty}^{\infty} \overset{0}{A} (k) e^{i (kz - ωt)} dk$ .............(1)

Here, $k$ is the propogation vector and $ω$ 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).

$\overset{0}{f} (z, 0) = \int_{- \infty}^{\infty} \overset{0}{A} (k) e^{i (kz - ω (0))} dk = \int_{- \infty}^{\infty} \overset{0}{A} (k) e^{ikz} dk$

Write the conjugate of the above expression.

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

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)

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

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

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

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

$\overset{0}{f} (z, t) = \int_{- \infty}^{\infty} \overset{0}{A} (k) (- iω) e^{i (kz - ωt)} dk \overset{0}{f} (z, t) = \int_{- \infty}^{\infty} [- iω \overset{0}{A} (k)] e^{ikz} dk \overset{0}{f} (z, t) = \int_{- \infty}^{\infty} [iω \overset{0}{A} (k)] e^{ikz} (- dk)$

On further solving, the above equation becomes,

$\overset{g}{f} {(z, 0)}^{*} = \int_{- \infty}^{\infty} [iω \overset{g}{A} {(- k)}^{*}] e^{- ikz} (dk) \overset{g}{f} (z, 0) = Re [\overset{g}{f} (z, 0)] \overset{g}{f} (z, 0) = \frac{1}{2} [\overset{g}{f} (z, 0) + \overset{g}{f} {(z, 0)}^{*}] . . . . . . . (4)$

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

$\overset{g}{f} (z, 0) = \int_{- \infty}^{\infty} \frac{1}{2} [- iω \overset{0}{A} (k) + iω \overset{0}{A} (- k)] e^{ikz} dk \overset{g}{f} (z, 0) = \frac{- iω}{2} [\overset{0}{A} (k) - \overset{0}{A} {(- k)}^{*}]$

The inversion theorem for Fourier transformation states that,

$\overset{g}{f} (z, 0) = \frac{1}{2 π} \int_{- \infty}^{\infty} \overset{g}{f} (z, 0) e^{ikz} dz$

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The "inversion theorem" for Fourier transforms states that
$ϕ (Z) = \int_{- \infty}^{\infty} ϕ (k) e^{ikz} dk \Leftrightarrow ϕ (k) = \frac{1}{2 π} \int_{- \infty}^{\infty} ϕ (z) e^{- ikz} dz$
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.

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The "inversion theorem" for Fourier transforms states thatϕ(Z)=∫-∞∞ϕ(k)eikzdk⇔ϕ(k)=12π∫-∞∞ϕ(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

Electricity

Atoms and Radioactivity

Geometrical and Physical Optics

Electricity and Magnetism

Nuclear Physics

Space Physics

Study anywhere. Anytime. Across all devices.

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