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Chapter 12: Problem 12

A \(50-\mathrm{MHz}\) uniform plane wave is normally incident from air onto the surface of a calm ocean. For seawater, \(\sigma=4 \mathrm{~S} / \mathrm{m}\), and \(\epsilon_{r}^{\prime}=78 .(a)\) Determine the fractions of the incident power that are reflected and transmitted. (b) Qualitatively, how (if at all) will these answers change as the frequency is increased?

Short Answer

Expert verified
Answer: As the frequency of the incident wave increases, the wave impedance of seawater decreases, causing the reflection coefficient to increase slightly. This leads to an increase in the fraction of reflected power and a decrease in the fraction transmitted into the seawater.

Step by step solution

01

Compute the wave impedance of seawater

To find the fractions of the incident power that are reflected and transmitted, we firstly need to compute the wave impedance of seawater. Wave impedance (Z) can be calculated using the formula: $$Z = \sqrt{ \frac{\mu_0}{\epsilon_0 \epsilon_r'} }$$ Since seawater is in a slightly conductive medium, we also need to add a term to account for the effect of the conductivity (σ): $$Z = \sqrt{ \frac{\mu_0}{\epsilon_0 \epsilon_r' + j\sigma\omega} }$$ where µ₀ is the permeability of free space (4π × 10⁻⁷ H/m), ε₀ is the permittivity of free space (8.854 × 10⁻¹² F/m), ε_r' is the relative permittivity of seawater (78), ω is the angular frequency (2π × frequency), and frequency is 50 MHz.
02

Calculate the reflection coefficient

The reflection coefficient (Γ) can be calculated as the ratio of the difference in impedance of air (Z₀) and the impedance of seawater (Z) to the sum of the impedances: $$\Gamma = \frac{Z - Z_0}{Z + Z_0}$$ where Z₀ is the wave impedance of air, which is approximately the impedance of free space (approximately 377 Ω).
03

Determine the fraction of reflected power

To find the fraction of incident power that is reflected, we can now use the formula: $$P_{reflected} = |\Gamma|^2$$
04

Determine the fraction of transmitted power

Since we know the total power is equal to 1, we can now find the fraction of incident power that is transmitted: $$P_{transmitted} = 1 - P_{reflected}$$
05

Discuss the impact of increasing frequency

As the frequency of the incident wave increases, the relationship between σ and ω becomes more significant. As a result, the wave impedance of seawater will decrease, causing the reflection coefficient to increase slightly. This means that the fraction of the reflected power will increase, whereas the fraction transmitted into the seawater will decrease, as the frequency increases.

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Most popular questions from this chapter

A \(10 \mathrm{MHz}\) uniform plane wave having an initial average power density of \(5 \mathrm{~W} / \mathrm{m}^{2}\) is normally incident from free space onto the surface of a lossy material in which \(\epsilon_{2}^{\prime \prime} / \epsilon_{2}^{\prime}=0.05, \epsilon_{r 2}^{\prime}=5\), and \(\mu_{2}=\mu_{0} .\) Calculate the distance into the lossy medium at which the transmitted wave power density is down by \(10 \mathrm{~dB}\) from the initial \(5 \mathrm{~W} / \mathrm{m}^{2}\).

Suppose that \(\phi\) in Figure \(12.17\) is Brewster's angle, and that \(\theta_{1}\) is the critical angle. Find \(n_{0}\) in terms of \(n_{1}\) and \(n_{2}\).

A uniform plane wave in region 1 is normally incident on the planar boundary separating regions 1 and 2. If \(\epsilon_{1}^{\prime \prime}=\epsilon_{2}^{\prime \prime}=0\), while \(\epsilon_{r 1}^{\prime}=\mu_{r 1}^{3}\) and \(\epsilon_{r 2}^{\prime}=\mu_{r 2}^{3}\), find the ratio \(\epsilon_{r 2}^{\prime} / \epsilon_{r 1}^{\prime}\) if \(20 \%\) of the energy in the incident wave is reflected at the boundary. There are two possible answers.

Over a small wavelength range, the refractive index of a certain material varies approximately linearly with wavelength as \(n(\lambda) \doteq n_{a}+n_{b}\left(\lambda-\lambda_{a}\right)\), where \(n_{a}, n_{b}\) and \(\lambda_{a}\) are constants, and where \(\lambda\) is the free-space wavelength. (a) Show that \(d / d \omega=-\left(2 \pi c / \omega^{2}\right) d / d \lambda\). (b) Using \(\beta(\lambda)=2 \pi n / \lambda\), determine the wavelength- dependent (or independent) group delay over a unit distance. ( \(c\) ) Determine \(\beta_{2}\) from your result of part \((b) .(d)\) Discuss the implications of these results, if any, on pulse broadening.

You are given four slabs of lossless dielectric, all with the same intrinsic impedance, \(\eta\), known to be different from that of free space. The thickness of each slab is \(\lambda / 4\), where \(\lambda\) is the wavelength as measured in the slab material. The slabs are to be positioned parallel to one another, and the combination lies in the path of a uniform plane wave, normally incident. The slabs are to be arranged such that the airspaces between them are either zero, one-quarter wavelength, or one-half wavelength in thickness. Specify an arrangement of slabs and airspaces such that \((a)\) the wave is totally transmitted through the stack, and \((b)\) the stack presents the highest reflectivity to the incident wave. Several answers may exist.
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