Introductory Circuit Analysis (13th Edition)
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ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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![### Wave Analysis
Consider the wave equation given by:
\[ \vec{E}(x, t) = -\hat{y} 9.25 \cos \left(10^5 \pi t + \frac{\pi}{18} x + \frac{\pi}{3} \right) \]
Let's extract some key properties of this wave:
- **The Phase Velocity:** The phase velocity \( v_p \) of the wave.
- **The Period of the Wave:** The period \( T \) of the wave.
- **The Wavelength:** The distance \( \lambda \) between successive peaks of the wave.
- **The Direction of Propagation:** The direction in which the wave is moving.
- **The Attenuation Number:** Any attenuation factor, if present, that signifies how the wave diminishes in amplitude over distance or time.
### Analysis
1. **The Phase Velocity ( \( v_p \) ):**
Phase velocity \( v_p \) is given by \( v_p = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wave number.
2. **The Period of the Wave ( \( T \) ):**
The period can be derived from the angular frequency \( \omega \) using \( T = \frac{2\pi}{\omega} \).
3. **The Wavelength ( \( \lambda \) ):**
The wavelength can be found using the wave number \( k \) with the relation \( \lambda = \frac{2\pi}{k} \).
4. **The Direction of Propagation:**
The wave propagates in a direction determined by the sign and coefficients of \( x \) and \( t \) in the cosine argument.
5. **The Attenuation Number:**
The wave equation presents no exponential decay factors, so there is no attenuation implied in the provided equation.
### Detailed Derivation
1. **Finding \( k \) and \( \omega \):**
From the given wave equation:
\[ \vec{E}(x, t) = -\hat{y} 9.25 \cos \left(10^5 \pi t + \frac{\pi}{18} x + \frac{\pi}{3} \right) \]
- \( \omega = 10^5 \pi \)
-](https://content.bartleby.com/qna-images/question/d5c9c582-d472-47d1-a3f0-643357e199d8/ff5b92ab-413a-4219-96bd-9f2ba2f1347b/5p1f6ai_processed.jpeg)
Transcribed Image Text:### Wave Analysis
Consider the wave equation given by:
\[ \vec{E}(x, t) = -\hat{y} 9.25 \cos \left(10^5 \pi t + \frac{\pi}{18} x + \frac{\pi}{3} \right) \]
Let's extract some key properties of this wave:
- **The Phase Velocity:** The phase velocity \( v_p \) of the wave.
- **The Period of the Wave:** The period \( T \) of the wave.
- **The Wavelength:** The distance \( \lambda \) between successive peaks of the wave.
- **The Direction of Propagation:** The direction in which the wave is moving.
- **The Attenuation Number:** Any attenuation factor, if present, that signifies how the wave diminishes in amplitude over distance or time.
### Analysis
1. **The Phase Velocity ( \( v_p \) ):**
Phase velocity \( v_p \) is given by \( v_p = \frac{\omega}{k} \), where \( \omega \) is the angular frequency and \( k \) is the wave number.
2. **The Period of the Wave ( \( T \) ):**
The period can be derived from the angular frequency \( \omega \) using \( T = \frac{2\pi}{\omega} \).
3. **The Wavelength ( \( \lambda \) ):**
The wavelength can be found using the wave number \( k \) with the relation \( \lambda = \frac{2\pi}{k} \).
4. **The Direction of Propagation:**
The wave propagates in a direction determined by the sign and coefficients of \( x \) and \( t \) in the cosine argument.
5. **The Attenuation Number:**
The wave equation presents no exponential decay factors, so there is no attenuation implied in the provided equation.
### Detailed Derivation
1. **Finding \( k \) and \( \omega \):**
From the given wave equation:
\[ \vec{E}(x, t) = -\hat{y} 9.25 \cos \left(10^5 \pi t + \frac{\pi}{18} x + \frac{\pi}{3} \right) \]
- \( \omega = 10^5 \pi \)
-
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