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524 1048 1 -1 L p(t)| 1 0 1048 1 718 -² 78 1 524 -1L 262 t in seconds

524 1048 1 -1 L p(t)| 1 0 1048 1 718 -² 78 1 524 -1L 262 t in seconds

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the Fourier coefficients for the attached sound wave and use them to write a function p(t) that describes the pressure. (You only need to write the first six terms)
The image depicts a waveform graph plotted over time, denoted by \( t \), measured in seconds. The function, \( p(t) \), is represented on the vertical axis. The waveform consists of a series of rectangular pulses, alternating between \( 1 \) and \(-1\). ### Description of the Waveform: 1. **Time Intervals:** - The first pulse starts at \( t = -\frac{1}{524} \) and increases to 1. - It drops to \(-1\) at \( t = -\frac{1}{1048} \). - It then returns to 0 at \( t = 0 \). 2. **Symmetrical Behavior:** - The behavior of the waveform post \( t = 0 \) is a mirror reflection of the pre-\( t = 0 \) characteristics. - From \( t = 0 \) to \( t = \frac{1}{1048} \), the function rises to \(\frac{7}{8}\). - From \( t = \frac{1}{1048} \) to \( t = \frac{1}{524} \), the function goes to 1. - Immediately after, it goes back to \(-1\) until \( t = \frac{1}{262} \), where it peaks at 1 once more. 3. **Repeating Pattern:** - The pattern appears to be repetitive, indicated by symmetrically occurring pulses across time intervals. ### Labels and Measurements: - Each time increment along the \( t \)-axis is clearly labeled with specific fractions, indicating the precise occurrence of the transitions in the waveform. - The vertical jumps are exactly at \( p(t) = 1 \), \(\frac{7}{8}\), and \(-1\). This waveform represents typical signals used in communications and control systems, illustrating periodic pulse modulation with specific timing and amplitude characteristics.
Transcribed Image Text:The image depicts a waveform graph plotted over time, denoted by \( t \), measured in seconds. The function, \( p(t) \), is represented on the vertical axis. The waveform consists of a series of rectangular pulses, alternating between \( 1 \) and \(-1\). ### Description of the Waveform: 1. **Time Intervals:** - The first pulse starts at \( t = -\frac{1}{524} \) and increases to 1. - It drops to \(-1\) at \( t = -\frac{1}{1048} \). - It then returns to 0 at \( t = 0 \). 2. **Symmetrical Behavior:** - The behavior of the waveform post \( t = 0 \) is a mirror reflection of the pre-\( t = 0 \) characteristics. - From \( t = 0 \) to \( t = \frac{1}{1048} \), the function rises to \(\frac{7}{8}\). - From \( t = \frac{1}{1048} \) to \( t = \frac{1}{524} \), the function goes to 1. - Immediately after, it goes back to \(-1\) until \( t = \frac{1}{262} \), where it peaks at 1 once more. 3. **Repeating Pattern:** - The pattern appears to be repetitive, indicated by symmetrically occurring pulses across time intervals. ### Labels and Measurements: - Each time increment along the \( t \)-axis is clearly labeled with specific fractions, indicating the precise occurrence of the transitions in the waveform. - The vertical jumps are exactly at \( p(t) = 1 \), \(\frac{7}{8}\), and \(-1\). This waveform represents typical signals used in communications and control systems, illustrating periodic pulse modulation with specific timing and amplitude characteristics.
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