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The Doppler Effect is a fundamental concept in physics, describing the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. This effect is commonly experienced in everyday life, such as when a passing vehicle sounds differently as it approaches and then moves away.

In the context of sound waves, the Doppler Effect explains how the detected frequency at a stationary point can change when the sound source is in motion. The formula \( f' = \frac{f \cdot v}{v \pm v_s} \) helps to calculate the observed frequency \( f' \), where \( f \) is the original frequency of the source, \( v \) is the speed of sound, and \( v_s \) is the velocity of the source.

The effect causes the frequency band observed by a detector, which is the range between the maximum and minimum frequencies, when a source moves in simple harmonic motion. This change in frequency can be used to calculate various properties of the source's motion, such as its time period.

A frequency band refers to the range of frequencies over which an oscillating system emits or responds. In simple harmonic motion, when a sound source moves, the frequency detected by a stationary observer changes due to the Doppler Effect. The difference between the maximum and minimum frequencies observed is known as the width of the frequency band.

In our example, the emitted frequency is \( 800 \mathrm{~Hz} \), and the width of the frequency band detected is \( 8 \mathrm{~Hz} \). This width represents how much the frequency shifts as the source moves toward and away from the detector. Understanding the frequency band is crucial in analyzing how quickly or slowly the source is moving and helps provide insight into the dynamics of the system in question.

Angular Frequency, noted as \( \omega \), is an important parameter in oscillatory motion, especially in simple harmonic motion (SHM). It represents how fast an object oscillates in a circular path, measured in radians per second. It is closely related to the concept of frequency but specifically deals with the angular displacement over time.

The relationship between angular frequency and frequency \( f \) is given by the equation \( \omega = 2\pi f \). In SHM, angular frequency is key to determining other aspects like the source's velocity; it can be calculated using the formula \( \omega = \frac{v_s}{A} \), where \( v_s \) is the source velocity and \( A \) is the amplitude.

Angular frequency is integral for solving problems involving oscillations, such as calculating the maximum speed of the source or its time period, as it provides a connection between linear motion parameters and circular motion metrics.

The time period \( T \) is the duration it takes for one complete cycle of oscillation, a significant parameter in analyzing oscillatory systems. It is inversely related to the frequency of the source. The relationship between the angular frequency and the time period is defined by \( T = \frac{2\pi}{\omega} \).

In solving time period problems, one must often determine the source velocity at maximum displacement within its simple harmonic motion. Given this velocity and amplitude, the angular frequency can be calculated, which then leads to the determination of the time period. In the provided solution, the time period is calculated using \( T = \frac{2\pi A}{v_s} \), where \( A \) is the amplitude and \( v_s \) is the known source velocity.

Understanding how to calculate the time period from parameters like amplitude and velocity equips learners with tools to solve a variety of oscillation-related problems efficiently.