College Physics II – Mechanics, Sound, Oscillations, and Waves
Definition
$f_o$ is the observed frequency of a wave, which is the frequency measured by an observer who is moving relative to the source of the wave. This term is particularly important in the context of the Doppler effect, which describes the change in observed frequency or wavelength of a wave due to the relative motion between the source and the observer.
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$f_o$ is the frequency of the wave as observed by the moving observer, and it is different from the emitted frequency ($f_s$) due to the Doppler effect.
The relationship between $f_o$, $f_s$, and the relative velocity $v$ is given by the Doppler shift equation: $f_o = f_s \left(\frac{c \pm v}{c}\right)$, where $c$ is the speed of the wave.
The Doppler shift can be either positive or negative, depending on whether the observer is moving towards or away from the source of the wave.
The Doppler effect is observed in various wave phenomena, including sound waves, electromagnetic waves, and even the motion of celestial objects.
Understanding the Doppler effect is crucial in fields such as astronomy, radar technology, and medical imaging, where it is used to measure the relative motion of objects.
Review Questions
Explain the relationship between $f_o$, $f_s$, and the relative velocity $v$ in the context of the Doppler effect.
The relationship between the observed frequency $f_o$, the emitted frequency $f_s$, and the relative velocity $v$ is given by the Doppler shift equation: $f_o = f_s \left(\frac{c \pm v}{c}\right)$, where $c$ is the speed of the wave. This equation shows that the observed frequency is different from the emitted frequency due to the Doppler effect, and the magnitude of the difference depends on the relative velocity between the source and the observer. If the observer is moving towards the source, the observed frequency is higher than the emitted frequency (positive Doppler shift), and if the observer is moving away from the source, the observed frequency is lower than the emitted frequency (negative Doppler shift).
Discuss the importance of understanding the Doppler effect in various scientific and technological applications.
The Doppler effect is a fundamental concept in many scientific and technological fields. In astronomy, it is used to measure the relative motion of celestial objects, such as stars and galaxies, which provides insights into the structure and evolution of the universe. In radar technology, the Doppler effect is used to detect the motion of objects, such as aircraft and weather patterns. In medical imaging, the Doppler effect is employed in techniques like Doppler ultrasound to measure blood flow and detect cardiovascular abnormalities. Understanding the Doppler effect is crucial in these applications because it allows researchers and engineers to accurately interpret the observed wave frequencies and extract valuable information about the motion and properties of the studied objects or phenomena.
Analyze how the Doppler effect and the observed frequency $f_o$ can be used to determine the relative velocity $v$ between the source and the observer.
The Doppler shift equation, $f_o = f_s \left(\frac{c \pm v}{c}\right)$, can be rearranged to solve for the relative velocity $v$ if the observed frequency $f_o$, the emitted frequency $f_s$, and the speed of the wave $c$ are known. Specifically, the equation can be manipulated to obtain $v = c \left(\frac{f_o}{f_s} - 1\right)$ for the case of the observer moving towards the source, or $v = c \left(1 - \frac{f_o}{f_s}\right)$ for the case of the observer moving away from the source. By measuring the observed frequency $f_o$ and knowing the emitted frequency $f_s$ and the speed of the wave $c$, the relative velocity $v$ between the source and the observer can be calculated. This allows for the determination of the motion and speed of objects in various applications, such as in astronomy, radar, and medical imaging.