Doppler Shift Formula:
The Doppler Shift, a phenomenon observed in various scientific fields, plays a crucial role in understanding the relative motion of objects in the universe. At its core, the Doppler Shift describes the change in frequency or wavelength of a wave as observed by an observer moving relative to the source of the wave. This article delves into the Doppler Shift Formula, a mathematical expression that quantifies this effect and is used in various applications.
Definition of Doppler Shift
Doppler Shift Formula
: Doppler Shift, named after the Austrian physicist Christian Doppler, is the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. When an object producing waves is moving toward an observer, the waves appear compressed, resulting in a higher frequency or shorter wavelength, known as "blueshift." Conversely, when the object is moving away, the waves appear stretched, resulting in a lower frequency or longer wavelength, known as "redshift."
Doppler Shift Formula
The Doppler Shift Formula provides a mathematical representation of this phenomenon. It can be expressed for both sound waves and electromagnetic waves. For sound waves, the formula is:
f' = f ((v + vo) / (v + vs))
where:
`f'` is the observed frequency.
`f` is the source frequency.
`v` is the speed of sound.
`vo` is the velocity of the observer.
`vs` is the velocity of the source.
For electromagnetic waves (e.g., light), the formula is slightly different and is given by:
f' = f (c + vo) / (c + vs)
where:
`f'` is the observed frequency.
`f` is the source frequency.
`c` is the speed of light.
`vo` is the velocity of the observer.
`vs` is the velocity of the source.
These formulas enable the calculation of observed frequencies when relative motion exists between the source and the observer.
Applications of Doppler Shift Formula
The Doppler Shift Formula finds application in various fields, including astronomy, meteorology, and medical diagnostics. In astronomy, it helps determine the motion of celestial objects and the expansion of the universe. In meteorology, it is used to study the movement of weather systems. Moreover, in medical diagnostics, it aids in the measurement of blood flow and the detection of abnormalities.
Limitations and Considerations
The Doppler Shift Formula, while powerful, has limitations. It assumes a constant velocity, and in reality, velocities can change, leading to more complex Doppler effects. Additionally, factors like the angle between the source, and observer, and their relative velocities can affect the observed frequency. These limitations emphasize the need for careful consideration when applying the formula in practical situations.
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Solved Examples Using Doppler Shift Formula
Let's explore a few solved examples to understand how the Doppler Shift Formula is applied in real-world scenarios.
Example 1:
Suppose an ambulance with a siren is moving towards you at a speed of 30 m/s, and the siren emits a sound with a frequency of 1000 Hz. Calculate the frequency of the sound observed by you.
Solution:
In this case, `v` (speed of sound) is approximately 343 m/s, `vo` (observer's velocity) is 0 m/s, `vs` (source's velocity) is 30 m/s, and `f` (source frequency) is 1000 Hz.
Plugging these values into the Doppler Shift Formula for sound waves:
f' = 1000 ((343 + 0) / (343 + 30))
f' ≈ 962.68 Hz
So, you would observe a frequency of approximately 962.68 Hz.
Example 2:
In astronomy, a distant galaxy is moving away from Earth with a velocity of 3,000 km/s. The galaxy emits light with a wavelength of 656 nm. Calculate the observed wavelength on Earth.
Solution:
In this case, `c` (speed of light) is approximately 299,792,458 m/s, `vo` (observer's velocity) is 0 m/s, `vs` (source's velocity) is 3,000,000 m/s, and the source wavelength (`λ`) is 656 nm, which is equivalent to 656 x 10^-9 meters.
Using the Doppler Shift Formula for electromagnetic waves:
λ' = λ (c + vo) / (c + vs)
λ' = (656 x 10^-9) ((299,792,458 + 0) / (299,792,458 + 3,000,000))
λ' ≈ 658.34 nm
The observed wavelength on Earth is approximately 658.34 nm, indicating a redshift.
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Example 3:
A weather station observes a tornado moving towards it with a velocity of 50 m/s. The tornado emits a sound with a frequency of 500 Hz. Calculate the frequency observed by the weather station.
Solution:
In this scenario, `v` (speed of sound) is approximately 343 m/s, `vo` (observer's velocity) is 0 m/s, `vs` (source's velocity) is 50 m/s, and `f` (source frequency) is 500 Hz.
Using the Doppler Shift Formula for sound waves
f' = 500 ((343 + 0) / (343 + 50))
f' ≈ 456.98 Hz
The frequency observed by the weather station is approximately 456.98 Hz, indicating a blueshift.
Example 4:
In medical diagnostics, an ultrasound probe is used to measure blood flow in a patient's artery. The probe moves towards the artery at a velocity of 1 m/s. The emitted ultrasound has a frequency of 5 MHz. Calculate the frequency observed in the ultrasound images.
Solution:
Here, `v` (speed of sound in tissue) is approximately 1540 m/s, `vo` (probe's velocity) is 1 m/s, `vs` (blood's velocity) can vary, and `f` (source frequency) is 5 MHz (5 x 10^6 Hz).
Using the Doppler Shift Formula for ultrasound:
f' = 5 x 10^6 ((1540 + 1) / (1540 + vs))
The observed frequency depends on the blood velocity (`vs`), which would need to be measured to calculate the frequency accurately. The Doppler Shift Formula is a valuable tool for assessing blood flow in medical imaging.
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Example 5:
An approaching car sounds its horn, emitting a frequency of 800 Hz. If the car is moving towards you at a speed of 25 m/s, calculate the frequency you would hear.
Solution:
In this case, `v` (speed of sound) is approximately 343 m/s, `vo` (observer's velocity) is 0 m/s, `vs` (source's velocity) is 25 m/s, and `f` (source frequency) is 800 Hz.
Using the Doppler Shift Formula for sound waves:
f' = 800 ((343 + 0) / (343 + 25))
f' ≈ 825.75 Hz
You would hear a frequency of approximately 825.75 Hz, indicating a blueshift.
These examples demonstrate the versatility of the Doppler Shift Formula in various fields, from astronomy to meteorology and medical diagnostics, providing essential insights into the relative motion of objects and sources. It's a fundamental tool for understanding the behavior of waves in different scenarios.
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