What is the Doppler effect? Explain, with the aid of labelled diagrams, how this phenomenon occurs - Leaving Cert Physics - Question 7 - 2010

The Doppler effect can be illustrated using two labelled diagrams:

1. Moving Source Approaching Observer: In this scenario, as the source of sound (e.g., a siren) approaches the observer, the waves are compressed, resulting in shorter wavelengths and a higher frequency perceived by the observer.

   

2. Moving Source Receding from Observer: Conversely, when the source moves away from the observer, the waves are stretched, leading to longer wavelengths and a lower frequency.

   

Overall, the change in frequency is directly related to the relative motion of source and observer, described mathematically as: $f' = f \frac{c + u}{c - u}$ where:

• $f'$ is the observed frequency,
• $f$ is the source frequency,
• $c$ is the speed of sound/light,
• and $u$ is the speed of the source.

A simple laboratory experiment can involve a sound source such as a buzzer attached to a spring. As the buzzer emits sound while being swung in a circular motion around the observer, the frequency of the sound heard by the observer will change based on the position of the buzzer relative to them.

When the source is moving towards the observer, a higher frequency is detected. When it moves away, a lower frequency is heard. This can be demonstrated effectively using a string to control the swinging motion.

The red shift in the spectrum of a distant star is primarily due to the Doppler effect, indicating that the star is moving away from Earth. As the star recedes, its light waves are stretched, resulting in a longer wavelength which falls into the red part of the spectrum.

The movement of the star can be deduced by comparing the laboratory wavelength (587 nm) with the observed wavelength from the star (590 nm). Since the observed wavelength is longer than the laboratory wavelength, it confirms that the star is receding from us.

To calculate the speed of the moving star, we can apply the formula for the Doppler effect:

$z = \frac{\lambda' - \lambda}{\lambda}$

Where:

• $z$ is the redshift,
• $\lambda'$ is the observed wavelength (590 nm),
• $\lambda$ is the rest wavelength (587 nm).

Substituting in the values: $z = \frac{590 - 587}{587} = 0.0051$

Thus, using the redshift formula: $u = z \cdot c$ Substituting $c$ with $3.00 \times 10^8 m/s$: $u = 0.0051 \cdot 3.00 \times 10^8 m/s = 1.5333 \times 10^6 m/s$

This indicates the speed at which the star is moving away from Earth.

Other applications of the Doppler effect include:

• Radar technology (measuring speed of moving objects),
• Medical imaging techniques (such as Doppler ultrasound),
• Underwater acoustics in sonar technology.