Doppler Effect Explained Police Car Siren Approaching Observer

The classic problem of a police car moving on a road at a speed of 40 m/s, approaching a stationary observer, with its siren emitting a sound at a frequency of 1200 Hz, provides a fascinating case study for understanding the Doppler effect. This phenomenon describes the change in frequency of a wave in relation to an observer who is moving relative to the wave source. In this scenario, the police car is the moving source of sound, the siren, and the observer is stationary. The speed of sound in the air is given as 340 m/s. The challenge lies in determining the frequency of the sound as perceived by the observer.

To unravel this problem, we delve into the principles of the Doppler effect for sound waves. The crucial factor here is the relative motion between the source of the sound (the police car) and the observer. When the source is approaching the observer, the sound waves are compressed in the direction of motion, leading to a higher frequency perception. Conversely, if the source were moving away, the waves would be stretched, and the perceived frequency would be lower. This principle has wide-ranging applications, from understanding astronomical phenomena like the redshift of distant galaxies to everyday applications like radar speed guns used by law enforcement.

The mathematical formulation of the Doppler effect for a moving source and a stationary observer is given by the formula: f' = f (v + vo) / (v + vs), where f' is the observed frequency, f is the source frequency, v is the speed of sound in the medium, vo is the velocity of the observer, and vs is the velocity of the source. In our case, the observer is stationary, so vo is 0. The police car is approaching, so we consider vs as positive. Plugging in the given values, we have f' = 1200 Hz * (340 m/s + 0 m/s) / (340 m/s - 40 m/s). This equation encapsulates the essence of the Doppler effect, allowing us to quantitatively determine how motion affects the perception of sound frequency. The formula highlights the interplay between the speed of sound, the speed of the source, and the original frequency in determining the shifted frequency observed.

Calculating the Observed Frequency

To calculate the observed frequency, we will use the Doppler effect formula, plugging in the values given in the problem. The formula is f' = f * (v + vo) / (v - vs), where:

• f' is the observed frequency.
• f is the source frequency (1200 Hz).
• v is the speed of sound in air (340 m/s).
• vo is the velocity of the observer (0 m/s, since the observer is stationary).
• vs is the velocity of the source (40 m/s, since the police car is approaching).

Substituting these values into the formula, we get: f' = 1200 Hz * (340 m/s + 0 m/s) / (340 m/s - 40 m/s). Simplifying the equation, we have f' = 1200 Hz * (340 m/s) / (300 m/s). Further simplification leads to f' = 1200 Hz * 1.133. Thus, the observed frequency f' is approximately 1360 Hz. This result illustrates the Doppler effect in action, where the frequency of the sound is perceived as higher due to the approaching source. The calculation underscores the importance of understanding the relative velocities of the source and the observer in determining the shift in frequency.

This calculated frequency of 1360 Hz is higher than the original frequency of 1200 Hz, which aligns with our understanding of the Doppler effect when a sound source approaches an observer. The observer perceives a higher pitch because the sound waves are compressed in front of the moving source. This change in frequency is a direct consequence of the relative motion between the source and the observer, and it is a fundamental concept in physics with numerous applications. The quantitative result provides a concrete example of how the Doppler effect manifests in a real-world scenario.

Determining the Wavelength

To determine the wavelength of the sound wave as perceived by the observer, we will use the relationship between wave speed, frequency, and wavelength: v = f'λ', where v is the speed of sound, f' is the observed frequency, and λ' is the observed wavelength. We already know the speed of sound (v = 340 m/s) and have calculated the observed frequency (f' ≈ 1360 Hz). Our goal now is to find λ'. Rearranging the formula, we get λ' = v / f'.

Substituting the values, we have λ' = 340 m/s / 1360 Hz. Performing the calculation, we find λ' ≈ 0.25 meters. This result indicates that the wavelength of the sound wave, as perceived by the observer, is approximately 0.25 meters. This wavelength is shorter than the wavelength of the sound emitted by the siren in its rest frame, which is another manifestation of the Doppler effect. The compression of sound waves due to the motion of the source results in a shorter wavelength and, consequently, a higher perceived frequency.

The observed wavelength of approximately 0.25 meters complements our understanding of the Doppler effect. As the police car approaches, the sound waves are compressed, leading to a higher frequency and a shorter wavelength. This inverse relationship between frequency and wavelength is a fundamental property of waves and is clearly demonstrated in this scenario. The calculation of the wavelength provides a complete picture of how the Doppler effect alters the characteristics of sound waves when there is relative motion between the source and the observer. This understanding is crucial in various applications, including medical imaging and radar technology.

Applications and Implications of the Doppler Effect

The Doppler effect isn't just a theoretical concept; it has numerous practical applications across various fields. In medicine, Doppler ultrasound is used to measure blood flow and detect abnormalities. Astronomers use the Doppler effect to measure the speeds of stars and galaxies, which is crucial for understanding the expansion of the universe. Weather forecasting relies on Doppler radar to track the movement of storms and predict weather patterns. Even speed cameras use the Doppler effect to measure the speed of vehicles, ensuring traffic safety.

The implications of the Doppler effect extend beyond these specific applications. It provides a fundamental understanding of how waves behave when there is relative motion between the source and the observer. This principle is applicable not only to sound waves but also to other types of waves, such as light waves and electromagnetic waves. The redshift of light from distant galaxies, for example, is a direct consequence of the Doppler effect and provides evidence for the expansion of the universe. The Doppler effect has revolutionized our understanding of the universe and has played a pivotal role in technological advancements across various fields.

Understanding the Doppler effect is also crucial for addressing real-world challenges. For instance, in the design of acoustic systems, it is important to consider the Doppler effect to ensure optimal sound quality in moving environments. In air traffic control, the Doppler effect is used to accurately track the speed and direction of aircraft. In emergency situations, understanding the Doppler effect can help in locating the source of a siren or other emergency signals. The Doppler effect is a ubiquitous phenomenon with far-reaching implications for our daily lives and for scientific advancements.

In conclusion, the problem of the police car siren and the stationary observer provides a clear illustration of the Doppler effect. By understanding the principles behind this phenomenon and applying the relevant formulas, we can accurately determine the observed frequency and wavelength of the sound wave. The Doppler effect has numerous applications in diverse fields, underscoring its importance in both theoretical physics and practical applications. From medical imaging to astronomical observations, the Doppler effect has transformed our understanding of the world around us and continues to be a valuable tool for scientific discovery and technological innovation. The ability to analyze and interpret the Doppler effect is essential for students and professionals alike, highlighting the enduring significance of this fundamental concept in physics.