Explain Newton's Formula For The Speed Of Sound In Air, Its Discrepancy, And Laplace's Correction. How Does The Frequency Of A 480 Hz Train Whistle Change For A Stationary Observer As The Train Approaches And Recedes Due To The Doppler Effect?

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Introduction

In the realm of physics, understanding the behavior of sound waves is crucial for various applications, from designing acoustic systems to comprehending natural phenomena. One fundamental aspect of sound wave behavior is its speed, which is influenced by the properties of the medium through which it travels. Air, being the most common medium for sound propagation, has been the subject of extensive study. This exploration delves into the historical development of the formula for the speed of sound in air, starting with Newton's initial proposal and the subsequent corrections made by Laplace. Understanding the evolution of this formula provides valuable insights into the nature of sound propagation and the importance of considering thermodynamic processes.

Newton's Expression for the Speed of Sound in Air

Sir Isaac Newton, a towering figure in the history of science, made significant contributions to our understanding of motion, gravity, and optics. In his seminal work, Principia Mathematica, published in 1687, Newton presented his formula for the speed of sound in air. His derivation was based on the assumption that sound propagation in air is an isothermal process. An isothermal process is one that occurs at a constant temperature. Newton reasoned that the compressions and rarefactions (regions of high and low pressure) that constitute a sound wave occur so slowly that the air has enough time to exchange heat with its surroundings, maintaining a constant temperature throughout the process. Based on this assumption, Newton derived the following expression for the speed of sound:

v = √(P/ρ)

where:

  • v represents the speed of sound.
  • P is the pressure of the air.
  • ρ is the density of the air.

Newton's formula essentially states that the speed of sound is proportional to the square root of the ratio of pressure to density. Using the experimental values for pressure and density of air at standard temperature and pressure (STP), Newton's formula yielded a value of approximately 280 meters per second for the speed of sound. However, this calculated value was significantly lower than the experimentally measured value of approximately 332 meters per second.

Discrepancy in Newton's Formula

The discrepancy between Newton's calculated value and the experimental value of the speed of sound posed a significant challenge to the scientific community. It was evident that Newton's formula, while based on sound reasoning, was missing a crucial element. The difference of over 50 meters per second could not be attributed to experimental errors or minor approximations. This discrepancy spurred further investigation into the nature of sound propagation and the underlying thermodynamic processes involved. Several scientists and mathematicians attempted to reconcile the theoretical and experimental values, but a satisfactory explanation remained elusive for several years.

The key issue with Newton's assumption lay in the fact that the compressions and rarefactions in a sound wave occur very rapidly. In reality, the air does not have sufficient time to exchange heat with its surroundings during these rapid fluctuations in pressure and density. As a result, the temperature of the air in the compression regions increases, while the temperature in the rarefaction regions decreases. This rapid change in temperature contradicts the isothermal condition assumed by Newton. The heat exchange process is not efficient enough to maintain a constant temperature throughout the wave propagation.

Laplace's Correction to Newton's Formula

The breakthrough in resolving the discrepancy in Newton's formula came from the work of Pierre-Simon Laplace, a renowned French mathematician and physicist. In 1816, Laplace proposed that the propagation of sound in air is not an isothermal process but rather an adiabatic process. An adiabatic process is one in which no heat is exchanged between the system and its surroundings. Laplace argued that the compressions and rarefactions in a sound wave occur so rapidly that there is negligible heat transfer. This means that the air does not have enough time to exchange heat with its surroundings during these rapid fluctuations in pressure and density.

Laplace's correction involved incorporating the adiabatic index (γ) into the formula. The adiabatic index represents the ratio of the specific heat at constant pressure (Cp) to the specific heat at constant volume (Cv) for a gas. For air, the adiabatic index is approximately 1.4. Laplace modified Newton's formula by multiplying the pressure term by the adiabatic index, resulting in the following expression:

v = √(γP/ρ)

where:

  • v represents the speed of sound.
  • γ is the adiabatic index.
  • P is the pressure of the air.
  • ρ is the density of the air.

By incorporating the adiabatic index, Laplace accounted for the fact that the temperature changes during the compressions and rarefactions of a sound wave. When Laplace's corrected formula is used with the same values for pressure, density, and the adiabatic index for air, the calculated speed of sound is approximately 331 meters per second. This value is in close agreement with the experimentally measured value of 332 meters per second, resolving the discrepancy that had plagued Newton's original formula.

Conclusion

The journey from Newton's initial formula for the speed of sound to Laplace's corrected version highlights the iterative nature of scientific progress. Newton's formula, while a significant step forward, was based on the simplifying assumption of an isothermal process. Laplace's recognition that sound propagation is an adiabatic process led to a more accurate formula that aligns with experimental observations. This correction underscores the importance of considering the thermodynamic processes involved in sound propagation. The corrected formula provides a robust framework for understanding and predicting the speed of sound in air and other gases. This understanding is crucial in many areas, including acoustics, meteorology, and engineering. The evolution of the formula for the speed of sound in air serves as a testament to the power of scientific inquiry and the ongoing quest for a deeper understanding of the natural world.

Train Whistle Frequency and Perceived Pitch: Exploring the Doppler Effect

Introduction

The Doppler effect is a fascinating phenomenon that describes the change in frequency of a wave in relation to an observer who is moving relative to the wave source. This effect is commonly observed with sound waves and light waves, and it has significant implications in various fields, including astronomy, medicine, and transportation. In the context of sound waves, the Doppler effect explains why the pitch of a siren or a train whistle changes as it approaches and then moves away from an observer. This exploration focuses on a specific scenario involving a train whistle emitting sound waves and how the perceived frequency changes due to the relative motion between the train and a stationary observer. Understanding the Doppler effect allows us to delve deeper into the physics of wave propagation and its practical applications.

The Doppler Effect: An Overview

The Doppler effect, named after Austrian physicist Christian Doppler, is the alteration in the observed frequency of a wave due to the motion of the source or the observer. When a sound source moves towards an observer, the sound waves are compressed in the direction of motion, resulting in a higher perceived frequency (higher pitch). Conversely, when the source moves away from the observer, the sound waves are stretched, leading to a lower perceived frequency (lower pitch). The magnitude of the frequency shift depends on the speed of the source and the observer relative to the medium through which the sound is traveling.

The Doppler effect can be quantified using the following formula for sound waves:

f' = f (v ± vo) / (v ± vs)

where:

  • f' is the observed frequency.
  • f is the source frequency (the actual frequency emitted by the source).
  • v is the speed of sound in the medium.
  • vo is the speed of the observer.
  • vs is the speed of the source.

In this formula, the plus and minus signs are chosen according to the direction of motion. If the observer is moving towards the source or the source is moving towards the observer, the signs are chosen to increase the observed frequency (higher pitch). If the observer is moving away from the source or the source is moving away from the observer, the signs are chosen to decrease the observed frequency (lower pitch).

Scenario: Train Whistle and a Stationary Observer

Consider a scenario where a train is moving and its whistle is generating sound waves with a frequency of 480 Hz. You are standing stationary beside the train tracks. As the train approaches you, the sound waves emitted by the whistle are compressed, and you perceive a higher frequency. Once the train passes you and moves away, the sound waves are stretched, and you perceive a lower frequency. The change in frequency is a direct consequence of the Doppler effect.

To analyze this scenario, we need to apply the Doppler effect formula. Since you are stationary, your speed (vo) is 0. The source frequency (f) is 480 Hz. The speed of sound in air (v) is approximately 343 meters per second at typical atmospheric conditions. Let's assume the train is moving at a speed (vs) of 30 meters per second. We can calculate the observed frequency as the train approaches and as it moves away.

Approaching Train

When the train is approaching, we use the plus sign in the numerator and the minus sign in the denominator of the Doppler effect formula:

f' = f (v + vo) / (v - vs)

Substituting the values:

f' = 480 Hz (343 m/s + 0 m/s) / (343 m/s - 30 m/s)

f' = 480 Hz (343 m/s) / (313 m/s)

f' ≈ 526 Hz

As the train approaches, you perceive a higher frequency of approximately 526 Hz. This increase in frequency is why the pitch of the whistle sounds higher as the train gets closer.

Receding Train

When the train is receding, we use the minus sign in the numerator and the plus sign in the denominator of the Doppler effect formula:

f' = f (v - vo) / (v + vs)

Substituting the values:

f' = 480 Hz (343 m/s - 0 m/s) / (343 m/s + 30 m/s)

f' = 480 Hz (343 m/s) / (373 m/s)

f' ≈ 442 Hz

As the train moves away, you perceive a lower frequency of approximately 442 Hz. This decrease in frequency explains why the pitch of the whistle sounds lower as the train moves farther away.

Conclusion

The scenario of a train whistle and a stationary observer vividly illustrates the Doppler effect. The perceived frequency of the sound waves emitted by the whistle changes depending on the relative motion between the train and the observer. As the train approaches, the frequency increases, resulting in a higher pitch, and as the train recedes, the frequency decreases, resulting in a lower pitch. This phenomenon has practical implications in various real-world applications. For example, police radar uses the Doppler effect to measure the speed of vehicles, and astronomers use it to determine the motion of stars and galaxies. Understanding the Doppler effect provides valuable insights into wave propagation and its applications in everyday life and scientific research.