Question

Suppose a long line of cars waits at an intersection for the light to turn green. Once the light turns, the cars in the back cannot move right away, but have to wait until the cars ahead of them move. Explain this effect (called a rarefaction wave) in terms of the wave physics that we have studied in this chapter.

Short answer

In summary, the rarefaction wave phenomenon in a traffic situation can be compared to the alternating areas of high and low pressure in a sound wave. When the cars at the front of the line move forward, they create a "compression," similar to increased pressure in a sound wave. As the cars at the back wait for their turn to move, the gap between them and the moving cars increases, creating a "rarefaction," analogous to the decreased pressure in a sound wave. This pattern of compressions and rarefactions progresses through the line of cars, illustrating the rarefaction wave in traffic, which resembles the behavior of a sound wave.

Step-by-step solution

Understanding the rarefaction wave in traffic situation

In the given situation, the cars at the back of the line cannot move immediately when the traffic light turns green. They have to wait for the cars in front of them to move. This leads to a spacing between the cars increasing as they start moving, which is called a rarefaction wave.

Understanding wave physics concepts

In wave physics, we have studied several types of wave phenomena. One example is the propagation of sound waves in air which consist of compressions (areas of high pressure) and rarefactions (areas of low pressure) that travel through the medium.

Analogy to wave physics

We can draw an analogy of the rarefaction wave seen in the traffic situation to the rarefaction in a sound wave. When the traffic light turns green, the cars at the front start moving and create a "compression" as they move through the intersection. This is similar to the compression in a sound wave where pressure increases. As the cars in the back of the line wait for their turn to move, the cars in front of them leave a more substantial gap between themselves and the stationary cars behind. This is similar to the rarefaction in a sound wave where pressure decreases, and the space between the particles in the medium increases. The transition from a compression to a rarefaction moving through the line of cars is the rarefaction wave generated at the intersection.

Conclusion

In conclusion, the rarefaction wave in the given traffic situation can be explained in terms of wave physics by drawing an analogy to a sound wave. The cars moving away from the traffic light create "compressions," and the waiting cars at the back create the "rarefactions" as the spacing between them starts to increase. This pattern progresses down the line of cars like a wave, similar to rarefaction in a sound wave.

Key concepts

Wave Physics

Wave physics is the branch of physics that deals with the study of waves and their properties. Essentially, waves are disturbances that transfer energy from one point to another without the transfer of matter.

Waves can be categorized into two main types: mechanical waves and electromagnetic waves. Mechanical waves, such as sound waves, require a medium (such as air, water, or a solid) to travel through, while electromagnetic waves do not need a medium and can travel through the vacuum of space. The rarefaction wave mentioned in the exercise is an example of a concept from mechanical wave physics.

In a rarefaction wave, a pattern of decreasing pressure allows areas of lower density or 'gaps' to propagate through a medium. This is associated with phenomena like sound waves, where rarefaction corresponds to the part of the wave cycle where the air particles are spread apart, creating a drop in pressure.

Sound Waves

Sound waves are a type of mechanical wave that are created by vibrating objects and are propagated through a medium (usually air). They are longitudinal waves, which means that the displacement of the medium is parallel to the direction of the wave's travel.

Components of a Sound Wave

The two main components of a sound wave are compressions and rarefactions. Compressions are regions where particles of the medium are close together, resulting in higher pressure. Rarefactions, on the other hand, are regions where particles are spread out, leading to lower pressure.

As sound waves propagate through a medium, they do so by alternating between these two states. The frequency and wavelength of these oscillations determine the pitch and tone of the sound heard. When discussing rarefaction waves in the context of sound, it's essential to understand that these lower-pressure areas are integral for creating the waves that we ultimately perceive as sound.

Propagation of Waves

Propagation of waves refers to the process by which waves travel through a medium. This process can be influenced by properties of the wave, such as its wavelength, frequency, and amplitude, as well as by the characteristics of the medium through which it moves.

Factors Affecting Wave Propagation

Several factors can affect how waves propagate, including:

• The medium's density: Denser media generally slow down mechanical waves.
• The medium's elasticity: A more elastic medium can propagate waves faster.
• The temperature of the medium: For sound waves, higher temperatures generally lead to faster propagation speeds due to increased particle energy.

The propagation of a rarefaction wave, like the traffic example, is analogous to how sound waves travel through air. Each car moving and stopping simulates the alternating high and low-pressure areas (compressions and rarefactions) and demonstrates how wave effects can be observed in everyday situations.