Problem 41 - University Physics with Modern Physics

Question

On the planet Arrakis a male ornithoid is flying toward his mate at 25.0 \(\mathrm{m} / \mathrm{s}\) while singing at a frequency of 1200 \(\mathrm{Hz}\) . If the stationary female hears a tone of 1240 \(\mathrm{Hz}\) , what is the speed of sound in the atmosphere of Arrakis?

Step-by-Step Solution

Answer

The speed of sound on Arrakis is 775 m/s.

Step 1: Understanding the Doppler Effect

The problem involves the Doppler Effect where the frequency of a wave changes based on the motion of the source and observer. In this case, the source (male ornithoid) is moving toward the observer (female ornithoid), which increases the frequency observed by the stationary female.

Step 2: Using the Doppler Effect Formula

The formula for observed frequency when the source is moving towards a stationary observer is \( f' = \frac{f \cdot (v + v_o)}{v - v_s} \) where \( f' \) is the observed frequency, \( f \) is the source frequency, \( v \) is the speed of sound, \( v_o \) is the observer's speed (0 since the female is stationary), and \( v_s \) is the source's speed (25.0 m/s).

Step 3: Substituting Known Values

Given the observed frequency \( f' = 1240 \, \mathrm{Hz} \), original frequency \( f = 1200 \, \mathrm{Hz} \), and source speed \( v_s = 25.0 \, \mathrm{m/s} \), substitute these into the formula: \[ 1240 = \frac{1200 \cdot (v)}{v - 25} \]

Step 4: Solving for Speed of Sound

Rearrange to solve for \( v \): \[ 1240(v - 25) = 1200v \] \[ 1240v - 31000 = 1200v \] \[ 40v = 31000 \] \[ v = \frac{31000}{40} \] \[ v = 775 \, \mathrm{m/s} \]

Step 5: Conclusion

The speed of sound on the planet Arrakis, under the given conditions, is 775 m/s.

Key Concepts

Speed of Sound

The speed of sound is an important factor that determines how quickly sound waves travel through a medium. On Earth, sound travels at approximately 343 meters per second in air at 20°C. However, this speed can vary based on the medium's properties like temperature, density, and humidity.
In our example on the planet Arrakis, we used the Doppler Effect to determine the speed of sound, yielding a value of 775 meters per second. This speed indicates a different atmospheric composition or conditions compared to Earth. When solving for the speed of sound, we must remember that:

• The medium's density and temperature can significantly influence the speed.
• Sound travels faster in liquids and solids compared to gases.

The Doppler Effect helped us calculate this specific speed when a sound source moves relative to an observer, showing how interconnected these concepts are.

Frequency Change

Frequency change refers to the alteration of the perceived pitch of a sound when either the source or the observer is moving. This change is due to the Doppler Effect, which describes how the frequency of waves varies based on relative motion.
Let's break it down with our ornithoid example:

• The male ornithoid sings at a frequency of 1200 Hz while flying toward a stationary female.
• The movement causes the female to perceive the frequency as 1240 Hz.

The frequency increases because the source is moving closer to the observer, resulting in compressed sound waves. This change in frequency can be calculated using the Doppler Effect formula, allowing us to determine other variables, like the speed of sound. Keep in mind:

• Higher relative speeds lead to more significant frequency shifts.
• If the source and observer are moving apart, the observer perceives a lower frequency.

This principle is widely used in radar, astronomy, and even healthcare.

Wave Propagation

Wave propagation refers to the manner in which sound waves travel through a medium. Sound waves are longitudinal waves, meaning their oscillations occur in the direction of the wave's travel. Understanding this concept helps explain how sound is transmitted differently through various environments.
In the context of Arrakis, we have observed how the propagation of sound waves is influenced by the atmosphere's characteristics, which affect the speed of sound. Some key points about wave propagation include:

• Sound waves require a medium to travel, unlike electromagnetic waves, which can travel through a vacuum.
• The efficiency of propagation depends on the medium's properties, such as its elasticity and density.

The Doppler Effect is a direct result of how waves interact with moving sources and observers. As the ornithoid moves, the wave propagation dynamics change, thus altering the perceived frequency by the female. This fundamental concept is crucial for understanding phenomena like echoes, reverberations, and even noise-canceling technology.