Question

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

Short answer

The speed of sound in the atmosphere of Arrakis is 775 m/s.

Step-by-step solution

Understand the Doppler Effect Formula

The Doppler Effect formula for observed frequency when the source is moving towards a stationary observer is given by $f^{\prime }=\frac{f\cdot (v+v_o)}{v-v_s}$, where $f^{\prime }$ is the observed frequency, $f$ is the source frequency, $v$ is the speed of sound, $v_o$ is the velocity of the observer (0 m/s in this case as the observer is stationary), and $v_s$ is the velocity of the source (25 m/s here).

Substitute Known Values

Substitute the given values into the formula: the observed frequency $f^{\prime }=1240$ Hz, the source frequency $f=1200$ Hz, and the velocity of the source $v_s=25$ m/s. We will solve for $v$, the speed of sound. The equation becomes $1240=\frac{1200\cdot v}{v-25}$.

Solve for Speed of Sound

Rearrange the equation to solve for $v$. Multiply both sides by $v-25$ to eliminate the fraction: $1240(v-25)=1200v$. This simplifies to $1240v-31000=1200v$. Simplifying further, we get $40v=31000$. Thus, $v=\frac{31000}{40}=775$ m/s.

Key concepts

Observed Frequency

The observed frequency is the frequency of a wave as perceived by an observer. In the Doppler Effect scenario, this frequency changes depending on the motion of the source or the observer. Therefore, if the source is moving towards a stationary observer, the observed frequency will be higher than the source frequency.

In the exercise from Arrakis, the female ornithoid is stationary, and she perceives the frequency as 1240 Hz, which is higher than the source frequency of 1200 Hz. This increase demonstrates the principle that the frequency increases when the source approaches the observer. Understanding this principle helps in using the Doppler Effect effectively to calculate other parameters, like the speed of sound.

Source Frequency

The source frequency is the original frequency emitted by the source of a wave, without any alterations. In the Doppler Effect, this is the baseline frequency before any movement-related changes are accounted for.

In our scenario, the male ornithoid is producing a source frequency of 1200 Hz. This frequency is crucial to calculating the observed frequency once you include all parameters such as source speed and the speed of sound. It serves as one of the known variables to resolve the equation involved in analyzing the Doppler Effect.

• Basic parameter needing no adjustments unless under influence.
• Starting point for understanding Doppler Effect in motion scenarios.

Speed of Sound

The speed of sound is critical in understanding how frequencies shift with motion in the Doppler Effect. It refers to how fast sound waves propagate through a medium, impacting how quickly differences in frequency can be perceived.

In the exercise, we calculated the speed of sound on the planet Arrakis by rearranging and solving the Doppler Effect formula. By substituting the known observed frequency (1240 Hz) and source frequency (1200 Hz) with the source speed (25 m/s), we were able to determine that the speed of sound is 775 m/s. This step underlines the interdependence of frequency perceptions and sound speed in the medium where the sound travels.

Stationary Observer

A stationary observer does not move relative to the medium through which sound is propagating. In the Doppler Effect, this means any frequency change observed is solely due to the motion of the source.

The stationary observer in our problem is the female ornithoid. Her lack of motion simplifies the Doppler Effect formula, as the observer's velocity term ( $v_o)$ is zero. This stationary condition is essential for calculating the exact shift in observed frequency without additional variables. Understanding the role of a stationary observer is key to correctly applying the Doppler Effect to determine changes in perceived sound frequency.