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Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey's speed. (The "horseshoe" that gives the bat its name is a depression around the nostrils that acts like a focusing mirror, so that the bat emits sound in a narrow beam like a flashlight.) A Rhinolophus flying at speed \(v_{\text {bat }}\) emits sound of frequency \(f_{\text {bat }} ;\) the sound it hears reflected from an insect flying toward it has a higher frequency \(f_{\text {refl }}\). (a) Show that the speed of the insect is $$ v_{\text {insect }}=v\left[\frac{f_{\text {ref }}\left(v-v_{\text {bat }}\right)-f_{\text {bat }}\left(v+v_{\text {bat }}\right)}{f_{\text {ref }}\left(v-v_{\text {bat }}\right)+f_{\text {bat }}\left(v+v_{\text {bat }}\right)}\right] $$ where \(v\) is the speed of sound. (b) If \(f_{\text {bat }}=80.7 \mathrm{kHz}, f_{\text {refl }}=83.5 \mathrm{kHz}\) and \(v_{\text {bat }}=3.9 \mathrm{~m} / \mathrm{s},\) calculate the speed of the insect.

Step by step solution

01

Understanding the Doppler Effect

The change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source is known as the Doppler effect. The student should know that when the source of the sound and the observer are getting closer, the frequency becomes higher since the waves are compressed. To calculate the modified frequency due to Doppler Effect, the formula is \[f' = f \frac{v ± v_o}{v ∓ v_s}\], where \(v\) is the speed of sound in air, \(v_o\) is the velocity of the observer, \(v_s\) is the velocity of the source, and \(f'\) is the observed frequency.

02

Applying the Doppler Effect

First, when the bat emits the sound, we consider the bat as the source and the insect as the observer. Applying the Doppler Effect formula, we get \[f’ = f_{\text{bat}} \frac{v + v_{\text{insect}}}{v - v_{\text{bat}}}\]. When the insect reflects the sound, we consider the insect as the source and the bat as the observer, we get \[f_{\text{refl}} = f’ \frac{v - v_{\text{bat}}}{v + v_{\text{insect}}}\]. We substitute the first equation into the second equation to get \[f_{\text{refl}} = f_{\text{bat}} \frac{v + v_{\text{insect}}}{v - v_{\text{bat}}} \cdot \frac{v - v_{\text{bat}}}{v + v_{\text{insect}}}\]. Simplifying, we get the equation given in the exercise.

03

Solving for the insect's speed

We use the equation derived in Step 2 and substitute the given values into it to calculate the speed of the insect. We have \(f_{\text{bat}} = 80.7\, \text{kHz}\), \(f_{\text{refl}} = 83.5\, \text{kHz}\), \(v_{\text{bat}} = 3.9\, \text{m/s}\), and \(v = 343\, \text{m/s}\) (speed of sound in air). After making substitutions and calculating, we find the speed of the insect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sound Frequency and the Doppler Effect

When we talk about the Doppler Effect, we're delving into how motion alters our perception of sound frequency. This phenomenon is a daily experience, like when an ambulance speeds past you, its siren's pitch drops suddenly as it moves away. This shift occurs because the frequency of sound waves changes depending on the relative speed of the wave source and the observer.

Sound frequency, measured in hertz (Hz), represents the number of sound wave cycles that pass a point per second. A higher frequency means a higher pitch sound. If a sound source like our Rhinolophus bat approaches an insect, the sound waves in front compress, increasing the frequency and pitch heard by the bat. The opposite occurs when the bat moves away; the waves spread out, decreasing frequency and pitch. The Doppler Effect is integral to bats using echolocation to hunt, allowing them to determine the speed and distance of their prey based on these frequency changes.

Wave Source and Observer Motion

How does movement come into play with the Doppler Effect? The key is in differentiating the roles of the wave source and the observer. In our exercise, the bat is initially the 'wave source', and the insect, as it receives the sound, is the 'observer'. Their relative speeds dictate the frequency change.

As the roles reverse when the sound reflects off the insect back to the bat, both participants have 'sounded' and 'listened'. The observed frequency is affected by whether they're moving towards each other, away, or one is stationary. This is why it's important to consider these roles when applying the Doppler Effect formula, as we are catering to two scenarios of sound emission and reflection. In these intricate natural dances, the math behind them helps us understand underlying principles for navigation and survival in animal behavior.

Determining Speed with the Speed of Sound

Now, let's focus on the speed of sound, which is crucial for solving our exercise. The speed of sound (\( v \) in our formulas) in air at standard temperature and pressure is approximately 343 meters per second (m/s). This is a constant that can really anchor our calculations in reality.

When the bat emits a sound that bounces off an object and returns, echolocation depends on knowing this constant speed of sound. Why? Because it allows the bat to process the time it takes for the sound to travel to and from the object, thus providing data on distance and object's speed. The faster the insect flies towards the bat, the shorter the time for sound to return, making the wave frequency higher. Through careful analysis of these wave patterns, the bat can even determine the precise speed of the insect, which is crucial for a successful hunt. Our exercise uses this exact principle, as we solve for the insect's speed based on the frequencies of the sound wave emitted and reflected.