Problem 87 - Precalculus Mathematics for Calculus

Question

As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the observer than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$ P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right) $$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(s_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{n}=440 \mathrm{Hz}\). Graph the function \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically? (IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

The vertical asymptote at 332 m/s indicates infinite pitch when the train reaches the speed of sound.

Step 1: Understand the Function

The function given is \( P(v) = P_0 \left( \frac{s_0}{s_0 - v} \right) \). In this case, \( P_0 = 440 \, \text{Hz} \) (the actual pitch of the whistle) and \( s_0 = 332 \, \text{m/s} \) (the speed of sound in air).

Step 2: Identify the Vertical Asymptote

The function has a vertical asymptote where the denominator equals zero. Hence, solve \( s_0 - v = 0 \) to find the value of \( v \), which yields \( v = s_0 = 332 \, \text{m/s} \).

Step 3: Interpret the Vertical Asymptote

The vertical asymptote at \( v = 332 \, \text{m/s} \) indicates that as the train's speed approaches the speed of sound (332 m/s), the pitch \( P(v) \) approaches infinity. This is because the sound waves are infinitely compressed at the speed of sound, leading to an infinitely high pitch due to the Doppler effect.

Step 4: Plot the Graph

Using a graphing device, plot \( y = P(v) = 440 \frac{332}{332-v} \). Notice the function will have a vertical asymptote at \( v = 332 \, \text{m/s} \), and \( y \) approaches infinitely large as \( v \) gets closer to this value.

Key Concepts

Sound Waves

Sound waves are disturbances that travel through a medium, like air, causing particles in the medium to oscillate back and forth, transferring energy as they go. This oscillation generates areas of compression and rarefaction, leading to the wave-like phenomena we hear as sound. The frequency of these waves determines the pitch, and therefore, plays a key role in phenomena like the Doppler effect.
In the context of the Doppler effect, sound waves become particularly interesting. As a source of sound, like a train whistle, moves toward an observer, the crests of the sound waves are pressed closer together because of the motion. This results in an increased frequency of the waves reaching the observer's ears, which in turn, makes the observed pitch higher than if the source were stationary. The science behind sound waves helps us better understand how their interaction with movement leads to real-world phenomena that even have technical applications in areas like radar and astronomy.

• Sound waves travel through different media, but their speed can vary depending on the medium.
• The Doppler effect is a vivid example of how the speed of an object and sound waves interact, altering perception.
• Real-world applications harnessing this effect include sonar, medical imaging, and speed detection by law enforcement.

Vertical Asymptote

A vertical asymptote in a function is a line where the function's value grows without bound as the input approaches some critical value. In simpler terms, as the input gets very close to a specific number, the value of the output becomes extremely large. In purely mathematical terms, this occurs when a denominator of a fraction equals zero at a particular point.
In our exercise, the function representing the pitch of the sound, given the speed of the train, is a classic case. We identified that at a speed (v) of 332 m/s, the denominator of our function equals zero, creating a vertical asymptote. This point is very significant; mathematically, as the train speeds up towards this specific velocity, our pitch function shoots towards infinity.

• Vertical asymptotes often denote a boundary that cannot be surpassed by the function’s output.
• They help highlight the behavior of functions around potential 'problem spots'.
• In practical scenarios, they can indicate a point of physical limitation or transition, as seen in the Doppler effect study.

Graphing Functions

Graphing functions provide a visualization of how a function behaves as inputs change, making abstract mathematical concepts more tangible. In graphing the function representing the Doppler effect in this exercise, we reveal insights into how the pitch changes as a train approaches or moves away from the speed of sound.
Details about the function being graphed include the actual pitch of the whistle, 440 Hz, and the speed of sound, 332 m/s. By plotting this function, you can observe how steeply the pitch rises as the train accelerates towards the speed of sound. This vertical rise refers back to our vertical asymptote discussion.

• Charts and graphs can help simplify complex mathematical relationships by presenting them visually.
• Understanding the general shape and behavior can also aid in anticipating how a function will act before any sophisticated calculations.
• The graph for this function illustrates how real-world phenomena, like sound wave compression, manifest mathematically.