Problem 79 - Algebra and Trigonometry

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 \(P_{0}=440 \mathrm{Hz}\) . Graph the function \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically?

Step-by-Step Solution

Answer

The vertical asymptote at \(v = 332 \text{ m/s}\) represents the train's speed matching the speed of sound, resulting in an undefined pitch due to infinitely compressed sound waves.

Step 1: Understanding the Problem

We're given the function \(P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right)\), which models the Doppler effect. Here, \(P_{0}=440 \text{ Hz}\) and \(s_{0}=332 \text{ m/s}\). We need to graph this function with respect to \(v\) and identify any vertical asymptotes.

Step 2: Identify Vertical Asymptote

A vertical asymptote occurs where the denominator of a function is zero. For the function \(P(v)=440\left(\frac{332}{332-v}\right)\), set the denominator \(332-v=0\). Solving for \(v\), we find \(v=332 \text{ m/s}\). This is where the vertical asymptote occurs.

Step 3: Graphing the Function

Using a graphing device, plot the function \(P(v)=440\left(\frac{332}{332-v}\right)\). You will see the graph climbing steeply upwards as \(v\) approaches 332 m/s from the left, indicating a vertical asymptote at \(v=332 \text{ m/s}\).

Step 4: Interpret the Vertical Asymptote Physically

The vertical asymptote at \(v=332 \text{ m/s}\) suggests a situation where the train's speed equals the speed of sound. Physically, this means the observer cannot hear the train's whistle as it approaches because the sound waves are infinitely compressed, resulting in an undefined or infinite frequency at this speed.

Key Concepts

Pitch

Pitch is how we perceive the frequency of a sound. In simpler terms, it helps us distinguish different sounds as low or high. For example, when a train approaches while blowing its whistle, the pitch might seem higher. This perceived change is due to the Doppler effect. The closer the source of the sound is moving towards the observer, the higher the pitch will be. Conversely, when it moves away, the pitch will seem lower.

Sound Waves

Sound waves are vibrations that travel through a medium such as air, water, or solids. In the Doppler effect, sound waves get compressed or stretched depending on the movement of the source.

• If a source moves towards you, the waves compress, making the sound seem higher.
• When the source moves away, the waves stretch, causing the sound to seem lower.

These changes in wave properties are essential in explaining why the sound of the train whistle changes as it moves past you.

Vertical Asymptote

A vertical asymptote is a point on a graph where lines approach a value but never actually touch it. In the case of the train's whistle, the function related to pitch becomes undefined at a certain speed.
For the function given, the vertical asymptote is at the speed of sound, 332 m/s. At this point, theoretically, the pitch becomes infinitely high. This is because the speed of the observer approaches the speed of sound, leading to a compression of sound waves that makes calculation of the pitch impossible by normal means.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium. In normal atmospheric conditions, this speed is approximately 332 m/s or 740 miles per hour.
When an object moves at this speed, sound waves can get dramatically compressed, leading to phenomena such as sonic booms or the disappearance of audible sound from an approaching source, a key aspect in understanding the Doppler effect and its implications in real-world situations.