Problem 86 - Pearson Physics

Question

You are heading toward an island in your speedboat when you see a friend standing on shore. You sound the boat's horn to get your friend's attention. Is the wavelength of the sound produced by the horn greater than, less than, or equal to the wavelength of the sound heard by your friend? Explain.

Step-by-Step Solution

Answer

The wavelength heard by your friend is less than the wavelength produced.

Step 1: Understand the Doppler Effect

The Doppler Effect describes how the frequency (and consequently the wavelength) of waves changes when the source of the wave is moving relative to the observer. When the source approaches the observer, the waves are compressed, leading to a higher frequency and a shorter wavelength. Conversely, when the source moves away, the waves are stretched, resulting in a lower frequency and a longer wavelength.

Step 2: Identify the Situation

In this case, you are on a boat moving toward the island where your friend is standing. You sound the horn (the source of the sound) as you move closer to your friend (the observer).

Step 3: Apply the Doppler Effect

Since you are moving toward your friend, the sound waves of your horn will be compressed as they travel to your friend. This will increase the frequency of the sound your friend hears compared to the frequency at which it was produced.

Step 4: Compare the Wavelengths

Using the relation between frequency and wavelength, \[ v = f \lambda \] where \( v \) is the speed of sound, \( f \) is the frequency, and \( \lambda \) is the wavelength. As the frequency perceived by your friend increases, the wavelength of the sound they hear decreases.

Key Concepts

Understanding Wavelength

The wavelength of a wave is the distance between successive crests, troughs, or identical points of the wave. It is often denoted by the Greek letter \( \lambda \). Wavelengths are crucial in understanding the properties of waves, including sound waves. In the context of the Doppler Effect, when the source of a sound wave is moving relative to an observer:

• The wavelength becomes shorter if the source approaches the observer.
• The wavelength becomes longer if the source moves away.

For the exercise scenario, as you drive your speedboat towards your friend, the sound waves of the horn are compressed. This means the wavelength of the sound reaching your friend is shorter than what was originally emitted.

The Role of Frequency

Frequency refers to how often waves pass a point in a given time period and is measured in hertz (Hz). It is a key factor in how we perceive sound, influencing the pitch of the sound. Relating this to the Doppler Effect:

• The frequency increases when the sound source approaches the observer.
• The frequency decreases when the source moves away from the observer.

Using the relationship \( v = f \lambda \), we can see that for a constant speed of sound \( v \), if the frequency \( f \) increases, the wavelength \( \lambda \) must decrease. That's exactly what happens in the exercise: as the source of the sound (the boat) moves closer to your friend, the frequency they perceive is higher, and the wavelength is consequently shorter.

Understanding Sound Waves in Motion

Sound waves are a type of mechanical wave that require a medium to travel through, such as air, water, or solid materials. These waves are created by the vibration of objects and propagate as longitudinal waves. In the context of this exercise, to truly grasp how the Doppler Effect and sound waves interact:

• Remember that sound waves from the boat's horn travel through the air to your friend onshore.
• As the boat moves toward the shore, the speed of sound remains constant, but the frequency and wavelength change due to relative motion.

The sound waves are compressed because of the boat's motion towards your friend, leading to a higher pitch sound than originally produced. This is a direct demonstration of the Doppler Effect in real-life situations, providing a clear understanding of how motion affects wave properties.