Problem 63 - Pearson Physics

Question

The wavelength of sound coming from a loudspeaker is doubled. Does the frequency of the sound waves increase, decrease, or stay the same? Explain. Does the speed of the sound waves increase, decrease, or stay the same? Explain.

Step-by-Step Solution

Answer

Doubling the wavelength decreases the frequency and the speed of sound remains the same.

Step 1: Understanding sound wave properties

Sound waves have two main properties: wavelength (\( \lambda \)) and frequency (\( f \)). These properties are related to the speed of sound (\( v \)) through the equation \( v = f \cdot \lambda \). In a medium like air, the speed of sound is constant at a given temperature and pressure.

Step 2: Analyzing the relationship between wavelength and frequency

Given the relationship \( v = f \cdot \lambda \), if the speed \( v \) is constant, any change in wavelength \( \lambda \) will inversely affect the frequency \( f \). Specifically, if \( \lambda \) is doubled, then \( f = \frac{v}{\lambda} \) implies that \( f \) is halved. Thus, doubling the wavelength decreases the frequency.

Step 3: Evaluating the speed of sound

The speed of sound \( v \) is determined by the properties of the medium, such as temperature and pressure, not by the frequency or wavelength. Therefore, when the wavelength of sound is doubled, the speed of sound remains the same because it is independent of these wave properties in a given medium.

Key Concepts

Wavelength of Sound Waves

The wavelength is a fundamental property of sound waves. Think of it as the distance between two consecutive crests or troughs of a wave. This distance affects how we perceive sound and is crucial in determining a wave's behavior.
In the context of sound, a longer wavelength implies that the wave travels a greater distance before repeating its cycle. This can impact how sound reflects off surfaces and is absorbed by materials. When you double the wavelength, you are making each cycle of the sound wave longer. But remember, wavelength is inversely related to frequency if the speed of sound remains constant.
Here's a handy way to visualize it:

• Longer wavelength: Lower frequency
• Shorter wavelength: Higher frequency

Understanding wavelength helps in various fields, from acoustics to music production.

Frequency of Sound Waves

Frequency refers to how often the crests or troughs of a wave pass a given point per second. It's measured in hertz (Hz), representing cycles per second. Frequency determines the pitch of the sound: the higher the frequency, the higher the pitch, and vice versa.
When the wavelength of sound is doubled, and because the speed of sound remains unchanged, the frequency decreases. Mathematically, this is expressed as:

• \[ f = \frac{v}{\lambda} \]
• where \( f \) is the frequency, \( v \) is the speed of sound, and \( \lambda \) is the wavelength.

The equation shows when wavelength \( \lambda \) doubles, frequency \( f \) halved, provided that the speed \( v \) is constant. This inverse relationship ensures that sound waves adjust either their frequency or wavelength to maintain constant speed in a given medium.

Speed of Sound in Different Media

The speed of sound refers to how quickly sound waves can travel through a medium. It's a constant in a given environment, affected by factors like medium type (air, water, steel) and conditions (temperature, pressure). Generally, sound travels faster in denser media, like water or steel, than it does through air.
The formula tying speed, frequency, and wavelength is:

• \[ v = f \cdot \lambda \]

When we double the wavelength and observe that the frequency halves, the speed of sound itself remains stable, assuming no change in the medium's conditions.
This constancy means:

• In the same medium, the speed of sound remains stable regardless of changes in frequency or wavelength.
• Audio engineers rely on this property to design better acoustics.
• In scientific experiments, knowing the medium ensures accurate speed measurements.

This understanding of sound speed helps us create environments where sound quality is optimized, invaluable in public speaking venues and recording studios.