Understanding Sound Speed and Frequency in a Medium

The speed of sound in a particular medium is a fundamental property that describes how quickly sound waves travel through it. This speed is determined by the physical characteristics of the medium itself, such as its elasticity (or stiffness) and density, as well as environmental factors like temperature and pressure.

The relationship between the speed of sound (\(v\)), its frequency (\(f\)), and its wavelength (\(\lambda\)) is given by the equation:

\[ v = f\lambda \]

This equation tells us that speed is the product of frequency and wavelength. However, it's crucial to understand what *causes* the speed to change and how frequency and wavelength behave.

How Medium Properties Affect Sound Speed

In a given, uniform medium under constant conditions (like temperature and pressure), the speed of sound (\(v\)) remains constant. For example, the speed of sound in dry air at 20°C is approximately 343 meters per second, regardless of whether the sound is a low-pitched hum or a high-pitched whistle.

When sound travels from one medium to another (e.g., from air to water), its speed *does* change because the properties of the medium change. Also, changes in temperature or pressure within the same medium can affect the speed.

Analyzing the Relationship Between Speed, Frequency, and Wavelength

For a constant speed (\(v\)) in a given medium, the frequency (\(f\)) and wavelength (\(\lambda\)) are inversely proportional. This means:

• If the frequency (\(f\)) of the sound wave increases, its wavelength (\(\lambda\)) must decrease to keep the product \(f\lambda\) equal to the constant speed \(v\).
• If the wavelength (\(\lambda\)) increases, the frequency (\(f\)) must decrease to maintain the constant speed \(v\).

Think of it like this: if waves are arriving more frequently (higher frequency), the distance between successive wave crests (wavelength) must be smaller if they are all traveling at the same speed.

Evaluating the Options

Let's examine the given options based on our understanding of sound speed in a given medium:

• Option 1: Speed of sound remains same at all frequencies
  This aligns with the principle that in a non-dispersive medium (like air for typical sound frequencies), the speed of sound is a characteristic of the medium and does not depend on the frequency of the wave.
• Option 2: Speed of sound is faster at higher frequencies
  This is incorrect. The speed is determined by the medium, not the frequency. A higher frequency sound wave will simply have a shorter wavelength while maintaining the same speed.
• Option 3: Speed of sound is slower at higher frequencies
  This is also incorrect. Similar to Option 2, the speed is independent of frequency in a non-dispersive medium. A higher frequency implies a shorter wavelength for the same speed.
• Option 4: Speed of sound is slower at higher wavelengths
  This statement confuses the relationship. Wavelength and frequency are inversely proportional for a constant speed (\(v = f\lambda\)). If the wavelength is higher, the frequency is lower, but the speed \(v\) remains constant in the given medium.

Conclusion

In a specific medium under constant conditions, the speed of sound is a property of that medium and is independent of the frequency or wavelength of the sound wave. The frequency and wavelength adjust according to the relationship \(v = f\lambda\) to maintain this constant speed.

Revision Table: Sound Speed Concepts

Additional Information on Sound Speed

While sound speed is generally independent of frequency in air (making air a non-dispersive medium for sound), there are some exceptions:

• Dispersion: In certain media or under specific conditions (like ultrasonic frequencies in some materials, or sound propagation over very long distances in the atmosphere due to non-uniform conditions), the speed of sound can show slight dependence on frequency. This phenomenon is called dispersion.
• Supersonic Speed: When an object travels faster than the speed of sound in a medium, it creates a sonic boom. This speed limit is the speed of sound itself, which is a fixed value for the given conditions.
• Factors Affecting Speed in Air: The speed of sound in air increases with increasing temperature. It is also slightly affected by humidity but is largely independent of atmospheric pressure at constant density.

Understanding the relationship between sound speed, frequency, and wavelength is key to comprehending wave mechanics. In the context of typical sound waves in common media like air or water, the speed remains constant for all frequencies and wavelengths.