Problem 13 - Pearson Physics

Question

The frequency of a sound is doubled. Does the wave speed of this sound increase, decrease, or stay the same? Explain.

Step-by-Step Solution

Answer

The wave speed stays the same.

Step 1: Understanding Wave Properties

In physics, the speed of a wave is determined by the medium through which it travels and is described by the equation: \( v = f \lambda \), where \( v \) is the wave speed, \( f \) is the frequency, and \( \lambda \) is the wavelength. For sound waves, the speed of sound in a given medium (like air, water, etc.) generally remains constant, regardless of changes in frequency.

Step 2: Analyzing the Effect of Frequency Doubling

If the frequency of the sound is doubled, \( f' = 2f \). According to the wave equation \( v = f \lambda \), if \( v \) is constant and \( f \) is doubled, then the wavelength, \( \lambda \), must be halved to maintain the equality, because \( v = f' \lambda' = 2f \cdot \frac{1}{2}\lambda = f \lambda \). Thus, the wave speed \( v \) remains unchanged.

Step 3: Conclusion

Since the wave speed \( v \) is a property of the medium and does not change when the frequency changes, the wave speed stays the same even when the frequency of the sound is doubled.

Key Concepts

Sound Frequency

When discussing sound, the frequency is a key concept that helps us understand the pitch of the sound. Frequency refers to the number of complete wave cycles passing a specific point per second, measured in Hertz (Hz).
For sound, a higher frequency means a higher pitch, as in the case of a whistle, while a lower frequency corresponds to a lower pitch, like a drum sound. Understanding how frequency affects sound is essential in fields ranging from music to acoustics. Changing the frequency does not alter the speed of sound in a medium, as wave speed depends on the medium's properties like temperature and pressure, not the sound's frequency.

Wavelength

Wavelength is another crucial component in understanding wave mechanics. It represents the distance between two corresponding points in consecutive cycles of a wave, such as crest to crest or trough to trough.
Wavelength is denoted by the Greek letter \( \lambda \) (lambda) and is measured in meters.An inverse relationship exists between frequency and wavelength. If a sound's frequency increases, its wavelength decreases, and vice versa. This inverse relationship keeps the wave speed constant in a particular medium, as the product of frequency and wavelength must equal the wave speed, expressed with the equation: \( v = f \lambda \).
Therefore, doubling a sound's frequency results in halving its wavelength, ensuring the speed of sound remains unchanged.

Wave Equation

The wave equation is a vital tool in understanding the relationship between wave speed, frequency, and wavelength. It is represented by the formula: \( v = f \lambda \). Here, \( v \) denotes wave speed, \( f \) stands for frequency, and \( \lambda \) symbolizes wavelength.The equation tells us that wave speed is the product of frequency and wavelength. In a given medium, like air or water, sound travels at a consistent speed. This means if one of the variables changes (say, frequency doubles), the other adjusts to keep the speed consistent. In our example, doubling the frequency leads to halving the wavelength.
Understanding these relationships allows us to predict how sound behaves in different environments and how changes to one property can affect others, while always maintaining equilibrium within the medium.