Understanding Sound Waves: Frequency, Wavelength, and Velocity

Let's explore the relationship between the properties of a sound wave: velocity, frequency, and wavelength. This relationship is fundamental to understanding wave behavior.

The velocity ($\mathbf{v}$) of a wave is the speed at which it travels through a medium. For sound, this speed depends on the properties of the medium, such as its temperature and density. The frequency ($\mathbf{f}$) of a wave is the number of wave cycles that pass a point per second, measured in Hertz (Hz). The wavelength ($\mathbf{\lambda}$) is the spatial period of the wave, or the distance over which the wave's shape repeats, measured in meters.

These three quantities are related by the following equation:

$\mathbf{v} = \mathbf{f\lambda}$

This equation tells us that the velocity of a wave is equal to the product of its frequency and its wavelength.

Analyzing the Relationship When Velocity is Constant

The question states that the velocity of the sound wave is given and constant. This means that if the frequency changes, the wavelength must also change in a way that keeps the product $f\lambda$ equal to the constant velocity $v$.

We can rearrange the formula to see how wavelength depends on frequency when velocity is constant:

$\mathbf{\lambda} = \frac{\mathbf{v}}{\mathbf{f}}$

From this rearranged formula, we can see that the wavelength ($\lambda$) is inversely proportional to the frequency ($f$) when the velocity ($v$) is held constant. This means if the frequency increases, the wavelength must decrease to keep the product constant. Conversely, if the frequency decreases, the wavelength must increase.

Evaluating the Options Based on the Inverse Relationship

Let's look at the given options in light of this understanding:

• Option 1: Wavelength will keep increasing and decreasing alternately. This is incorrect. With a constant velocity and a steady change in frequency (an increase in this case), the wavelength will change consistently, not alternately increasing and decreasing.
• Option 2: The wavelength will not be affected. This is incorrect. Since $\mathbf{v} = \mathbf{f\lambda}$ and $v$ is constant, if $f$ changes, $\lambda$ must also change.
• Option 3: Its wavelength will increase. This is incorrect. Because wavelength is inversely proportional to frequency ($\mathbf{\lambda} = \frac{\mathbf{v}}{\mathbf{f}}$) when velocity is constant, an increase in frequency leads to a decrease in wavelength.
• Option 4: Its wavelength will decrease. This is correct. When the frequency ($f$) of a sound wave increases and its velocity ($v$) remains constant, the wavelength ($\lambda$) must decrease according to the formula $\mathbf{\lambda} = \frac{\mathbf{v}}{\mathbf{f}}$.

Conclusion on Wavelength and Frequency

When the speed of sound in a medium is constant, an increase in the frequency of the sound wave results in a corresponding decrease in its wavelength. This inverse relationship is a key concept in wave physics.

Revision Table: Sound Wave Properties

Additional Information: Factors Affecting Sound Wave Velocity

While the question specifies a given, constant velocity, it's important to know that the speed of sound itself depends on the medium. For example:

• In air, the speed of sound changes with temperature. It is approximately 343 m/s at 20°C.
• Sound travels faster in liquids than in gases.
• Sound travels fastest in solids.

So, if the medium or its conditions (like temperature for air) were to change, the velocity ($v$) would change, which would then affect the relationship between frequency and wavelength for a wave passing through it.