Understanding the Behavior of Standing Waves in a Fixed String and Calculating Their Frequency

Understanding the physics behind standing waves occurring in a string fixed at both ends is crucial in many practical applications, from vibrations analysis to musical instruments. In this article, we delve into the behavior of a standing wave, defining its characteristics and illustrating how to calculate its frequency. We will solve a specific problem to demonstrate the application of this knowledge.

What is a Standing Wave?

A standing wave is a wave pattern that appears to remain in a fixed position. The energy in a standing wave does not travel along the string but is confined within the region where it is produced. In such a wave, different particles move with different amplitudes. The particles at the nodes always remain at rest, while particles at points of maximum displacement (antinodes) oscillate between their maximum and minimum positions. This oscillation creates a distinctive pattern where the wave appears to stand still, hence the name standing wave.

Rather than providing a direct answer to the problem at hand, let us first explore the properties of standing waves, which are:

Disturbances in a standing wave are confined to the region where it is produced. Vibrating particles move with different amplitudes. Particles at the nodes always remain at rest. All particles between two successive nodes reach their extreme positions together, thus moving in phase. The energy in one region of the standing wave is always confined within that region.

Problem: Calculating the Frequency of a String

Given that the velocity of a wave in a string is 2 m/s and the string forms standing waves with nodes 5.0 cm apart, let us calculate the frequency of vibration of the string.

Step-by-Step Calculation

To solve this problem, we need to use the relationship between frequency, velocity, and wavelength in a standing wave. The formula that relates these variables is:

[ f frac{v}{lambda} ]

Where:

( f ) is the frequency (in Hertz, Hz) ( v ) is the velocity of the wave (in meters per second, m/s) ( lambda ) is the wavelength (in meters, m)

From the problem statement, we know:

The velocity of the wave, ( v ), is 2 m/s. The distance between two nodes is 5 cm, which is half the wavelength. Therefore, the wavelength ( lambda ) is twice the distance between two nodes.

Let's calculate the wavelength:

[ lambda 2 times 5 text{ cm} 10 text{ cm} 0.1 text{ m} ]

Now, substitute the values into the formula:

[ f frac{2 text{ m/s}}{0.1 text{ m}} 20 text{ Hz} ]

Thus, the frequency of vibration of the string is 20 Hz.

Conclusion

The frequency of vibration of the standing wave in this string is 20 Hz, based on the given wave velocity and node distance. By understanding the properties of standing waves, we can calculate important parameters such as frequency, which is essential for various applications in physics, engineering, and music.

Further Reading

Further investigation into the behavior of standing waves can provide deeper insights into their applications in different fields. For more detailed information and additional problems, refer to resources such as Khan Academy and other reputable sources.