Understanding the Frequency of a Wave in Its 2nd Harmonic for a Given Wavelength

Harmonics are a crucial concept in wave physics, and understanding the frequency of waves in these harmonics is essential for numerous applications, from music to physics experiments. In this article, we delve into the specifics of a wave vibrating in its 2nd harmonic, given a fixed wavelength of 4.80 meters.

What is a Harmonic?

A harmonic is a component frequency of a periodic wave that is an integer multiple of the fundamental frequency and whose wavelength is a fraction of the wavelength of the fundamental wave. The fundamental frequency is the lowest frequency of a wave that can be produced by a vibrating object, and higher frequency waves are known as harmonics.

Harmonic Frequency Calculation

For a string that is fixed at both ends, the harmonic frequencies are given by the equation:

fn n * (v / λn)

Where:

fn is the frequency of the nth harmonic, n is the harmonic number (1 for the fundamental, 2 for the 1st overtone, 3 for the 2nd overtone, etc.), v is the wave velocity in the string, λn is the wavelength of the nth harmonic.

For the 2nd harmonic, the wavelength is half that of the fundamental. Mathematically, this can be expressed as:

λ2 λf / 2

Example Calculation

To calculate the frequency of the 2nd harmonic, we need additional information, such as the wave velocity in the string. The wave velocity in a string can be calculated using the following formula:

v √(T / μ)

Where:

T is the tension in the string, μ is the mass per unit length of the string.

Given a wavelength of 4.80 meters for the 1st harmonic, the wavelength for the 2nd harmonic would be:

λ2 4.80 / 2 2.40 meters

Impact of String Properties

The frequency and wavelength of the harmonic are greatly affected by the properties of the string, particularly the tension and the mass per unit length. A higher tension or a lower mass per unit length will result in a higher frequency for the 2nd harmonic.

Conclusion

Understanding the frequency of a wave in its 2nd harmonic, given a specific wavelength, requires additional information such as the wave velocity in the string. The wave velocity itself is dependent on the tension and the mass per unit length of the string. By grasping these relationships, one can accurately predict and analyze wave phenomena in a wide array of applications.

Key Points to Remember:

The wavelength of the 2nd harmonic is half that of the fundamental. The frequency of the 2nd harmonic is twice the frequency of the fundamental. The wave velocity in the string determines the exact frequency of the harmonic.

For further reading and detailed calculations, the following resources are recommended:

Duffy's Online Physics Textbook: Waves Physics Classroom: Harmonic Frequency