Derive v = 2 l_nf_n for the nth harmonic

In summary, the equation for the nth harmonic of a vibrating string is v = 2 l_nf_n, where v is the velocity, l_n is the length of the string, and f_n is the frequency of the nth harmonic. This equation is derived from the fundamental principle of standing waves, which states that the wavelength of a standing wave on a string must be equal to twice the length of the string. The letter "n" represents the harmonic number, indicating the number of nodes or antinodes present in the standing wave. The harmonic number is squared because it represents the number of nodes present in the standing wave. This equation is used in various fields to study the properties of vibrating strings and other systems that exhibit standing waves.
  • #1
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Homework Statement


Derive the expression v = lnfn where ln is the shortest distance between nodes for the nth harmonic.


Homework Equations


v = wave speed
ln = shortest distance between nodes for the nthharmonic
fn = frequency of the nth harmonic


The Attempt at a Solution



Is it asking me to take the derivative of the wave speed, which I believe would give me the acceleration of the wave?

So,

wave acceleration = 2*ln

Am I making any sense?
 
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  • #2
nth harmonics of fo is n times fo. Similarly ln is equal to (lambda)o/2n.
Now proceed.
 

What is the equation for the nth harmonic of a vibrating string?

The equation is v = 2 l_nf_n, where v is the velocity, l_n is the length of the string, and f_n is the frequency of the nth harmonic.

How is this equation derived?

This equation is derived from the fundamental principle of standing waves on a string, which states that the wavelength of a standing wave on a string must be equal to twice the length of the string.

What does the letter "n" represent in this equation?

The letter "n" represents the harmonic number, which indicates the number of nodes or antinodes present in the standing wave. For example, the first harmonic (n=1) has one node and two antinodes, while the second harmonic (n=2) has two nodes and three antinodes.

Why is the harmonic number squared in this equation?

The harmonic number is squared because it represents the number of nodes that are present in the standing wave. Since standing waves occur when there is a fixed number of nodes and antinodes, the harmonic number must be squared to properly account for the number of nodes.

How is this equation used in scientific research?

This equation is used to calculate the velocity of a standing wave on a string, which is important in understanding the behavior of vibrations and waves. It is also used in various fields such as acoustics, music, and engineering to study the properties of vibrating strings and other systems that exhibit standing waves.

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