Velocity of a Wave on a String

The velocity of a wave on a string fixed at both ends is related to several factors, as we have seen earlier. The equation that relates all the variables that affect the velocity is: v = (F/μ)^1/2 where F = tension in the string, in N and μ = the linear mass density of the string.

 

• In other word, μ is the mass/unit length, in units of kg/m.

 

• According to the above formula, as the tension increases the velocity increases and as the mass increases the velocity decreases.

 

• We can substitute fλ for v and we now have fλ = (F/μ)^1/2.

Fundamental Frequency and Harmonics

When a string vibrates in its lowest frequency of vibration, it is said to be at its fundamental (or first harmonic). For a string clamped at both ends, this means that only a wavelength vibrates on the length of the string. Therefore , λ = 2L.

• All other natural vibrations are multiples of this fundamental frequency, or harmonics.

 

• By increasing the frequency of the vibrating string, the string will now vibrate in the second harmonic and one complete wavelength will fit on the string. Therefore, λ = L.

 

• Increasing the frequency again, the string will vibrate in the third harmonic and 1.5 wavelength will fit on the string of length L. Therefore, λ = 2L/3.

 

• If the fundamental frequency had been 50 Hz, the second harmonic would be 100 Hz, and the third harmonic would be 150 Hz, etc.

 

• Since this standing wave has points of no movement, called nodes, the number of nodes is equal to one greater than the harmonic. For the fundamental, there are two nodes, for the second harmonic there are three nodes, etc.

 

Resonance

Resonance is the maximum amplitude reached when the frequency of the driving force equals the natural frequency of the system.  Forced Vibration is the frequency set up from the driving force.  Open and closed tubes resonant at the natural frequency of the tube and whole number multiples (harmonics).  Musical instruments that are made of an air column, such as the flute, depend on the physics of resonance for different lengths of the column.

Open Tubes

An open tube is one in which the tube is open at BOTH ends. An antinode exists at each end, so the node(s) must exist somewhere in the length of the air column.

• The fundamental (first harmonic) has one node in the middle and antinodes at each end. Therefore, 1/2 of a wavelength fits in the tube and L = λ/2 and λ ₁ = 2L.

 

• The 2^nd harmonic has antinodes at both ends and two nodes equally spaced. Therefore, one wavelength fits in the tube and L = λ₂.

 

• The 3^rd harmonic has antinodes at both ends and three nodes equally spaced. Therefore, 1.5 wavelengths fits in the tube and L =3/2λ and λ₃ = 2L/3.

 

• The expression that relates the frequency, harmonic number, velocity, and length of the air column is :

 

f_n = nv/2L

 

Where n = harmonic #, whole numbers, no units

 

v = velocity of sound wave in air in the tube, in m/s

 

L = length of the air column, in m

Closed Tubes

A closed tube is one in which the tube is open at only ONE end and closed at the other. The closed side has a node and the open side has an antinode.

• The fundamental (1^st harmonic) has only the node at one end and the antinode at the other end. Therefore, only 1/4 of a wavelength fits in the tube and L= λ/4 and λ₁ = 4L.

 

• Closed Tubes can only accommodate odd numbered harmonics; it is physically impossible to have a node and an antinode at each end for even harmonics.

 

• The 3^rd harmonic has a node and antinode at each end and another node inside the tube. Therefore, 3/4 of a wavelength fits in the tube and L=3/4λ and λ₃ = 4L/3.

 

• The expression that relates frequency, harmonic number, velocity and length of the air column is:

f_n = nv/4L

 

Where n = harmonic #, only odd numbers, no units

 

  v = velocity of sound wave in air in the tube, in m/s

 

L = length of the air column, in m

(source)