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Standing waves on a string

When a string under tension is set into vibration, transverse harmonic waves propagate along its length (Fig. 3). When the length of string is fixed, reflected waves will also exist. The incident and reflected waves will superimpose to produce transverse stationary waves in a string
 
33561.png
Fig. 3
 
Incident wave, y1 = a sin 31890.png(vt + x)
 
Reflected wave, 31884.png
31878.png
 
According to superposition principle:
y = y1 + y2 = 2 a cos 31872.png
 
General formula for wavelength λ = 2L/n, where n = 1, 2, 3, … correspond to 1st, 2nd, 3rd, modes of vibration of the string.
  • First normal mode of vibration, 31859.png
     
    ⇒ 31853.png
     
    This mode of vibration is called the fundamental mode and the frequency is called fundamental frequency. The sound from the note so produced is called fundamental note or first harmonic.
     
    33584.png
     
    Fig. 4
  • Second normal mode of vibration:
     
    31741.png
     
    This is the second harmonic or first over tone.
     
    33613.png
     
    Fig. 5
  • Third normal mode of vibration:
     
    31677.png
     
    This is the third harmonic or second overtone.
     
    33633.png
     
    Fig. 6

Position of nodes

31490.png
 
For first mode of vibration, x = 0, x = L [two nodes]
 
For second mode of vibration, x = 0, x = L/2, x = L [three nodes].
 
For third mode of vibration, x = 0, x = L/3, x = 2L/3, x = L [four nodes].

Position of antinodes

31472.png
 
For first mode of vibration, x = L/2 [one antinode].
 
For second mode of vibration, x = 31458.png, [two antinode].




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