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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
Fig. 3
Incident wave, y1 = a sin 31890.png(vt + x)
Reflected wave, 31884.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.
    Fig. 4
  • Second normal mode of vibration:
    This is the second harmonic or first over tone.
    Fig. 5
  • Third normal mode of vibration:
    This is the third harmonic or second overtone.
    Fig. 6

Position of nodes

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

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

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