When a string under tension is set into vibration, transverse harmonic rays propagate along its length. When the size of string is constant, reflected rays will also exist. The operation and reflected waves will superimpose to produce transverse stationary waves in a string.
Incident wave y1 = a sin2π/λ(vt + x)
Reflected wave y2 = a sin2π/λ[(vt - x)+] = -asin2π/λ(vt - x)
According to superposition principle : y = y1 + y2 = 2 a cos2πvt/λ sin2πx/λ
General formula for wavelength λ=2L/n where n = 1, 2, 3, ... correspond to 1st , 2nd, 3rd modes of vibration of the string.
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(1) First normal mode of vibration : n1= v/λ1 = v/2L n1=1/2L√T/M
This mode of vibration is called the fundamental mode and the frequency is called fundamental frequency. The voice from the note so given is called fundamental note or first harmonic.
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(2) Second normal mode of vibration :
n2= v/λ2 = v/L 2n1
This is second harmonic or first over tone.
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(3) Third normal mode of vibration : n3= v/λ3 = 3v/2L = 3n1
This is third harmonic or second overtone.
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Position of nodes : x = 0,L/n, 2L/n,3L/n.............L
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 : x = 0,L/2n, 3L/2n,5L/2n.............(2n-1)L/2n
For first mode of vibration x = L/2 [One antinode]
For second mode of vibration x = L/4,3L/4 [Two antinode]
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