When a wave passes through a point on a string, that point
oscillates. 

If the wave later returns to the same point (for example, after
reflection), interference will occur between the existing
oscillation and that due to the reflected wave. 

The interference will be constructive if
the "effective distance" moved by the wave is equal to a
whole
number of wavelengths. 

When a wave is reflected at a fixed end of a string, a
phase change occurs. 

A phase change of 180° makes a wave "look as if" it has
travelled an extra half wavelength. 

So an effective distance of
λ
could be due to the wave actually moving λ/2
but also experiencing a phase change of 180° or
πradians (see here for
illustration) 

A given string at a given tension can resonate at a
(theoretically infinite) series of frequencies. 

The lowest frequency of resonance is called the fundamental
frequency or the first harmonic of the string. 



Fundamental Frequency of Resonance
(First Harmonic) 





As the animation shows, all points oscillate in phase
but with different amplitudes of oscillation. 



Consider the string to be disturbed at A (the centre). 

Waves travel towards B (and C) and are reflected with with 180°
phase change. 

If the effective distance travelled by the waves is
λ, then resonance occurs. 

This means that, for the first resonance, the distance
A→B→A
(or A→C→A)
must be equal to λ/2. 

Therefore, at the fundamental frequency 



We conclude that the distance between adjacent nodes of
a stationary wave is equal to half the wavelength of the
travelling waves which are producing the stationary wave. 

The frequency at which this first resonance occurs is given the
symbol f_{o} 



Relation Between f_{o}
and the Length of the String 

If the distance A→B→A
(= L) is λ/2
then the time taken by the waves to move this distance is half
of a time period, T/2. 

This follows from the definition of one wavelength. 



If the wave velocity is v then 



and, remembering the relation between frequency and time period 



this gives us 



It can easily be shown that the velocity of propagation, v of
waves on a string under tension is given by 



where Te is the tension and m
is the mass per unit length of the string. 

So, the fundamental frequency, f_{o}
is given by 



and the frequencies of the higher harmonics are given
by 



where n = 1, 2, 3… etc 



The diagram below represents the second
harmonic (also sometimes called the first overtone) 



Assuming that this represents the same string under the same
tension as the first diagram, we can immediately see that the
wavelength is half that of the fundamental so f = 2f_{o}
(as shown in the animation). 
