Principle of Superposition of Wave
We have so far discussed the propagation of a single wave in a medium. In practice, however, one often comes across the possibility of two or more waves travelling through a medium simultaneously. A boatman may receive the sound wave from the siren of a ship directly from it as well as the wave reflected by the seawater. A floating log may simultaneously receive water waves created by two or more boats. The transverse wave on a string will eventually get reflected at the fixed end and we have two waves travelling in opposite directions on the string. We use the term interference to describe the physical effects of superposition of two or move waves travelling through the same region of space.
The Superposition Principle
Consider two waves travelling in a medium. Let us focus our attention on the motion of one particle of the medium. When a wave reaches this particle, it suffers one displacement. When two waves simultaneously cross this particle, it suffers two displacements, one due to each wave. The resultant displacement of the particle is given by the algebraic sum of the individual displacements given to it by the two waves. This is the principle of superposition.
Superposition of Two pulses
Let us apply the superposition principle to find out what happens when two pulses of identical shapes travelling in opposite directions on a string cross each other. Figure (i) shows the two pulses moving towards each other with a speed of 1 m s^{1}. Figures 13.10a to f show the wave form after every second. They cross each other between t = 2 s and t = 3 s as shown in Figures (i) (c) and (d). These figures show that when the two pulses overlap (or superpose) the resultant wave profile is the algebraic sum of the displacements due to each pulse.
Figure (i)  Superposition of two identical pulses travelling in opposite directions 
Figure (ii) shows what happens when two equal and opposite pulses moving in opposite directions cross each other. In Figure (ii) (c) we see that there is an instant when the string appears undisturbed. In this situation the positive pulse overlaps the negative pulse and seem to cancel each other. This again shows that the resultant wave profile is the algebraic sum of individual wages.
From Figures (i) and (ii) we notice that two pulses continue to retain their individual shapes after crossing each other. However, at the instant they cross each other, the appearance of the wave profile is different from the shape of either individual pulse.
Figures (i) (c) and (d) are examples of constructive interference and Figure (ii) (c) is an example of destructive interference.
Figure (ii)  Superposition of two equal and opposite pulses travelling in opposite directions 
Superposition of Harmonic Waves
You must have observed the interference pattern produced when circular waves, resulting from dropping two stones into a pond of water, cross over each other. In the laboratory this interference can be demonstrated by attaching a stylus to each prong of a tuning fork and placing it over a big ripple tank. The fork is set into vibration and each stylus is brought into contact with water. Two sets of concentric waves start simultaneously from the point of contact of each stylus (Fig. 13.12). The wave trains travel in the form of crests and troughs on the surface of water. The solid circles represent crests and the dotted ones troughs.
Figure (iii)  Interference pattern 
At points like Q where two crests intersect, a larger crest is formed, the displacements of the particle due to two waves adding up. This is called constructive interference. At points like P where a crest of one wave and a trough of the other intersect, they cancel each other. At these points the water surface is level and the interference is said to be destructive. At points like R two troughs intersect, the two negative displacements of the same particle add up to give a bigger trough. As these waves spread out and reach the walls of the tank, they get reflected and we get a stationary interference pattern.
We have given above a graphical description of the phenomenon of superposition of waves. A mathematical description of the superposition can be given in terms of addition of wave functions. If two or move waves travelling in a medium superpose, the resultant wave function is obtained by the sum of the wave functions of the individual waves, i.e. if the wave functions of the waves are
y_{1}(x, t) = f_{1}(Ï…t  x)
y_{2}(x, t) = f_{2}(Ï…t â€“ x)
. .
. .
. .
y_{n}(x, t) = f_{n}(Ï…t â€“ x)
then the resultant wave is give y by
y = y_{1}(x, t) + y_{2}(x, t) + â€¦ + y_{n }(x, t)
y = f_{1}(Ï…t â€“ x) + f_{2}(Ï…t  x) + â€¦ +f_{n }(Ï…t  x)
This is the principle of superposition. This principle holds only if the amplitude of the disturbance is not too large. In the following two sections we will use the principle of superposition to discuss standing waves and beats.
Interference Of Waves
Suppose we send two sinusoidal waves of the same wavelength and amplitude in the same direction along a stretched string. The superposition principle applies. What resultant wave does it predict for the string?The resultant wave depends on the extent to which the waves are in phase (in step) with respect to each other, that is, how much one wave form is shifted from the other wave form. If the waves are exactly in phase (so that the peals and valleys of one are exactly aligned with those of the other without any shift), they combine to double the displacement of either wave acting alone. If they are exactly out of phase (the peaks of one are exactly aligned with the valleys of the other), they cancel everywhere and the string remains straight. We call this phenomenon of combining and canceling of waves interference, and the waves are said to interfere. (These terms refer only to the displacements of the waves; the travel of the waves is unaffected)
Let one wave traveling along a stretched string be given by
y_{1}(x.t) = y_{m}sin (kx  Ï‰ t)
and another, shifted from the first, by
y_{2}(x,t) = y_{m}sin (kx  Ï‰ t + Ï• )
v = (speed)  (i)
yâ€™ (x, t) = y_{1}(x,t) + y_{2}(x,t)  (ii)
These waves have the same angular frequency Ï‰ (and thus the same frequency f), the same angular wave number k (and thus the same wavelength Î» ), and the same amplitude y_{m}. They travel in the same direction, that of increasing x, with the same speed, given by Equation (i). They differ only by a constant angle Ï• , which we call the phase constant. These waves are said to be out of phase by Ï† or to have a phase difference of Ï† , or one wave is said to be phase shifted from the other by Ï† .
From the principle of superposition Equation (ii), the combined wave has displacement
yâ€™ (x,t) = y_{1}(x,t) + y_{2}(x,t) = y_{m}sin (kx  Ï‰ t) + y_{m}sin (kx  Ï‰ t + Ï† )  (iii)
We can write the sum of the sines of two angles Î± and Î² as
sin Î± + sin Î² = 2 sin
Applying this relation to Equation (iii) yields
yâ€™ (x,t) = [2y_{m}cossin (kx  Ï‰ t +  (iv)
The resultant wave is thus also a sinusoidal wave traveling in the direction of increasing x. It is the only wave you would actually obtain.
The resultant wave differs from the original waves in two respects:
 its phase constant is and
 its amplitude yâ€™_{m} is the quantity in the brackets in Equation (iv)
yâ€™_{m} = 2y_{m}cos  (v)
If Ï† = 0 rad (or 0^{0}), the two combining waves are exactly in phase. Then we get
yâ€™ (x,t) = 2y_{m}sin(kx  Ï‰ t) (Ï† = 0) (vi)
Note that the amplitude of the resultant wave is twice the amplitude of either combining wave. This is the greatest amplitude the resultant wave can have, because the cosine term in Eq.15.18 and Eq.15.19 has its greatest value (unity) when Ï† = 0. Interference that produces the greatest possible amplitude is called fully constructive interference.
If Ï† = Ï€ rad (or 180^{0}), the combining waves are exactly out of phase. Then cos becomes cos , and the amplitude of the resultant wave is zero. We the have for all values of x and t,
yâ€™(x,t) = 0 (Ï† = Ï€ rad)
Figure  Two identical waves, y1(x,t) and y2(x,t), traveling in the same direction along a string, interfere to give a resultant wave yâ€™(x,t). 
 If the waves are exactly in phase, they undergo fully constructive interference and produce a resultant wave of twice their own amplitude.
 If they are exactly out of phase, they undergo fully destructive interference and the string remains straight.
Now, although we sent two waves along the string, we see no motion of the string. This type of interference is called fully destructive interference.
A phase difference Ï† = 2Ï€ rad (or 360^{0}) corresponds to a shift of one wave relative to the other wave by a distance of one wavelength. So phase differences can also be described in terms of wavelengths. For example, in Figure (b) the waves may be said to be 0.50 wavelength out of phase. Table 1 shows some other examples of phase differences and the interference they produce. Note that when interference is neither fully constructive nor full destructive, it is called intermediate interference. The amplitude of the resultant wave is then intermediate between 0 and 2y_{m}.
Two waves with the same wavelength are in phase if their phase difference is zero or any integer number of wavelengths. Thus the integer part of any phase difference expressed in wavelengths may be discarded. For example, a phase difference of 0.40 wavelength is equivalent in every way to one of 2.40 wave lengths, and so the simpler of the two numbers can be used in computations.
Phase Differences and Resulting Interference Tyres
Phase Difference, In 
Amplitude of Resultant Wave 
Type of Interference 

Degrees 
Radians 
Wavelengths 

0 120 180 240 360 865

0 2Ï€ /3 Ï€ 4Ï€ /3 15.1 
0 0.33 0.50 0.67 1.00 2.40 
2y_{m} 0 Y_{m } 2y_{m } 0.60y_{m} 
Fully constructive Intermediate Fully destructive Intermediate Fully constructive Intermediate 