When a wave is incident on the boundaries of the medium, totally the wave is reflected and partially transmitted (or refracted).
When you flip the end of a rope whose far end is tied to a rigid support, a pulse travels the length of the rope and is reflected back to you. The incident and reflected waves overlap.
This overlapping of waves is called interference.
When there are two boundary points or surfaces, such as a guitar string, that’s tied down at both ends we get repeated reflections.
Later we will see that in such situations, sinusoidal waves can occur only for certain special frequencies; which are called natural frequencies.
In this chapter our focus will be on the interference of mechanical waves. But interference is also important in nonmechanical (i.e. Electromagnetic) waves.
It explains the colours seen in soap bubbles.
Boundary Conditions :
Reflection of a wave pulse from some boundary depends on the nature of the boundary.
The following points should be remembered regarding reflection or refraction of any type of wave (whether mechanical or nonmechanical).
(a) The phase of a displacement wave changes by π in case of reflection from a denser medium, fixed end or rigid support.
(As discussed earlier also the difference of rare and denser medium for a wave is through its speed, not density of medium) i.e. if incident wave is given by
Yi = Ai Sin (ωt – kx)
The reflected wave from a denser medium will be
Yr = Ar Sin (ωt + kx + π)
(b) In case of reflection from a rare medium or free end there is no phase change. Plus (+) sign is now used between wt and kx because the reflected wave travels in negative x direction. Here is one exception: longitudinal pressure waves suffers a phase change of π on reflection from a free or open end and no change in phase from rigid boundaries.
(c) No phase change takes place in case of refraction.
(d) The frequency of the wave remains unchanged both in reflection and refraction.
I = Incident wave
T = Transmitted or refracted wave
R = Reflected wave
i.e. If Yi = Ai sin (ωt – k1x)
Yr = Ar sin (ωt + k1x + π) = −Ar sin (ωt + k1x) from a rigid support
= Ar sin (ωt + k1x) from a free support
Yt = At sin (ωt – k2x).
Note : Since k is the wave number, it does not change in case of reflection, because medium is same and hence speed, frequency and wavelength (or k) do not change.
On the other hand in case of transmitted or refracted wave since medium changes and hence speed, wavelength (or k) changes but frequency (ω) remains the same.
Principle of Superposition
When two waves overlap, the actual displacement at some point is obtained by adding the displacement the point would have if only the first wave were present and the displacement it would have if only the second wave were present.
In other words the wave function y(x,t) that describes the resulting motion in this situation is obtained by adding the two wave functions for the two separate waves.
It is called the principle of superposition. i.e.
y (x ,t) = y1(x , t) + y2(x , t) + y3 (x , t) + …..
e.g. suppose two waves 1 and 2 are passing through point P.
Let at some instant the displacement (y1) of P due to wave 1 is + 4 cm and the displacement (y2) of point P due to wave 2 is −6 cm then at the same instant the displacement of point P would be −6 + 4 = −2 cm.
Similarly , let at some other instant y1 was + 2 cm and y2 was −2 cm, then net displacement of P at that instant will be zero.
Interference , Diffraction , Beats , Stationary waves etc. all are based on the principle of superposition.