Learning object 2

Understanding the Interference of waves travelling in the same

directionWave M

Wave L

Resultant wave

How can a resultant wave have an amplitude of 0? Read on to find out how this destructive interference comes about.

The Concept• Interference of waves occurs due to the superposition of waves

travelling in the same direction. When two waves are travelling in opposite directions it is known simply only as superposition, when they are travelling in the same direction, interference occurs.

• The resultant wave that occurs has an amplitude at any point x that corresponds to the sum of the displacements of the travelling waves at that point x.

• So if we define wave one as: D1(x,t)=A sin (kx-wt) and wave two described as: D2(x,t)=A sin (kx-wt +Φ)

• We can see that the two waves have the same frequency, wavelength, amplitude and direction and both are moving forwards.

• So the resultant wave Dr(x,t)=D1+D2

Understanding the mathematics

• We can use the trigonometric identity of sin(a)+sin(b)=2sin((a+b)/2) cos((a-b)/2) to simplify the problem. If we let D1=sin(a) and D2=sin(b) then we are left with:

• Dr=sin(a)+sin(b)=A sin(kx-wt)+A sin(kx-wt+Φ)=2A cos(-Φ/2) sin (kx-wt+Φ/2)

• It is important to note in this case that we can replace cos(-Φ/2) with cos(Φ/2) because the graph of cos θ it symmetrical on both sides of a graph as seen below.

So the equation for Dr can be re-written as

Dr=2Acos(Φ/2) sin(kx-wt+Φ/2)

From this we can now see that the amplitude of the resultant wave will be = 2Acos(Φ/2) as the maximum value that the sine function sin(kx-wt+Φ/2) can take is 1.

Important notes• We can re-write the equation of wave two to give us a different expression of

the phase constant.

• D2=A sin (kx-wt +Φ) = A sin(k(x+Φ/k)-wt). But we know k=2π/λ. So D2 = Asin(k(x+(Φλ/2π))-wt).

• When we compare this with D1=A sin (kx-wt) we see that the only difference is the (Φλ/2π) expression in the kx term. So the second wave is shifted in relation to the first wave by (Φλ/2π), (Graphically this would represent shifting the graph to the left).

• It is also very important to note that the equations given represent the interference of waves with the same wavelength, amplitude, frequency and direction. If one of these factors were to change this method of obtaining the displacement function would be less viable.

• Take for example if the amplitudes of the waves were to change, say A1=3 and A2=2. It would then be more difficult to apply the trigonometric identity necessary to find the displacement function Dr.

Final note on resultant amplitude

• Because the resultant amplitude is = 2Acos(Φ/2), looking at the graph of cosθ below, we can see that if Φ/2=π/2 or –π/2, then the resultant wave will have a amplitude of 0. So if the two waves are out of phase by π or an odd multiple of π, then the resultant wave will have an amplitude of zero as the two effectively “cancel each other out”. The opposite is true of even multiples of π, if two waves have a phase difference of 2π, then they are in phase, so if the phase constant is 2π or some multiple of it then the waves will reinforce each other and the amplitude will be 2A. This relationship between phase difference and resultant amplitude is represented graphically on the next slide.

π

-π/2 π/2

-π

• If the phase difference is some intermediary between the two multiples, which it most likely is, then the absolute value of the resultant amplitude will be some value of the original amplitude between: 0A<Ar<2A

Graph of how the resultant amplitude changes in relation to the phase difference

Graph of an example of constructive interference.

Both photographs take from “Physics for Scientists and Engineers An Interactive Approach revised custom volume 1 PHYS 101” page 221

HAWKES, IQBAL, MANSOUR, MILNER-BOLTON and WILLIAMS