Summing waves and interference

Figure 2.3: Addition of waves- red and blue waves add to give the purple wave

The left hand figure demonstrates that when the waves are the same, or ‘in-phase’, i.e. the maxima in the waves coincide, as do the minima. Consequently, they add together and produce a much larger wave. This is known as constructive interference. This wave will have twice the amplitude of the previous wave. As can be seen the waves will be in-phase when they are a whole number of wave-lengths apart: 

In the second case, the blue wave is now slightly out of phase with the red wave. We still obtain a wave result, whose magnitude is slightly larger than the original waves (∼ √2). In the third case the waves are separated by exactly half a wavelength (a phase of π) and so are perfectly ‘out of phase’- i.e. the low points in one wave meet the high points in the next and so perfectly cancel. The result is no overall wave- this is known as destructive interference and occurs when the

When we need to add waves together, this vector property previously described becomes important. For example, consider two waves of the same maximum amplitude and wavelength, which have travelled distances x₁ and x₂. The resultant wave can be calculated by adding the waves as vectors. This has been done in Fig. 2.4. In the arrangement shown the resultant is no longer equal to the original maximum amplitude, but is a new vector Ψ. 

Figure 2.4: Adding Waves as Vectors. In this case the amplitude of the two initial waves (shown in red and blue) are the same.