(5-5)    A peak of constructive interference will occur when each of the waveforms has a coincident peak where y = ±1. Peaks occur at these positions (positive values given but corresponding negative values would also have peaks, because of the symmetry of cosine curves around x=0):

y = cos (0.5x) = +1 when 0.5x  = 0, 2π, 4π, 6π etc, so x = 0, 4π, 8π, 16π, ……….. .

                      = -1 when 0.5x  = π, 3π, 5π etc, so x = 2π, 6π, 10π, ………..  

y = cos (x) = +1 when x  = 0, 2π, 4π, 6π, ………..

           = -1 when x  = π, 3π, 5π, ………..

y = cos (1.5x) = +1 when 1.5x  = 0, 2π, 4π, 6π etc, so x = 0, (4/3)π, (8/3)π, 4π, (16/3)π ……….. .

                      = -1 when 1.5x  = π, 3π, 5π etc, so x = (2/3)π, 2π, (10/3)π, ………..

y = cos(2x) = 1 when 2x = 0, 2π, 4π, 6π etc, so x = 0, π, 2π, 3π, 4π ……….. .

           = -1 when 2x = = π, 3π, 5π, etc, so x = (1/2)π, (3/2)π, (5/2)π, ………..

y = cos(0x) = 1 at all values of x.

Comparing these waveforms, you can see that all of them have a coincident positive peak at x=0, as we already know. The next value they all have in common is again a positive peak, at x = 4π (and this will be repeated at subsequent intervals of 4π). There is no value of x for which all of the curves have coincident negative peaks, so there will be no strong negative peak in the wave train.

And here is the graph of y = (cos(0x) + cos(0.5x) + cos(x) + cos(1.5x) + cos(2x))/ 5, just to prove that you’re right: