To do this I would use a mapping node and a noise texture:
The key here is that the mapping node is stretching the noise texture along the x axis by having the 'x' value of the scale to be very small. The closer to 0 it gets the more stretched out it is.
You can then increase the scale value on the noise to increase the number of 'bands' in the noise.
I then used the 'Fac' value of the noise instead of the 'Color' as the output for the shader to display.
Note: I used 'Generated' coordinates but this should work with other coordinates, like UVs. The axis you need to stretch with the mapping node may change depending on the orientation of those coordinates.