Traveling wave and Young's double slit experiment?

Question

1. The sum of two sinusoidal traveling waves is also a sinusoidal traveling wave only if:
A. Their amplitudes are the same and they travel in the same direction.
B. Their amplitude are the same and they travel in opposite direction.
C. Their frequency are the same and they travel in the same direction.
D. Their frequency are the same and they travel in opposite direction.
The answer is C. Why? Does it mean that is their frequency aren’t the same, no traveling wave will be formed?

2. When 450 nm light is incident normally on a certain double slit system, the number of interference maxima within the central diffraction maximum is 5. When 900 nm light is incident on the same slit system the number is:
A. 2
B. 3
C. 5
D. 9
Please show solution and explanation, thanks!

Answer 1

1. No, a traveling wave will be formed but if it's the sum of two different frequency waves the result won't be a sine wave since it will contain both frequency components. A pure sine wave is a single frequency. See ref. 1. Incidentally, answer 1 is a little off the mark on frequencies of the beat wave. The high frequuency is (f1 + f2) / 2 and the beat frequency is (f1 - f2) / 2, resulting in amplitude modulation at a rate of two maxima per beat-frequency cycle or f1 - f2.
2. Answer is B (3). With 450 nm you have a central maximum and two maxima to either side of the center. With 900 nm the maxima will be spaced twice as far apart, so there will be a central maximum and one maximum to either side for a total of three. The important equation here is y ~= m * lambda * D / d, where y is the location of the mth maximum, lambda is wavelength, and d and D are respectively the slit separation and the distance from the slits to the viewing surface. See ref. 2.

Answer 2

When the frequency of two sinusoidal waves is not the same, then what we get depends on the difference of the frequency of the two waves. If the diff. in their frequency is not much then we get what are known as beats. It consists of a sinusoidal wave with frequency equal to the greater of the two frequencies, but the amplitude varies sinusoidally with frequency equal to the smaller of the two frequencies.As the frequency difference increases, the resultant wave obtained gets more complex in nature.