Frequencies with Constructive/Destructive Interference, Help plz!

[DrBTS]

So I uploaded a picture here of a question, and the answer is D. Clearly it's the 'best' answer but I'm a little cloudy due to the fact that the frequency seems to not be nearly as definitive as it should be for a right answer, especially with regard to the alternative contender C. If you look closely answer choice D's frequency seems be off by a (1/4)f with regard to the reference figure's freq.

The explanation says that "the frequency of the resultant wave should be the same as each individual wave." How is that even possible, ie if you have two in phase waves that have significant frequency differences? Could someone clarify this statement for me as well? THANKS.

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[stgt9202]

I think the trick here is that they still have the same wavelength and frequency just one is shifted slightly. The trick here is the wording in the question. They're still additive but since they're slightly shifted their sum would be slightly less than if they were smack on top of each other.
That's how I looked at it.

 

[DrBTS]

I'm sorry but this reasoning is incorrect. There's no real shift in one direction. It seems as if the wave is stretched out (shifted out in both directions) because of the delay in constructive addition and then a delay in destructive subtraction going back down, so it makes the area under the wave wider...agreed...but this in turn changes the frequency. So I can see why D is correct. Theres gotta be a more direct mathematical way of relating to the resultant frequency. Thus, the answer explanation boggles me.

Also this raises another issue about how I'm confused in that you can have different wavelengths but the same frequency (which seems to be happening here)?

 

[SSerenity]

I don't understand that explanation at all.

However, I know from a previous example that when two waves propagate down the same material, the combined form is actually the Least Common Multiple of both their respective frequencies.

So I imagine in this problem, the combined wave must have a higher frequency than both of these waves and therefore a shorter wavelength. Thats how I arrived at C...

I would love to know how they got D.

 

[DrBTS]

Well C is clearly wrong because if you look closely (sorry the resolution is really bad in the picture) the amplitude has not increased, whereas the two individual frequencies are of the same amplitude so should have twice the amplitude change.

 

[SSerenity]

OK Try this,

The two wavelengths ARE of the same frequency and wavelength. The starting points are just shifted, so one wave just has to travel further. If they were offset by any WHOLE number of wavelengths, the overlap would be completely constructive. If they were offset by a half wavelength, they would be completely destructive.

Now, our combined wave must have a higher amplitude, since this is mostly constructive.

A & C can be eliminated based on the marking on their amplitude.

Next B should be eliminated because our resultant wave should have the same frequency (according to another forum Im reading, ignore my last message).

Leaving us with D.

 

[DrBTS]

Yeah thanks it didnt occur to me that the two individual frequencies were the same. that helped clarify the explanation of the explanation from the book that the resultant frequency must be the same frequency (i guess specifically to this case) but what I dont understand is how D is the same frequency (it shows it as being a little lower. I'm ready to just write this off as a really vague question.