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IB past paper doubt

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Can't do this wave question, can someone solve this clearly please? thanks

 

Because you know that the 2 sources vibrate in phase and you have to deduce that there's no sound at P, it's quite easy to realize that this question is about constructive & destructive interference. For such a question, the first step to do is always to figure out the path difference (pd) from point P to each of the sources. This can be done by simply drawing the triangles, and then use the Pythagoras theorem to figure out the difference in length between side XP and YP:

gif.latex? pd = XP - YP = \sqrt{4^2 + (1

 

Let 'n' be an integer, then I'm sure that you know that:

- For constructive interference, then gif.latex? pd = \lambda n

- For destructive interference, then gif.latex? pd = \lambda \left(n+\frac{1}

 

Because gif.latex? pd \div \lambda = 0.5499 \div, which is not an integer, but a number half way between 2 integers, so at point P, there's a destructive interference. That's why there's no sound at that point.

If there's anything unclear in my answer, feel free to ask.

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You can calculate the distance each wave needs to travel to reach point P using trigonometry. Then, divide that by wavelength to find the no. of cycles taken to get there. The no. of cycles of the 2 sources will differ by a multiple of 0.5, meaning the path difference a multiple of half of wavelength, which results in destructive interference. 

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