A question in a new physics textbook we are using in school tasks readers with determining the path difference and them commenting on whether it would result in constructive or destructive interference. The path difference in this question comes out to be 0.4m. The textbook says that this results in constructive interference. How do we decide this when the path difference is not, 0, 0.5, 1, 1.5, etc? The soundwaves have a wavelength of 0.4m.

Many thanks



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Total constructive interference happens when the path difference is a whole integer multiple of the wavelength, or $$\Delta p=n\lambda, n\in\mathbb{Z}$$ Total destructive interference happens when the path difference is a half-integer multiple of the wavelength, or $$\Delta p=\left(n+\frac{1}{2}\right)\lambda, n\in\mathbb{Z}$$ If $\Delta p$ is somewhere in between, then the effect will be somewhere in between constructive and destructive. In those cases, the problem becomes finding out what happens when you add two otherwise identical sine waves of differing phase.

For understanding the edge cases however, the intuition is that if the peaks and valleys all line up with each other they will constructively interfere; however, if the peak of one wave corresponds with a valley of another wave, then they will cancel each other out and destructively interfere.

  • $\begingroup$ Thank you. It sounds like the textbook I am using made a judgement call then on what the final wave 'looks' like given that it's path partway between constructive and destructive interference. I just assumed that 0.4 is closer to 0.5 than to 0 and the final result would be destructive. $\endgroup$ – greenplasticdave Aug 27 at 4:48
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    $\begingroup$ @greenplasticdave Hang on, this is total constructive interference, not anything in between. It is given that $\lambda=0.4\mathrm{m}$, and the path difference is exactly that, meaning that the resulting interference is totally constructive. When we refer to 0, 0.5, 1, 1.5, etc., we are referring to multiples of the wavelength. Watch out that this isn't a source of confusion for you. $\endgroup$ – DanDan0101 Aug 27 at 6:02
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    $\begingroup$ This makes perfect sense to me now and a point the textbook glossed over. Thank you very much. $\endgroup$ – greenplasticdave Aug 27 at 22:56

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