# What to find the path difference in constructive interference?

Hi there,

I am unsure of how to calculate the path difference in questions relating to constructive and destructive interference.

I understand that when two waves are exactly in phase with each other, constructive interference occurs, so I labeled the intersecting points as areas where constructive interference occurs. But the answer scheme states that at the points where the waves are in phase, the path difference is $$nλ$$, while at the points where destructive interference occurs, the path difference is $$(n − 1/2)\lambda$$.

Where do these equations come from?

Re. "Where do these equations come from?"

Note that $$sin(\theta + n\lambda)$$ = $$sin(\theta)$$ so a phase shift of $$n\lambda$$ leaves the two waves in phase so they reinforce.

Whereas $$sin(\theta + (n-1/2)\lambda)$$ = $$-sin(\theta)$$ so a phase shift of $$(n-1/2)\lambda$$ leaves the two waves 180 degrees out of phase so the cancel.

You probably know that the total wave in space is simply the sum of the waves from points $$A$$ and $$B$$. What matters for the strength of a signal is the intensity of a wave, as its squared modulus. Then, the maximum of constructive interference (point X) occurs when two peaks - represented by red circles - or two troughs - represented by green circles - meet. On the other hand, the minimum (point Y) will be where a peak meets a trough.

As you can see here, in a somewhat different problem: at $$x=0$$, both waves have values $$w_1(0)=w_2(0)=1$$, so they interfere constructively, while at $$x=\pi$$ we have $$w_1(\pi)=-w_2(\pi)=-1$$, so the intensity is 0 and the interference is destructive.

The other part of your question is about the path differences. In your specific case, notice that the peaks happen at integer multiples $$n\lambda$$ of the wavelength, and the troughs happen at semi-integers $$(n-1/2)\lambda$$ away from the source. If a maximum is at the intersection of the red circles, the path difference is $$d = n_A\lambda - n_B\lambda = (n_A-n_B)\lambda=n\lambda,\ \text{ with }n\text{ integer},$$ and if it is at a intersection of green circles, $$d = \left(n_A-\frac{1}{2}\right)\lambda - \left(n_B-\frac{1}{2}\right)\lambda = \left(n_A-n_B\right)\lambda=n\lambda,\ \text{ with }n\text{ integer}.$$

If it is at a minimum, it is at a point where a red circle meets a green circle, so we have a situation like $$d = \left(n_A-\frac{1}{2}\right)\lambda -n_B\lambda = \left(n_A-n_B-\frac{1}{2}\right)\lambda=\left(n-\frac{1}{2}\right)\lambda,\ \text{ with }n\text{ integer}.$$ Notice that the $$-1/2$$ might be on the other parentheses, it doesn't change the result.

Let say you have two electromagnetic wave

$$y_{1}=A\sin(\omega t-kx_{1}),\,y_{2}=A\sin(\omega t-kx_{2}),$$

Now phase difference denoted, by variable $$h$$, so we get $$kx_{1}-kx_{2}=h\tag{1}$$

Where $$k$$ denote wave number, because we measure the distance traveled by wave in terms of wavelength, so we get the wave number.

Where $$(x_{1}-x_{2})$$ is the path difference. Two waves interfere like a vector did.

So when the phase difference is integral of $$2n(\pi)$$,then wave will show constructive interference, if it $$(n+1/2)(\pi)$$ then they show destructive interference, after this value in equation 1, you will get path difference between waves. As a function of $$n$$.

They come from the realisation that a sine or cosine function changes sign twice every wavelength.