1. The document discusses constructive and destructive interference that occurs when two waves meet at a point. 2. For constructive interference to occur, the path length difference (Δd) between the waves must be an integer multiple of the wavelength. For destructive interference, Δd must be a half-integer multiple of the wavelength. 3. The document uses examples of waves that are in-phase or out-of-phase to show how the equations for Δd change depending on whether the interference is constructive or destructive.

1. Exploring Interference
By: Eugenie Kwong
Learning Object 7
PHYS 101-202
Section LC2

2. Interference
Constructive Destructive
How can we determine WHERE these interferences occur mathematically?
First, let us consider 2 waves that are in phase over the next two slides.

3. To have constructive interference,
either:
1. The waves meet at a point when they
are both at the greatest positive
amplitude.
OR
2. The waves meet a point when they are
both at the greatest negative
amplitude.
Since we are considering 2 waves that are in phase (such as the diagram
above), the waves will constructively meet up only when there is an integer
number of λ between each source and the point (in order for the waves to
meet up at the peaks).
So if d is the path length from a source to the point where the two waves
meet, then for the 2 sources, ∆d=nλ, where n is any integer (…-1,0,1,2…)

4. To have destructive interference:
The waves meet at a point when one is at
the greatest positive amplitude, while the
other wave is at the greatest negative
amplitude.
Since we are considering two waves that
are in phase with each other, the only way
that the two waves will meet up in that
configuration is when:
- One of the waves has a path length of
d1=λn
- Other wave has a path length that is
d2=(0.5λ+nλ)=(0.5+n)λ.
(where n is any integer)
Thus, ∆d=(n+0.5)λ

5. Concept tester:
Which type of interference is at A?
The speed at which the wave travels is 420m/s, and the frequency is 210Hz.
The waves are in phase, and both sources emit the same type of wave.

6. 1. From the given information, we
want to find whether or not the
path length difference (∆d) agrees
with the equation for constructive
or destructive interference.
Constructive: ∆d=nλ
Destructive: ∆d=(n+0.5)λ
2. We know that d1=10m, and d2=13m, so
∆d=13m-10m=3m
Given frequency and velocity, we can find the
wavelength using v=fλ.
λ=v/f
λ=420(m/s)/210(Hz)
λ=2 m
3. Now substitute into either the constructive or
destructive equations for ∆d, and we find:
For constructive: 3=n(2), n=1.5 is not an integer.
For destructive: 3=(n+0.5)(2), n=1, which is an
integer.
Therefore, the interference at point A is
destructive.
Another way to solve this problem is to compare
the individual path lengths. 10 is an integer of
the wavelength 2, and 13 is a half integer
multiple of the wavelength 2. This satisfies the
conditions for destructive interference.

7. What if the two waves interfering are OUT OF PHASE?
How does this affect our calculation?
Let’s take a look at a simple situation:

8. If the second wave is rad out of
phase, then this would result in ∆d
having an extra rad as well.
So from the original ∆d which was
constructive (∆d=nλ), we add the
extra phase shift term, which makes it
∆d=nλ+ .
Since rad=half a wavelength, then
∆d=nλ+0.5λ
=λ(n+0.5)
And this corresponds to the path
length difference equation for
destructive interference!
In general, if there is a phase shift between the two waves, then we add
the phase shift to the original ∆d equation.

9. Concept tester:
Which type of interference is at A if the two
point sources are /2 radians out of phase?
The speed at which the wave travels is 420m/s, and the frequency is 210Hz.
The waves are in phase, and both sources emit the same type of wave.

10. /2 radians out of phase situation
1. Put this phase shift in terms of λ.
Since λ=2 , then =λ/2 and so the waves will be (λ/2)/2 = λ/4
out of phase.
2. Add this phase shift to the original ∆d equation:
∆d=13-10+(λ/4)
=3+(λ/4)
=3+(2/4)
=3.5
3. Compare ∆d=3.5 with the ∆d for constructive and
destructive equations.
Constructive: ∆d=nλ
Destructive: ∆d=(n+0.5)λ
3.5 is not an integer multiple of λ=2m, therefore not
constructive.
If you plug in 3.5 into the destructive equation, n=6.5 which is
not an integer and therefore is not destructive.
(What we know from
the given info: λ=2)
Conclusion: the
interference would be
somewhere between
constructive and
destructive.