1) Destructive interference
The condition for constructive interference to occur is:
[tex]\delta = m\lambda[/tex] (1)
where
[tex]\delta =|d_1 -d_2|[/tex] is the path difference, with
[tex]d_1[/tex] is the distance of the point from the first source
[tex]d_2[/tex] is the distance of the point from the second source
m is an integer number
[tex]\lambda[/tex] is the wavelength
In this problem, we have
[tex]d_1 = 78.0 m\\d_2 = 143 m\\\lambda=26.0 m[/tex]
So let's use eq.(1) to see if the resulting m is an integer
[tex]\delta =|78.0 m-143 m|=65 m\\m=\frac{\delta }{\lambda}=\frac{65 m}{26.0 m}=2.5[/tex]
It is not an integer so constructive interference does not occur.
Let's now analyze the condition for destructive interference:
[tex]\delta = (m+\frac{1}{2})\lambda[/tex] (2)
If we apply the same procedure to eq.(2), we find
[tex]m=\frac{\delta}{\lambda}-\frac{1}{2}=\frac{65.0 m}{26.0 m}-0.5=2[/tex]
which is an integer: so, this point is a point of destructive interference.
2) Constructive interference
In this case we have
[tex]d_1 = 91.0 m\\d_2 =221.0 m[/tex]
So the path difference is
[tex]\delta =|91.0 m-221.0 m|=130.0 m[/tex]
Using the condition for constructive interference:
[tex]m=\frac{\delta }{\lambda}=\frac{130.0 m}{26.0 m}=5[/tex]
Which is an integer, so this is a point of constructive interference.
3) Destructive interference
In this case we have
[tex]d_1 = 44.0 m\\d_2 =135.0 m[/tex]
So the path difference is
[tex]\delta =|44.0 m-135.0 m|=91.0 m[/tex]
Using the condition for constructive interference:
[tex]m=\frac{\delta }{\lambda}=\frac{91.0 m}{26.0 m}=3.5[/tex]
This is not an integer, so this is not a point of constructive interference.
So let's use now the condition for destructive interference:
[tex]m=\frac{\delta}{\lambda}-\frac{1}{2}=\frac{91.0 m}{26.0 m}-0.5=3[/tex]
which is an integer: so, this point is a point of destructive interference.
