The ship in Figure P14.35 travels along a straight line parallel to the shore and a distance d = 500 m from it. The ship’s radio receives simultaneous signals of the same frequency from antennas A and B, separated by a distance L = 800 m. The signals interfere constructively at point C, which is equidistant from A and B. The signal goes through the first minimum at point D, which is directly outward from the shore from point B. Determine the wavelength of the radio waves.
To determine the wavelength of the radio waves, we need to understand the concept of constructive interference and the condition for the first minimum.
Constructive interference occurs when two waves of the same frequency and amplitude meet at a point and their amplitudes add up. This results in a stronger wave at that point. The condition for constructive interference is that the path difference between the two waves must be an integer multiple of the wavelength.
In this scenario, we have two antennas, A and B, that emit signals of the same frequency. These signals interfere constructively at point C, which is equidistant from A and B. The path difference between the signals from A and B at point C is:
Path difference = AB - AC
Since C is equidistant from A and B, the path difference becomes:
Path difference = L/2
So we have:
L/2 = n * λ --- Equation 1
Where n is an integer representing the number of complete wavelengths in the path difference and λ is the wavelength we want to find.
Now, the signal goes through the first minimum at point D, which is directly outward from the shore from point B. This means that the path difference between the signals from A and B at point D is equal to one-half of the wavelength.
Path difference at point D = L/2 + AD
Since D is directly outward from the shore from point B, we have:
AD = AB - BD
Given that BD = d (distance of the ship from the shore), we have:
AD = AB - d
Substituting this into the path difference equation:
Path difference at point D = L/2 + AB - d
Since this path difference is equal to λ/2 (one-half of the wavelength), we have:
L/2 + AB - d = λ/2 --- Equation 2
Now, we can solve Equations 1 and 2 simultaneously to find the wavelength.
Using Equation 1:
L/2 = n * λ
Using Equation 2:
L/2 + AB - d = λ/2
We can rearrange Equation 2 to solve for AB:
AB = λ/2 - L/2 + d
Substituting this into Equation 1:
L/2 = n * λ
L/2 = n * (λ/2 - L/2 + d)
Simplifying:
L = nλ - nd + d
Rearranging:
nλ = L + nd - d
Finally, solving for λ:
λ = (L + nd - d) / n
Substituting the given values:
L = 800 m
d = 500 m
Let's substitute these values into the equation to calculate the wavelength: