Radio waves of frequency 30 kHz are received at a location 1500 km from a transmitter. The radio reception
temporarily “fades” due to destructive interference between the waves travelling parallel to the ground and the
waves reflected from a layer (ionosphere) of the earth’s atmosphere.
(i)What is the minimum distance that the reflected waves should travel for destructive interference to occur
at the receiver?
(ii) The layer at which the waves are reflected is at a height h above the ground. Calculate the minimum
height of this layer for destructive interference to occur at the receiver.
what is 1500+lambda/2 ?
lambda=speedlight/frequency.
for ii0, draw the figure. use right triangle relationships, the hypotensue is 750+lambda/4, the base is 750, so figure h.
lambda works out to be 10^4m
so 1500+10^4/2=1505km
but i'm still stuck on how this is , why do you add the 1500?
ok i'm using pythagoras theorem but I'ts not working out right for me , thanks
To find the minimum distance that the reflected waves should travel for destructive interference to occur at the receiver, we can use the concept of path difference.
(i) Minimum Distance for Destructive Interference:
We know that for destructive interference to occur, the path difference between the waves traveling parallel to the ground and the reflected waves should be equal to an odd multiple of half the wavelength.
The formula for path difference (Δx) in terms of wavelength (λ) is given by:
Δx = (2n - 1) * (λ/2)
In this case, since the frequency of the radio waves is 30 kHz, we can calculate the wavelength (λ) using the formula:
λ = c/f
where c is the speed of light in a vacuum (approximated to 3 x 10^8 m/s) and f is the frequency of the waves. Substituting the values, we get:
λ = (3 x 10^8 m/s) / (30 x 10^3 Hz) = 10^4 m
Let's assume the minimum distance for destructive interference to occur is represented by d.
Thus, the path difference can be given by:
Δx = d - 2d = -d
Now, we equate the path difference to an odd multiple of half the wavelength:
-d = (2n - 1) * (λ/2)
To find the minimum distance, we set n = 1, since it is the odd multiple closest to zero:
-d = (2 * 1 - 1) * (10^4 / 2)
Simplifying the equation:
-d = 10^4 / 2
-d = 5 x 10^3
d = -5 x 10^3 m
However, distance cannot be negative, so taking the absolute value, we have:
d = 5 x 10^3 m
Therefore, the minimum distance that the reflected waves should travel for destructive interference to occur at the receiver is 5 km.
(ii) Minimum Height of the Reflecting Layer:
The minimum height (h) of the reflecting layer can be found by considering the path taken by the radio waves.
The distance traveled by the reflected waves is twice the height of the reflecting layer (2h) plus the distance traveled from the transmitter to the reflecting layer (1500 km).
Using the information given, we can write the following equation:
2h + 1500 km = 5 km
Converting all distances to meters:
2h + 1.5 x 10^6 m = 5 x 10^3 m
Rearranging the equation to solve for h:
2h = 5 x 10^3 m - 1.5 x 10^6 m
2h = -1.495 x 10^6 m
h = (-1.495 x 10^6 m) / 2
h = -7.475 x 10^5 m
Again, height cannot be negative, so taking the absolute value, we have:
h = 7.475 x 10^5 m
Therefore, the minimum height of the reflecting layer for destructive interference to occur at the receiver is 7.475 x 10^5 meters above the ground.