# Consider a ship that travels along a straight line parallel to the shore and a distance d = 500 \...

## Question:

Consider a ship that travels along a straight line parallel to the shore and a distance {eq}d = 500 \ m {/eq} from it. The ship's radio receives simultaneous signals of the same frequency from antennas {eq}A {/eq} and {eq}B {/eq}, separated by a distance {eq}L = 800 \ m {/eq}. The signals interfere constructively at point {eq}C {/eq}, which is equidistant from {eq}A {/eq} and {eq}B {/eq}. The signal goes through the first minimum at point {eq}D {/eq}, which is directly outward from the shore from point {eq}B {/eq}. Determine the wavelength of the radio waves.

## Sound Interference:

Sound waves are known to undergo interference, whether constructively or destructively, depending on their phase difference. This, in turn, is dependent upon their relative displacement from their respective sources.

## Answer and Explanation:

In order for destructive interference to occur, the following condition must be satisfied:

{eq}\displaystyle l - d = \left( m + \frac{1}{2} \right) \lambda {/eq}

where {eq}l {/eq} and {eq}d {/eq} are distances as given in the figure, {eq}m {/eq} is the order of the minimum (m=1 for the first-order minimum), and {eq}\lambda {/eq} is the wavelength.

Here, we first solve for {eq}l {/eq} by applying the Pythagorean theorem:

{eq}\displaystyle l = \sqrt{ (800\ \rm m)^2 + d^2} = \sqrt{ (800\ \rm m)^2 + (500\ \rm m)^2} = \sqrt{ 890, 000\ \rm m^2 } = \sqrt{ 890, 000} \ \rm m {/eq}

Now, we wolve for the wavelength:

{eq}\displaystyle \begin{align} l - d &= \left( m + \frac{1}{2} \right) \lambda\\ \sqrt{ 890, 000} \ \rm m - 500\ \rm m &= \left( 1 + \frac{1}{2} \right) \lambda \\ 443.398\ \rm m &= \frac{3}{2} \lambda \\ \lambda & \approx 296\ \rm m \end{align} {/eq}