Question

A sound source is located somewhere along the x-axis. Experiments show that the same wave front simultaneously reaches listeners at x = -7.0 m and x = +3.0 m. a. What is the x-coordinate of the source? b. A third listener is positioned along the positive y-axis. What is her y-coordinate if the same wave front reaches her at the same instant it does the first two listeners?

Short answer

Part a

X- coordinate of the source is $x=-2.0m$.

Part b

Y-coordinate of the third listener is $y=4.6m$.

Step-by-step solution

Given information

Wavefront simultaneously reaches at$x=-7.0m$and$x=+3.0m$.

Part a

As the same wavefront simultaneously reaches listeners at $x=-7.0m$and $x=+3.0m$, role="math" localid="1649951372478" $∆\varphi =0rad$which implies that

$$\begin{gathered} \frac{2\mathrm{\pi }}{\lambda }\left(r_2-r_1\right)=0 \\ {r}_2=r_1 \end{gathered}$$

Thus, the source is at $x=-2.0m$so that it is equidistant from the two listeners.

Part b

The third person is also $5.0m$away from the source.

Her y-coordinate is thus

$$\begin{gathered} y=\sqrt{5^2-2^2} \\ =\sqrt{25-4} \\ =\sqrt{21} \\ =4.6m \end{gathered}$$

Therefore, the y-coordinate of the third listener is$4.6m$.