Question

Two in-phase loudspeakers, which emit sound in all directions,

are sitting side by side. One of them is moved sideways by 3.0 m, then forward by 4.0 m. Afterward, constructive interference is

observed $\frac{1}{4}$and $\frac{3}{4}$of the distance between the speakers along the line that joins them. What is the maximum possible wavelength of the sound waves?

Short answer

The maximum possible wavelength of the sound waves is$5m.$

Step-by-step solution

Given data

Out of two loudspeakers, of them is moved sideways by 3.0 m and then forward by 4.0 m.

A constructive interference is observed $\frac{1}{4}$and $\frac{3}{4}$of the distance between the speakers along the line that joins them.

Therefore, the position of the loudspeakers should be as follows:

Determination of the maximum possible wavelength

The final distance between the speakers can be obtained by applying the Pythagorean theorem as,

$$\begin{gathered} d=\sqrt{3^2+4^2}m \\ d=\sqrt{25}m \\ d=5m \end{gathered}$$

Therefore, the position of the loudspeaker should be as given below:

Now, the distance between the two spots producing constructive interference is,

$$\begin{gathered} D=\frac{3}{4}\left(5m\right)-\frac{1}{4}\left(5m\right) \\ D=\frac{5}{2}m \end{gathered}$$

As the constructive interference happens when the distance between the two spots is, $D=\frac{\lambda }{2}$

Thus, the longest wavelength possible in this situation must be$$\begin{gathered} \frac{\lambda }{2}=\frac{5}{2}m \\ \lambda =5m \end{gathered}$$