Question:

Two speakers, A and B, are placed 3 meters apart on a straight line. The speakers are emitting sound waves of the same frequency with a phase difference of 180 degrees. At a point P located on the line perpendicular to the line joining the speakers and passing through the midpoint of the line segment joining speakers A and B, the waves from both speakers interfere. The distance between point P and speaker A is 4 meters.

a) Calculate the path difference between the waves from speakers A and B at point P.

b) Determine the phase difference between the waves from speakers A and B at point P.

c) Given that the speed of sound in air is 340 m/s, calculate the frequency of the sound waves emitted by the speakers such that a destructive interference occurs at point P.

Answer:

a) The path difference between the waves from speakers A and B at point P is given by:

\begin{equation*} \text{Path Difference} = d_2 - d_1 \end{equation*}

Where:

• $d_1$ is the distance between speaker A and point P
• $d_2$ is the distance between speaker B and point P

From the given information, $d_1 = 4$ meters and $d_2 = 3$ meters.

Therefore, \begin{align*} \text{Path Difference} &= d_2 - d_1 \ &= 3 - 4 \ &= -1 , \text{meters} \end{align*}

b) The phase difference between the waves from speakers A and B at point P can be calculated using the formula:

\begin{equation*} \text{Phase Difference} = \frac{\text{Path Difference}}{\lambda} \times 360^\circ \end{equation*}

Where:

• $\lambda$ is the wavelength of the sound waves

Since the phase difference is given as 180 degrees, we can set up the equation:

\begin{align*} \frac{-1}{\lambda} \times 360^\circ &= 180^\circ \ -1 \times 360^\circ &= 180^\circ \times \lambda \ \lambda &= -2 , \text{meters} \end{align*}

Therefore, the wavelength of the sound waves is 2 meters.

c) Destructive interference occurs when the waves are out of phase by 180 degrees. This corresponds to a half-wavelength path difference (i.e., $\frac{\lambda}{2}$ path difference).

Using the formula for path difference, we can set up the equation:

\begin{align*} \frac{\lambda}{2} &= d_2 - d_1 \ \frac{2}{2} &= 3 - 4 \ 1 &= -1 \end{align*}

This is not possible as it results in an inconsistency.

Therefore, it is not possible for a destructive interference to occur at point P.

Note: The above calculation suggests that there is a mistake in the problem setup or the given phase difference (180 degrees). Typically, speakers that are 180 degrees out of phase would result in constructive interference at the given distances.