A college student is at a concert and really wants to hear the music, so she sits between two in-phase loudspeakers, which point toward each other and are 60.8 m apart. The speakers emit sound at a frequency of 428.75 Hz. At the midpoint between the speakers, there will be constructive interference, and the music will be at its loudest. At what distance closest to the midpoint (and along the line connecting the loudspeakers) could she also sit to experience the loudest sound? (Use 343.0 m/s for the speed of sound.)

Question

A college student is at a concert and really wants to hear the music, so she sits between two in-phase loudspeakers, which point toward each other and are 60.8 m apart. The speakers emit sound at a frequency of 428.75 Hz. At the midpoint between the speakers, there will be constructive interference, and the music will be at its loudest. At what distance closest to the midpoint (and along the line connecting the loudspeakers) could she also sit to experience the loudest sound? (Use 343.0 m/s for the speed of sound.)

Answer 1

To find the distance closest to the midpoint where the student will experience the loudest sound, we need to consider the concept of constructive interference between two sound waves.

Constructive interference occurs when two sound waves with the same frequency and in-phase (their crests and troughs align) combine to produce a resultant wave with a greater amplitude. In this case, the two loudspeakers emit sound waves that reach the student simultaneously.

The distance between the loudspeakers is given as 60.8 m, and the frequency of the sound emitted by the loudspeakers is 428.75 Hz. The speed of sound is given as 343.0 m/s.

To find the distance closest to the midpoint where the student will experience the loudest sound, we need to first calculate the wavelength of the sound wave. We can use the formula:

wavelength = speed of sound / frequency

wavelength = 343.0 m/s / 428.75 Hz

Now, we can calculate the distance from each loudspeaker to the midpoint where constructive interference occurs. Since the waves are in-phase, the distance from each loudspeaker to the midpoint will be an integer multiple of half-wavelength. Let's represent this distance as d:

d = (n * wavelength) / 2 (where n is an integer)

However, the midpoint distance we're interested in lies on the line connecting the loudspeakers, so the total distance from the first loudspeaker to the student will be (60.8 / 2) - d.

To find the distance closest to the midpoint, we need to determine the smallest possible positive value for (60.8 / 2) - d. We can do this by considering the case where n = 0.

d = (0 * wavelength) / 2 = 0

So, the distance closest to the midpoint will be:

(60.8 / 2) - d = 60.8 / 2 = 30.4 m

Therefore, the student should sit at a distance of 30.4 meters from either loudspeaker (along the line connecting the loudspeakers) to experience the loudest sound.