Hi guys hope someone here can maybe help me out.

Two loudspeakers are placed at either end of a gymnasium, both pointing toward the center of the gym and equidistant from it. The speakers emit 286 Hz sound that is in phase. An observer at the center of the gym experiences constructive interference.

How far toward either speaker must the observer walk to first experience destructive interference?

I suspect if he walked a quarter wavelength, one wave would be delayed for a quarterwavelength, and the other advanced for a quarter wavelength, making a total shift of..

To determine the distance at which the observer will first experience destructive interference, we need to consider the concept of path difference.

Path difference is the difference in the distance traveled by the sound waves from each speaker to the observer. In the case of constructive interference, the path difference is an integer multiple of the wavelength, while for destructive interference, it is a half-integer multiple.

Let's start by finding the wavelength of the sound emitted by the speakers. The formula to calculate the wavelength is:

wavelength = speed of sound / frequency

The speed of sound is approximately 343 meters per second. Plugging in the given frequency of 286 Hz into the formula, we have:

wavelength = 343 m/s / 286 Hz
≈ 1.199 meters

Now, let's analyze the situation. Since the two speakers are equidistant from the center of the gym, the path difference will change when the observer moves directly towards one speaker. The observer will experience destructive interference when the path difference is a half-integer multiple of the wavelength.

Let's assume the observer starts at the center of the gym and walks a distance 'x' towards one of the speakers. The path difference is then given by:

path difference = 2x

For destructive interference, the path difference should be equal to a half-integer multiple of the wavelength:

2x = (n + 1/2) * wavelength

where 'n' is any whole number including zero.

Now, let's solve for 'x' to find the distance the observer needs to walk to experience destructive interference. Rearranging the equation:

x = ((n + 1/2) * wavelength) / 2

We can substitute the values we have:

x = ((n + 1/2) * 1.199 meters) / 2

Calculating for different values of 'n', we can find the respective distances towards each speaker where the observer will first experience destructive interference.