Two sound sources radiating in phase at a fre-
quency of 540 Hz interfere such that maxima
are heard at angles of 0◦ and 21◦ from a line
perpendicular to that joining the two sources.
To understand why maxima are heard at specific angles in interference patterns, we need to consider the concept of interference in waves.
Interference occurs when two or more waves interact with each other. In the case of sound waves, interference can occur when two sound sources emit waves that overlap and combine.
The interference pattern depends on the phase difference between the waves at a given point in space. The phase difference determines whether the waves will interfere constructively (adding up) or destructively (canceling out) at that point.
In your scenario, two sound sources are radiating in phase at a frequency of 540 Hz. This means that the peaks and troughs of the two waves are aligned when they leave the sources.
The maxima are heard at angles of 0° and 21° from a line perpendicular to the line connecting the two sources. This implies that at those angles, the waves from the two sources interfere constructively, resulting in an increased sound intensity.
To determine why these specific angles result in constructive interference, we can use the concept of path difference. The path difference is the difference in distance traveled by the waves from the two sources to reach a particular point.
For constructive interference to occur, the path difference between the two waves must be an integer multiple of the wavelength. In other words, the waves must reach the point in phase.
Let's assume that the distance between the two sources is d. At an angle of 0°, the point is equidistant from both sources, so the path difference between the waves is 0.
At an angle of 21°, the distance between the point and one source is d*sin(21°). The distance between the point and the other source can be approximated as d. Therefore, the path difference is approximately d*sin(21°) - d.
To find the angles where the path difference is an integer multiple of the wavelength λ, we can set the path difference equal to mλ, where m is an integer.
d*sin(21°) - d = mλ
Now you can solve this equation for m to find the different possible values. For each value of m, you can calculate the corresponding angle using trigonometry.
By solving the equation and calculating the angles, you will find the specific angles at which maxima are heard in your interference pattern.