Two point sources emit sound waves of 1.0-m wavelength. The sources, 2.0 m apart, as shown below, emit waves which are in phase with each other at the instant of emission. Where, along the line between the sources, are the waves out of phase with each other by pi radians?
I know that at the antinodes, the diffrence is pi radians and i got 0.5, 0.5, 1 and 1.25 for the position of antinodes. but the answer is x = 0.75 m, 1.25 m.
what am i doing wrong
To determine where the waves are out of phase with each other by pi radians, we need to consider the concept of path difference. Path difference is the difference in the distance traveled by two waves from their respective sources to a given point.
In this case, the two point sources emit waves of 1.0-m wavelength, and they are 2.0 m apart. We want to find the positions along the line between the sources where the waves are out of phase by pi radians.
To calculate the path difference at a specific point, we first need to determine the phase difference between the two waves. Since the sources emit waves in phase at the instant of emission, the phase difference is initially zero.
The general formula for calculating path difference is:
Path Difference = d * sin(theta)
where d is the distance between the sources (2.0 m in this case), and theta is the angle between the point of interest and the line joining the two sources.
To find the positions where the waves are out of phase by pi radians, we need to calculate the path difference and set it equal to half a wavelength (pi radians corresponds to half a wavelength phase difference).
So, we have:
d * sin(theta) = lambda/2
where lambda is the wavelength (1.0 m in this case).
Now, let's solve for theta.
sin(theta) = lambda/2d
sin(theta) = (1.0 m)/(2 * 2.0 m)
sin(theta) = 0.25
To find the values of theta that satisfy this equation, you can use the inverse sine function (sin^(-1)) on a calculator to find the angle whose sine value is 0.25.
sin^(-1)(0.25) = 14.48 degrees
Now, we have the angle. To find the positions along the line between the two sources, we need to consider the geometry of the setup.
From each source to a point on the line between them, the angle forms a triangle. The sum of the angles in a triangle is 180 degrees, so the angle between the line joining the sources and the line between them is:
180 degrees - 2 * 14.48 degrees = 150.04 degrees
Now, we can calculate the distance from one of the sources to the point where the waves are out of phase by pi radians:
x = d * sin(150.04 degrees)
Using x = 0.75 m in the formula, we get:
x = 2.0 m * sin(150.04 degrees)
x ≈ 1.73 m
Therefore, one of the positions along the line between the sources where the waves are out of phase by pi radians is x ≈ 1.73 m.
Similarly, for the second position, we need to consider the other angle formed:
180 degrees - (180 degrees - 2 * 14.48 degrees) = 29.96 degrees
Using x = 0.75 m in the formula, we get:
x = 2.0 m * sin(29.96 degrees)
x ≈ 0.66 m
Therefore, the other position along the line between the sources where the waves are out of phase by pi radians is x ≈ 0.66 m.
Hence, the correct answers are x = 0.75 m and x = 0.66 m, not 0.5 m and 1.25 m as you initially thought.
I hope this clarifies the issue and helps you understand the correct solution.