To determine the phase difference between the two sound waves arriving at a specific point, we need to consider the geometry of the situation and the properties of wave interference. Let's break this down step by step for both scenarios you've mentioned.
Understanding the Setup
We have two point sources of sound, separated by a distance of 10 cm, vibrating in phase. The wavelength of the sound waves produced by these sources is 5.0 cm. This means that the waves will constructively or destructively interfere depending on the path difference between them when they reach a given point.
Calculating Phase Difference
The phase difference (\(\Delta \phi\)) between two waves can be calculated using the formula:
- \(\Delta \phi = \frac{2\pi}{\lambda} \Delta x\)
where \(\Delta x\) is the path difference between the two waves at the point of interest, and \(\lambda\) is the wavelength.
Scenario (a): On the Line Joining the Sources
Let’s consider a point that is 20 cm from one of the sources. Since the sources are 10 cm apart, the distance from the second source to this point will be:
- Distance from Source 1 (S1) = 20 cm
- Distance from Source 2 (S2) = 20 cm - 10 cm = 10 cm
The path difference (\(\Delta x\)) is:
- \(\Delta x = |20 \, \text{cm} - 10 \, \text{cm}| = 10 \, \text{cm}\)
Now, substituting into the phase difference formula:
- \(\Delta \phi = \frac{2\pi}{5.0 \, \text{cm}} \times 10 \, \text{cm} = 4\pi\)
Since \(4\pi\) radians is equivalent to \(720^\circ\), we can simplify this to \(0\) radians (or \(0^\circ\)) because phase differences are periodic with a period of \(2\pi\). Thus, the phase difference at this point is:
- Phase Difference (a): 0 radians
Scenario (b): On the Perpendicular Bisector
Now, let’s analyze the point located on the perpendicular bisector of the line joining the two sources. This point is equidistant from both sources. Since the sources are 10 cm apart, the distance from each source to this point will be:
- Distance from S1 = Distance from S2 = \(\sqrt{(5 \, \text{cm})^2 + (h)^2}\)
Here, \(h\) is the height from the midpoint to the point on the perpendicular bisector. However, since we are looking for the phase difference, we can note that the path difference (\(\Delta x\)) is:
- \(\Delta x = 0 \, \text{cm}\) (because the distances from both sources are equal)
Substituting this into the phase difference formula gives:
- \(\Delta \phi = \frac{2\pi}{5.0 \, \text{cm}} \times 0 \, \text{cm} = 0\)
Thus, the phase difference at this point is:
- Phase Difference (b): 0 radians
Summary of Findings
In both scenarios, the phase difference between the two waves arriving at the specified points is:
- On the line joining the sources: 0 radians
- On the perpendicular bisector: 0 radians
This indicates that at both locations, the waves arrive in phase, leading to constructive interference. This is a fundamental concept in wave physics, illustrating how the geometry of wave propagation affects interference patterns.
