Two point sources produce waves of the same wavelength and are completely out of phase (that is produces a crest at the same time as the other produces a trough). At a point midway between the sources, would you expect to find a node or an antinode? Why?
Please show work,
Thank you
antinode. Why?
To determine whether we would expect to find a node or an antinode at a point midway between the two point sources, we need to consider the interference of the waves produced by the sources.
When two waves interfere, their amplitudes can either add up constructively (creating an antinode) or cancel each other out destructively (creating a node).
In this case, since the two sources are completely out of phase (producing a crest at the same time as the other produces a trough), the waves they produce will interfere destructively.
To understand this, let's assume the two wave sources are labeled A and B. At the halfway point between them (point P), when wave A is at a crest, wave B will be at a trough. This means that at point P, the crest of wave A will perfectly coincide with the trough of wave B, causing the magnitudes of these waves to cancel each other out.
To confirm this, we can use the principle of superposition to calculate the resultant displacement at point P. Since the waves from sources A and B are out of phase, we can write their equations as:
Wave from A: A*sin(kx - ωt) (eq. 1)
Wave from B: -A*sin(kx - ωt) (eq. 2)
Adding these two equations together, we get:
Resultant displacement: (A*sin(kx - ωt)) + (-A*sin(kx - ωt))
The terms in parentheses are identical but have opposite signs, resulting in cancellation:
Resultant displacement: 0
Therefore, the two waves completely cancel each other out at point P, resulting in a node.
Hence, at a point midway between the two point sources, we would expect to find a node. This is because the waves produced by the sources interfere destructively when they are completely out of phase, producing a net displacement of zero.