Question

1. When transverse positive and negative pulses that have the same symmetric shape and size but travel in opposite directions meet, is it necessary that there be a moment when the string or wire on which they move is flat?
   1. If so, how do the pulses "know" to continue moving on the string?
2. Two people on each end of a long rope send off a wave pulse. If both wave pulses are on the same side of the rope, describe what happens when the pulses meet.
   1. What about when the pulses are on opposite sides?
   2. What happens when one pulse is oriented at a 90^o angle to the second pulse?

Please include equations and concepts :)

Answer 1

Solution :

Given that

Answer 2

When transverse pulses of the same symmetric shape and size but traveling in opposite directions meet on a string or wire, they create a phenomenon known as interference. Whether the string becomes flat or not depends on the nature of the interference.

1. Same Side (In-Phase) Meeting: When both wave pulses are on the same side of the rope (in-phase meeting), they will superpose constructively, resulting in a larger amplitude at the point of meeting. The string will not be flat at any moment during this interference. Instead, the amplitudes of the two pulses will add up momentarily, creating a temporary wave with a larger amplitude.

Mathematically, if the two pulses have the same amplitude A and are given by the functions y₁(x) and y₂(x), their superposition (y_total) at any time t and position x is given by: y_total(x, t) = y₁(x, t) + y₂(x, t) = 2A * cos(kx - ωt)

Here, k is the wave number, ω is the angular frequency, x is the position, and t is the time.

1. Opposite Side (Out-of-Phase) Meeting: When the pulses are on opposite sides of the rope (out-of-phase meeting), they will superpose destructively, resulting in a moment when the string is flat. This occurs because the crests of one pulse coincide with the troughs of the other pulse, leading to complete cancellation at that point.

Mathematically, if the two pulses are given by the same functions y₁(x) and y₂(x) as before, their superposition (y_total) at any time t and position x is given by: y_total(x, t) = y₁(x, t) - y₂(x, t) = 0

At the point of interference, the string remains flat momentarily.

1. 90° Angle Meeting: When one pulse is oriented at a 90° angle to the second pulse, they will superpose in a way that creates circular or elliptical motion at the point of meeting. This type of interference is called polarized interference. The string will oscillate vertically and horizontally, creating a circular or elliptical wave pattern at the point of intersection.

The concept behind how the pulses "know" to continue moving on the string lies in the wave equation and the principle of superposition. Waves follow the wave equation, and when they encounter each other, they interfere based on their respective amplitudes and phases. The principle of superposition states that when two waves meet at a point, their displacements add up algebraically at that point, leading to constructive or destructive interference.

In summary, when transverse pulses meet:

• In-phase meeting results in constructive interference, increasing the amplitude temporarily.

• Out-of-phase meeting results in destructive interference, causing a moment of flatness on the string.

• 90° angle meeting results in polarized interference, creating circular or elliptical motion at the point of intersection.