Anthony sends a series of pulses of amplitude 24 cm down a string that is attached to a post at

Anthony sends a series of pulses of amplitude 24 cm down a...

Anthony sends a series of pulses of amplitude $24 \mathrm{cm}$ down a string that is attached to a post at one end. Assuming the pulses are reflected with no loss of amplitude, what is the amplitude at a point on the string where two pulses are crossing if a. the string is rigidly attached to the post? b. the end at which reflection occurs is free to slide up and down?

Anthony sends a series of pulses of amplitude $24 \mathrm{cm}$ down a string that is attached to a post at one end. Assuming the pulses are reflected with no loss of amplitude, what is the amplitude at a point on the string where two pulses are crossing if
a. the string is rigidly attached to the post?
b. the end at which reflection occurs is free to slide up and down?

Key Concepts

Boundary Conditions in Wave Reflection

When a wave encounters a boundary, the conditions at the boundary determine how the wave is reflected. For a fixed or rigid boundary, the reflected wave undergoes a phase change (inversion) while maintaining its amplitude. In contrast, at a free boundary, there is no phase inversion. These differing boundary conditions are crucial for predicting the interference outcomes when incident and reflected pulses meet.

Constructive and Destructive Interference

The phenomenon where overlapping waves combine is known as interference. Constructive interference occurs when waves are in phase, producing an amplitude equal to the sum of the individual amplitudes. Destructive interference happens when waves are out of phase, potentially reducing the overall amplitude. The particular outcome depends on the phase relationship dictated by the reflection properties of the boundary.

Superposition Principle

This principle states that when two or more waves overlap in a medium, the resulting displacement is the sum of the displacements of each individual wave. It is fundamental to understanding interference phenomena, where the combined effect of multiple waves can lead to increased or decreased amplitudes depending on their phase relationship.

Transcript

00:01 So anthony sends a series of pulses down a string and it reaches a wall with a rigid post.

00:09 Or in one scenario, it's a rigid post.

00:12 The other scenario, the post is allowed to move up and down.

00:16 So we want to know what's the amplitude of the pulses after the wave is reflected from that post.

00:26 We want to look at this question in the first sense where the post is rigid.

00:36 And we see here that when anthony sends the pulses down, there's an amplitude of 24 centimeters.

00:45 So let's write it 24 centimeters.

00:51 Okay.

00:53 And this is measured from the equilibrium point to the highest displacement of the wave.

00:59 Now when there's a rigid point, a rigid pulse, the reflection of the wave from that point is going to be exactly that.

01:10 Going to be a reflection.

01:11 So you can see in the first image, it's reflected to be the one on the second image here.

01:18 You begin with a crest and then end with a trough.

01:22 That trough is reflected to become a crest and then a trough, going in the opposite direction that anthony sends his pulses.

01:30 So in a rigid scenario, in a rigid post, you have this.

01:43 You have a wave traveling like this to the right.

01:47 But you also have a wave traveling like this to the left.

01:54 When you add these two, from previous questions, you can see that the superposition of this two ways would give a deconstructed wave, deconstructive interference, which gives us a plain way without any crest or troughs, because they all canceled each other out...

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