These two waves travel along the same string: y1(x, t)=(4.60 mm) sin(2 πx-400 πt) y2(x

step 1 So in this question, we have two waves. Both of them has the same form, but one has an amplitude

These two waves travel along the same string: y1(x,...

These two waves travel along the same string: $$ \begin{aligned} &y_{1}(x, t)=(4.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t) \\ &y_{2}(x, t)=(5.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+0.80 \pi \mathrm{rad}) \end{aligned} $$ What are (a) the amplitude and (b) the phase angle (relative to wave 1 ) of the resultant wave? (c) If a third wave of amplitude $5.00 \mathrm{~mm}$ is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave?

These two waves travel along the same string:
$$
\begin{aligned}
&y_{1}(x, t)=(4.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t) \\
&y_{2}(x, t)=(5.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+0.80 \pi \mathrm{rad})
\end{aligned}
$$
What are (a) the amplitude and (b) the phase angle (relative to wave 1 ) of the resultant wave? (c) If a third wave of amplitude $5.00 \mathrm{~mm}$ is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave?

Key Concepts

Phasor Addition

Phasor addition is a method used to add sinusoidally varying quantities by representing them as vectors (phasors) in the complex plane. Each wave is represented by a magnitude (its amplitude) and an angle (its phase). The addition of these phasors allows one to compute the resultant amplitude and phase of the combined wave effectively.

Superposition Principle

The superposition principle states that when two or more waves traverse the same medium, the resultant wave at any point is the algebraic sum of the individual wave displacements at that point. This concept is fundamental in understanding how waves interact when overlapping, whether they are in phase, out of phase, or somewhere in between.

Phase Difference

Phase difference refers to the difference in the phase angles of two or more waves of the same frequency. It is a key factor in determining whether the waves interfere constructively or destructively. The phase difference affects the resultant amplitude when the waves are superimposed, as it determines the relative alignment of the wave peaks and troughs.

Constructive Interference

Constructive interference occurs when two or more waves combine to produce a wave with a larger amplitude than any of the individual waves. This typically happens when the waves are in phase or have a phase difference that leads to their peaks aligning. Adjusting the phase of additional waves can be used to maximize the overall amplitude of the resultant wave.

Transcript

00:01 So in this question, we have two waves.

00:03 Both of them has the same form, but one has an amplitude of 4 .6, and the millimeter, and the other has an amplitude of 5 .6 millimeter.

00:12 And they are 0 .8 pi apart.

00:15 We want to use the technique of phasers to figure out what is the amplitude of the resultant of these two waves, traveling all the same stream.

00:25 Then what is the face shift? what is the face of the resultant wave with respect to? the first wave and then for part c we have another wave similar to these two waves but with the amplitude of 5 millimeter again these are all millimeters and how should we add the third wave so that the final wave has a maximum amplitude possible so let's first start from part a for part a we're trying to add these two together use a master of phasers so for phasers we pretty much think of each wave as a vector so we can draw the first one y1 is a vector of 4 .6 millimeter and then we can draw out y2 as a vector of 5 .6 millimeter with 0 .8 pi .0 .8 pi is like 144 degree with respect to the first one.

01:18 So this is y1, y2 would be something like this.

01:24 So this is 144 degree and this is 5 .6.

01:30 And then the result in ym will be this arrow.

01:34 So use our standard factor addition technique, we can work out the amplitude of this.

01:41 So this distance is 4 .6, this distance, let me just...

01:48 This distance is 4 .6 minus 5 .6 times cosine 36 degree.

02:00 Because this is 36 degree and 5 .6 times cosine 36 degree is this much.

02:07 So 4 .6 minus this much is this green little line over here.

02:12 So this is, and then we need to, look at what this vertical one is.

02:17 This vertical one is just, oops, it's 5 .6 times sine 36 degree square...