Two waves are described by y1=0.30 sin[π(5 x-200 t)] and y2=0.30 sin[π(5 x-200 t)+π/ 3] where

step 1 So in question 89, we have two different standing waves. So the first standing wave, Y1, equals

Two waves are described by y1=0.30 sin[π(5 x-200 t)] and ...

Two waves are described by $$y_{1}=0.30 \sin [\pi(5 x-200 t)]$$ and $$\quad y_{2}=0.30 \sin [\pi(5 x-200 t)+\pi / 3]$$ where $y_{1}, y_{2},$ and $x$ are in meters and $t$ is in seconds. When these two waves are combined, a traveling wave is produced. What are the (a) amplitude, (b) wave speed, and (c) wavelength of that traveling wave?

Two waves are described by
$$y_{1}=0.30 \sin [\pi(5 x-200 t)]$$
and $$\quad y_{2}=0.30 \sin [\pi(5 x-200 t)+\pi / 3]$$ where $y_{1}, y_{2},$ and $x$ are in meters and $t$ is in seconds. When these two
waves are combined, a traveling wave is produced. What are the (a)
amplitude, (b) wave speed, and (c) wavelength of that traveling wave?

Key Concepts

Superposition Principle

This concept explains that when two or more waves travel through the same medium, the total displacement at any point is the algebraic sum of the displacements due to the individual waves. It is fundamental for understanding how complex waveforms are formed by the addition of simpler, individual waves.

Phase Difference

Phase difference refers to the offset in phase between two sinusoidal waves, which affects how they interfere with each other. The phase difference determines whether the interference will be constructive, destructive, or somewhere in between, and it directly influences the amplitude of the resultant wave.

Trigonometric Identity (Sum-to-Product Formula)

The sum-to-product formulas are trigonometric identities used to combine two sinusoidal functions into a single sinusoid. These identities simplify the analysis of the resulting wave's characteristics by transforming the sum of sinusoids into a product form that reveals the amplitude modulation from the phase difference.

Wave Parameters (Amplitude, Angular Frequency, Wave Number)

These parameters describe the characteristics of a wave. The amplitude measures the maximum displacement, the angular frequency relates to the oscillation rate in time, and the wave number relates to the spatial periodicity. They are essential in defining the behavior and properties of the wave.

Wave Speed

Wave speed is the rate at which a wave propagates through space, and it is calculated by dividing the angular frequency by the wave number. Understanding wave speed is important for connecting the temporal and spatial features of a wave, and for predicting how the wave will evolve over time.

Wavelength

Wavelength is the spatial period of a wave, representing the distance between consecutive points in phase, such as crests. It can be determined from the wave number as the wavelength is inversely proportional to the wave number. This parameter is critical for understanding the spatial characteristics of the wave.

Transcript

00:01 So in question 89, we have two different standing waves.

00:03 So the first standing wave, y1, equals 0 .3, sine is pi 5x minus 500 t.

00:19 And the second standing wave is exactly the same, except that it has an extra phase of plus pi over 3.

00:32 So right now we have y1, y2, and they are travel that combined to produce a traveling wave.

00:41 So we want to understand what is the amplitude, what is wave speed, and what is the wavelength of the new wave.

00:48 So we have, in this section, we learned that when we have two different waves, when they are in the form of ym sine, kx plus omega -t, and when another one has an extra phase difference, plus omega t plus five, we will end up getting a wave that looks something like this.

01:14 2 times ym, cosine 5 over 2, sine, kx plus omega t, plus 5 over 2.

01:29 So this is the equation from this chapter, and this is also obtained.

01:34 You can derive it by using the half angle formula.

01:37 If you think about, if you try to consider this as an angle of sine x plus omega -t plus 5 over 2 minus 5 over 2.

01:46 And try to think of this is plus 5 over 2 plus 5 over 2.

01:51 And then you can divide both these angles as sine a plus b for the first case and sign a minus b for the second case.

01:59 And then you decompose them and you can combine them again.

02:02 If you are not familiar with this process, you might want to go back to the chapter and review.

02:07 This is a formula we get and i think this is also formula 1650.

02:16 This is equation 1650.

02:19 So now let's go back to this actual yyy2 .y5 from this question and ym is 0 .3.

02:27 So we have 0 .6 over here, cosine, 5 is pi over 3, so this is 5 over 6...

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