A sinusoidal wave is propagating along a stretched string...

A sinusoidal wave is propagating along a stretched string that lies along the $x$ -axis. The displacement of the string as a function of time is graphed in Fig. E15.11 for particles at $x=0$ and at $x=$ $0.0900 \mathrm{~m}$. (a) What is the amplitude of the wave? (b) What is the period of the wave? (c) You are told that the two points $x=0$ and $x=0.0900 \mathrm{~m}$ are within one wavelength of each other. If the wave is moving in the $+x$ -direction, determine the wavelength and the wave speed. (d) If instead the wave is moving in the $-x$ -direction, determine the wavelength and the wave speed. (e) Would it be possible to determine definitively the wavelengths in parts (c) and (d) if you were not told that the two points were within one wavelength of each other? Why or why not?

A sinusoidal wave is propagating along a stretched string that lies along the $x$ -axis. The displacement of the string as a function of time is graphed in Fig. E15.11 for particles at $x=0$ and at $x=$ $0.0900 \mathrm{~m}$. (a) What is the amplitude of the wave?
(b) What is the period of the wave?
(c) You are told that the two points $x=0$ and $x=0.0900 \mathrm{~m}$ are within one wavelength of each other. If the wave is moving in the $+x$ -direction, determine the wavelength and the wave speed.
(d) If instead the wave is moving in the $-x$ -direction, determine the wavelength and the wave speed. (e) Would it be possible to determine definitively the wavelengths in parts (c) and (d) if you were not told that the two points were within one wavelength of each other? Why or why not?

Key Concepts

Sinusoidal Wave

A sinusoidal wave is a smooth, periodic oscillation that can be described mathematically using sine or cosine functions. These waves are characterized by properties such as amplitude, period, wavelength, and phase, and are fundamental in understanding how energy and signals propagate through various media.

Amplitude

The amplitude of a wave is the maximum displacement of points on the wave from its equilibrium (rest) position. It is a measure of the energy carried by the wave—the greater the amplitude, the more energy the wave transports.

Period

The period of a wave is the duration of time it takes for one complete cycle of the oscillation to occur. This time interval is crucial, as it is inversely related to the frequency, indicating how many cycles occur per unit time.

Wavelength

Wavelength is the spatial distance between successive points that are in the same phase on the wave, such as the distance between two consecutive crests or troughs. It defines the scale of the wave's spatial repetition and is a key factor in determining the wave’s propagation characteristics.

Wave Speed

Wave speed is the rate at which a wave travels through a medium and is defined by the relationship between the wavelength and the period (or frequency). Specifically, the wave speed can be calculated by multiplying the wavelength by the frequency or by dividing the wavelength by the period.

Phase Difference

Phase difference refers to the amount by which one point on a wave is out of step with another point along the wave. It is determined by the distance between the points relative to the wavelength and is essential for understanding the spatial relationship and timing of wave oscillations.

Transcript

00:01 In this question, it is about reading the waves, obtaining information from the displacement time curve, and we are given the displacement time curve for a wave at two different positions at x equals to 0 and at x equals to 0 .09 meters.

00:23 Okay, there are five parts in this question.

00:26 And in part a we want to find out the amplitude of the wave.

00:33 Okay, so this can be read all from the graph.

00:39 So this is 4 m m.

00:41 The period can also be read off from the curve from the graph as well.

00:46 This is 0 .04 seconds.

00:50 Okay.

00:57 Then in part c, we are given that the two points are within one wavelength of feature.

01:04 And now that the wave is moving in the positive x direction you want to find out the with length and the wave speed okay so to do this question will be using the wave function y x t is a sine omega t minus k x yeah we are using the sign function here because at t equals zero at x equals is zero the and the y is zero okay and then since the two points, we didn't know with length.

01:55 Okay, the peak at t equals 0 .01 seconds for x equals to 0 .0.

02:05 And the peak at t equals to 0 .35 seconds at x equals 0 .09 meters, the same.

02:21 Okay, so, yeah, so which means that the peak that appears at x equals to zero will appear later at x equals 0 .09, right? so, and since they are within the same wavelength, then the peak that shows up immediately after the peak that shows that x equals 0 will be the same peak.

02:55 Okay, so, yeah, so in this case, we'll say, so x1 equals to 0 .09 meters and t1 equals to 0 .035 seconds.

03:18 Okay, and then at peak, so we have omega t1 minus kx1 equals to pi or 2, right? so i'm going to pull out the 2 pi, t1 over the period minus x1 divided by lambda is equal to pi over 2.

03:43 And so t1 divide by t minus x1 divided by lambda is one quarter.

03:51 And so x1 over lambda is equal to 0 .035 divided by 0 .04 minus a quarter.

04:01 And then this is 5 over 8 so lambda is 8 over 5 x1 which is 8 over 5 times 0 .909 is 0 .14 meters.

04:21 Okay so this is a wavelength and then the wave speed is equal to lambda divide by t 0 .14 .4 divided by 0 .04 .4 and this is 3 .6 meters per second...

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