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. 15.30 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 wavelength 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

Wave Speed

Wave speed is the rate at which a wave propagates through a medium, calculated as the ratio of the wavelength to the period (or by multiplying wavelength and frequency). It indicates how quickly the wave energy is transmitted across the medium.

Phase Difference

Phase difference refers to the difference in the phase of the wave between two points along its path. This concept is essential when comparing displacements at different positions or times, and it can be used to deduce properties such as wavelength and propagation direction when the distance between observation points is known.

Wavelength

Wavelength is the spatial period of a wave, defined as the distance over which the wave's shape repeats. It is a critical parameter in describing how a wave propagates in space, and together with the frequency, it determines the wave speed.

Propagation Direction

The propagation direction of a wave indicates the direction in which the wave energy travels. Determining the wave’s propagation direction is important for understanding the dynamics of the system, as the sign of the wave speed and the phase relationship between points are directly influenced by the direction of travel.

Amplitude

Amplitude is the maximum displacement of a vibrating particle from its equilibrium position in a wave. In the context of sinusoidal waves, the amplitude represents the peak value of oscillation and is directly related to the energy carried by the wave, with larger amplitudes indicating larger oscillations and typically more energy.

Sinusoidal Wave

A sinusoidal wave is a type of periodic oscillation characterized by a smooth, continuous, and repetitive oscillation that can be described mathematically by a sine (or cosine) function. This kind of wave is ubiquitous in physics and engineering because many natural phenomena can be approximated or modeled using sinusoidal functions due to their simplicity and well-understood properties.

Period

The period of a wave is the amount of time it takes for one complete cycle of the oscillation to occur. It is a fundamental temporal property of periodic motion and is inversely related to the frequency, which counts how many cycles occur in one second.

Transcript

00:01 So from the problem given the figure 15 .30, we can solve the part a of this problem by observing the graph.

00:09 By observing the graph of the wave, we can actually find the maximum displacement y.

00:16 And the maximum displacement y is actually the magnitude of the wave, of the wave itself, or the amplitude of the wave.

00:27 So, from the graph we obtain that the amplitude is equal to 4 millimeters.

00:38 In similar manner, for the part b of the problem, we can find the period, which is the period is the time that passes between two consecutive points, having the same value of y.

00:53 So if we have, for example, wave that looks like this, this is the visual.

01:03 Of the wave and we have of course with the figure given this problem so if this is one amplitude here maximum amplitude then this is another maximum amplitude the time and then this is the time direction this is the why direction this here is going to be the period so from the given figure we can say that for the two points that we are given that the problem given us for these two points the period is equal to 0 .04 seconds.

01:43 These two pieces of data are very important for solving this problem.

01:50 So for the part c of this problem, we are asked to find the velocity.

01:58 So here we will find the velocity from the standard kinematic formula that says that velocity is the distance transverse over some t.

02:09 Or is a distance.

02:09 In this case it would be delta x or delta t since the wave is traveling in the horizontal direction.

02:19 So first we need for the first point x equals 0, we need to find that the maximum disturbance or the peak is occurring at t1 that equals 0 .01 seconds.

02:37 And for another point that is given for this part of the problem to be x equals 0 .09 meters.

02:48 The corresponding time point is 0 .035 seconds.

02:54 So we can immediately substitute that and say that the velocity equals.

03:01 Now we'll have 0 .09 meters minus 0 meters over the time.

03:11 Point difference so 0 .035 minus 0 .01 meters per second and this equals 3 .6 meters per second so the answer is for the part c the velocity is 3 .6 meters per second this is all the right from the given figure the given diagram in in the problem with little calculations on our side.

03:42 Now we can also find the wavelength from the formula that says that the velocity is actually a wavelength of a period.

03:51 Since we know the period, then we can find the wavelength by simply multiplying velocity with the period.

04:04 So in this case, lambda would be equal to 3 .6 meters per second.

04:19 Times 0 .04 seconds which turns out to be equal to 0 .144 meters per just meter this is the way length so this is for the part c and we can see that we can actually calculate both the velocity and the wavelength now for the part d we have a very similar problem but now the way is traveling in the negative direction, negative horizontal direction...

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