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11.3.2.1: Illustrations

  • Page ID
    34148
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    Illustration 1: Wave Types

    The four animations represent a particle description of three waves on a string and a wave on a spring (position is given in meters and time is given in seconds). For the waves on a string the motion of the red circle is shown as a function of time. Restart.

    Animation 1 and Animation 2 depict transverse waves (Animation 1 shows a wave pulse and Animation 2 shows the creation of a sinusoidal traveling wave). The waving is in the \(y\) direction, while the wave propagation (the direction of the wave velocity) is in the \(x\) direction. If you have ever done "the wave" at a football or a basketball game you have been a part of a transverse wave! ("The wave" is a special example of a traveling wave called a pulse, in that every part of the medium that supports the wave does not always wave.) Note that the individual particles that make up the string go up and down, yet do not move in the \(x\) direction (just as during the wave you just stand up and then sit down). Watch the red particle in each animation and also view the graph showing the red particle's motion in the \(y\) direction.

    Animation 3 represents a longitudinal wave. An example of a longitudinal wave is sound. In a longitudinal wave, the waving of the medium (here the string particles) is in the direction of the propagation of the wave. Watch the red particle in this animation and also view the graph showing the red particle's motion in the \(x\) direction. Notice that it oscillates back and forth instead of up and down. Animation 4 represents a wave on a spring. Is it a transverse or longitudinal wave? It is both! Can you tell why this is so?

    In Animation 5 water waves are depicted by showing the individual motion of the water molecules (position is given in meters and time is given in seconds). What type of wave is depicted by the animation?

    Illustration authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.
    Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

    Illustration 2: Wave Functions

    A traveling wave is shown in black at time \(t = 0\) seconds (position is given in meters). Three sliders are given that change certain properties of the wave. Restart. In general, we would write the wave function for a right-moving wave as

    \[y(x, t)=A\cos (kx-\omega t+\phi )=A\cos ((2\pi /\lambda )x-(2\pi /T)t+\phi )\nonumber\]

    However, we are looking at the wave at \(t = 0\) and we cannot determine the wave speed or frequency (where \(v =\lambda f = \omega /k\)), so we just have:

    \[y(x, t)=A\cos (kx+\phi )=A\cos ((2\pi /\lambda )x+\phi )\nonumber\]

    Which slider changes which quality of the wave? Well, there are three sliders and three parameters in the wave function. Try each slider and see what happens. Slider A controls the phase shift, \(\phi\), since it shifts the function to the left or right. Slider B controls the wavelength of the wave and therefore the wave number \(k\), since \(k = 2\pi /\lambda\). Clearly Slider C controls the amplitude, \(A\), of the wave function.

    If what was discussed above has made sense, you should be able to identify the wave parameters (find the value of the phase shift, wavelength and amplitude) using the sliders for this wave function (shown in red).

    Illustration authored by Mario Belloni.
    Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

    Illustration 3: Superposition of Pulses

    One of the most interesting phenomena we can explore is that of a superposition of waves. In this Illustration we consider a superposition of two traveling pulses, while in Illustration 17.4 and Illustration 17.7 we consider the superposition of two traveling waves. (In Illustration 16.5 and Illustration 16.6 we considered the addition of multiple periodic functions in a Fourier series) Restart.

    A superposition of two waves is nothing more than the arithmetic sum of the amplitudes of the two underlying waves. We can represent the amplitude of a transverse wave by a wave function, \(y(x, t)\). Notice that the amplitude, the value \(y\), is a function of position on the \(x\) axis and the time. If we have two waves moving in the same medium, we call them \(y_{1}(x, t)\) and \(y_{2}(x, t)\), or in the case of this animation, \(f(x, t)\) and \(g(x, t)\). Their superposition, arithmetic sum, is written as \(f(x, t) + g(x, t)\).

    This may seem like a complicated process, so we often focus on the amplitude at one point on the \(x\) axis, say \(x = 0\text{ m}\) (position is given in meters and time is given in seconds). So now let's consider Animation 1, which represents waves traveling on a string. The top panel represents the right-moving Gaussian pulse \(f(x, t)\), the middle panel represents \(g(x, t)\), the left-moving Gaussian pulse, and the bottom panel represents what you would actually see: the superposition of \(f(x, t)\) and \(g(x, t)\). As you play the animation, focus on \(x = 0\text{ m}\). Until the tail of each wave arrives at \(x = 0\text{ m}\), the amplitude there is zero. Watch what happens during the time that the two waves overlap. They add together in the way you would expect. As time goes on, the waves "separate" and move along the string as if they had not "run into" each other.

    What does the superposition in Animation 2 look like at \(t = 10\text{ s}\)? The two waves add together and exactly cancel there. As time goes on, the waves "reappear" (they were always there) and move along the string as if they had not "run into" each other.

    Illustration 4: Superposition of Traveling Waves

    In Illustration 17.3 we considered the superposition of two traveling pulses. In this Illustration we consider a superposition of two traveling sinusoidal waves. (In Illustration 16.5 and Illustration 16.6 we consider the addition of multiple periodic functions in a Fourier series). Restart.

    Let's begin by considering Animation 1, which represents two waves traveling on a string (position is given in meters and time is given in seconds). As you play the animation, focus on \(x = 0\text{ m}\). Until each wave arrives at \(x = 0\text{ m}\), the amplitude there is zero. Watch what happens during the time that the two waves overlap, \(t\geq 8\text{ s}\). They add together in the way you would expect. Given that the two waves always have the opposite amplitude at \(x = 0\text{ m}\), the superposition of the two traveling waves at \(x = 0\text{ m}\) will always be zero. This point that never moves is called a node. The resulting wave is called a standing wave. It is created when we have two identical waves traveling in opposite directions in a particular medium (here the medium is a string, but we can set up standing waves in air as well).

    What does the superposition in Animation 2 look like at \(t\geq 8\text{ s}\)? The two waves add together and exactly cancel at \(x = 0\text{ m}\). As time goes on, the waves "reappear" (they were always there) and move along the string as if they had not "run into" each other. Given that the two waves always have the same amplitude at \(x = 0\text{ m}\), the superposition of the two traveling waves at \(x = 0\text{ m}\) will always be changing. This point is called an anti-node. The resulting wave is still a standing wave. Note that it is shifted in comparison to Animation 1.

    In Animation 3, we have a traveling wave that is incident on a wall located at \(x = 15\text{ m}\). The wave travels and is then reflected by the wall. By reflected we mean that the direction of propagation of the wave changes (the right-moving wave is now a left-moving wave) and its amplitude is now the negative of what it was before. So we now have a right-moving wave and a left-moving wave that resemble the superposition shown in Animation 1. In that animation, the node was at \(x = 0\text{ m}\); here in Animation 3 the node is at \(x = 15\text{ m}\).

    Illustration authored by Mario Belloni.
    Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

    Illustration 5: Resonant Behavior on a String

    Thus far we have considered either a traveling wave or a traveling pulse. The wave traveled off to infinity unencumbered by any barrier. Here we consider a pulse on two strings, but the strings are pulsed multiple times. To complicate matters the two pulse frequencies are different. Run the animation and consider the results (position is given in meters and time is given in seconds). Restart.

    How can we understand what is happening? First we notice that the pulse is reflected at the wall. Second we notice the effect of good and bad timing. Which animation has the good timing and which one the bad timing?

    In the bottom animation the timing is awful! The waves add up in a way that does not yield a maximum wave amplitude. All we get is a jumbled-up mess.

    The top animation shows the effect of good timing. All of the pulses add constructively to the returning reflected wave to give the largest wave amplitude possible. Whenever we get successive contributions to the wave adding in this way, we call it a resonance. It is like pushing a swing. If you push a swing at just the right frequency (good timing), large amplitude motion will result. If you apply the same force, but at a different frequency (bad timing), not a lot usually happens. In order to get a large amplitude you must push at the same frequency as the natural frequency of the swing.

    Illustration authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni. Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

    Illustration 6: Plucking a String

    A green string of length \(L = 28\text{ cm}\) (position is given in centimeters) is shown plucked to \(x = 6\text{ cm}\) and \(y = 3\text{ cm}\). The unstretched position of the string is shown in gray. Changing the slider changes this plucking point along the length of the string in the \(x\) direction (the \(y\) point of the pluck remains the same). You may also look at the Fourier components that make up the green stretched string by clicking on an n value. The relative size of these sine waves is depicted by the graph on the right. Restart.

    We have thus far looked at using a Fourier series to describe an arbitrary periodic wave (see Illustration 16.5 and Illustration 16.6). For the plucked string, we must consider a different way to add up waves to get the Fourier series. Here we must consider any wave that is zero at the ends of the string (since the plucked string, like a standing wave, has ends that are tied down). Therefore, we find that our plucked string can be described in terms of a Fourier series as

    \[f(x)=\sum A_{n}\sin (n\ast\pi\ast x/L)\nonumber\]

    where in the animation \(L = 28\text{ cm}\) (see Illustration 16.5 and Illustration 16.6 for more details on the periodic case).

    When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

    Illustration authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.
    Script authored by Morten Brydensholt, Wolfgang Christian, and Mario Belloni.

    Illustration 7: Group and Phase Velocity

    So what do we mean by the velocity of a wave? This may seem like a simple question. When we talk about a wave on a string (or a sound wave) we can talk about the velocity as \(v = \lambda f\). We can rewrite this expression in terms of the wave's wave number, \(k\), and angular frequency, \(\omega\), given that \(\lambda = 2\pi /k\) and that \(f = 2\pi /\omega\). We therefore find that \(v = \omega /k\). We note here that the velocity of the wave is also fundamentally related to the medium in which the wave propagates.

    But what happens when we want to add several traveling waves together? In this case we are interested in several waves traveling in the same direction. We can change the wave number and angular frequency for each wave, but we must ensure that the wave speeds are identical. In this animation we add the red wave to the green wave to form the resulting blue wave (position is given in meters and time is given in seconds). Restart.

    Consider what happens when we change \(k_{1}\) to \(8\text{ rad/m}\) and \(\omega_{1}\) to \(8\text{ rad/s}\). Note the interesting pattern that develops in the superposition. Notice that there is an overall wave pattern that modulates a finer-detailed wave pattern. The overall wave pattern is defined by the propagation of a wave envelope with what is called the group velocity. The wave envelope has a wave inside it that has a much shorter wavelength that propagates at what is called the phase velocity. For these values (of \(k\) and \(\omega\)), the phase and group velocities are the same.

    Now consider \(k_{1} = 8\text{ rad/m}\) and \(\omega_{1} = 8.4\text{ rad/s}\). What happens to the wave envelope now? It does not move! This is reflected in the calculation of the group velocity. The finer-detailed wave has a phase velocity of \(1.02\text{ m/s}\). Now consider \(k_{1} = 8\text{ rad/m}\) and \(\omega_{1} = 8.2\text{ rad/s}\). The group velocity is now about half that of the phase velocity (certain water waves have this property). Now consider \(k_{1} = 8\text{ rad/m}\) and \(\omega_{1} = 7.6\text{ rad/s}\). The group velocity is now about twice that of the phase velocity.

    For a superposition of two waves the group velocity is defined as \(v_{\text{group}} = \Delta\omega /\Delta k\) and the phase velocity as \(v_{\text{phase}} =\omega_{\text{avg}}/k_{\text{avg}}\). In general, the group velocity is defined as \(v_{\text{group}} =\partial\omega /\partial k\) and the phase velocity as \(v_{\text{phase}} =\omega /k\).

    So what velocity do we want? The physical velocity is that of the wave envelope, the group velocity. For waves on strings we got lucky: the phase and group velocities are the same (these are harmonic waves).

    Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.


    This page titled 11.3.2.1: Illustrations is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Wolfgang Christian, Mario Belloni, Anne Cox, Melissa H. Dancy, and Aaron Titus, & Thomas M. Colbert.