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Standing waves

Page history last edited by Joe Redish 6 years, 11 months ago

Working Content > Oscillations and Waves > Waves in 1D > Waves on an elastic string > Superposition of waves in 1D

 

Prerequisites:

 

In our study of Beats we considered the superposition of two traveling waves of slightly different frequencies going in the same direction. We found the interesting phenomenon that the alternating adding up and canceling of the two waves resulted in broad wave packets that oscillated at a much lower frequency (basically the difference of the two frequencies). If we consider the case of two waves of the same frequency but moving in opposite directions, we find another (and perhaps even more important) phenomenon: standing waves.

 

In our analysis of beats we relied on the trigonometric identity

 

sin(a) + sin(b) = 2sin((a+b)/2) cos((a-b)/2)

 

Let's consider what this tells us if we have two identical waves but going in opposite directions. From our reading on sinusoidal waves, we know that the wave

 

propagates in the positive x direction, and  

 

is the wave that propagates in the negative x direction.

 

 The notation might look a bit strange at first, but it makes sense if you think about what it's telling you. The wave labelled "y+" has the negative signs inside its arguments -- but that means that as time grows, x has to increase to keep up with it. So if you want to stay at the same point on the wave, you have to move towards more positive values of x. The wave labelled "y-" has the positive signs inside its arguments -- but that means that as time grows, x has to decrease to keep up with it. So if you want to stay at the same point on the wave, you have to move towards more negative values of x.   

 

If we add y+ and y- together and use our trig identity with a=kx-ωt and b=kx+ωt, we get 

 

 

This is an interesting result. If we consider how it depends in space, it always looks like Asin(kx) -- the same shape. But its amplitude is 2Acos(ωt). It varies with time! It doesn't look as if anything is moving left or right at all, even though we built this out of two moving waves.

 

To see what this looks like, run the PhET simulation of a standing wave in a 10 bead model. To get the particular case shown at the left, slide the slider in the green box to the maximum number of beads and move the amplitude up under "normal mode 1" as shown. Then run the simulation by pressing the "Start" button.

 

You'll see the shape oscillating -- pinned down at the ends, but never changing its shape. 

 

The trick is that because the shape always looks like sin(kx), if you choose k so that sin(kL) = 0, then the oscillating string will ALWAYS be 0 both at 0 and at L. You will be able to pin the string down there and it won't matter that you built the original solution out of traveling waves that are assumed to be running on a string of unlimited length.

 

Choosing sin(kL) = 0 gives many different possibilities. Since sin(nπ) = 0 for any n, we can choose a whole range of different values of k that will produce an oscillation that keeps the endpoints fixed. The n-th one is:

 

knL = nπ     n = 1, 2, 3, ...      or     kn = nπ/L

 

(Why doesn't n=0 work?) These different values are called the normal modes of the system and the frequencies associated with them,  ωn = knv0 are called the natural frequencies of the system.   

 

We can figure out what this means by looking for the wavelength corresponding to each particular k. Since making sense of sinusoidal waves shows us that

 

λ = 2π/k

 

we see that for the n-th mode, 

λn = 2π/k= 2πL/nπ = 2L/n.

  

This means for the first mode (n = 1) the wavelength is twice L -- so one half of a wavelength fits perfectly between 0 and L. For the second mode (n =2), the wavelength = L, so a whole wavelength fits in. For the third mode (n = 3), the wavelength = 2L/3 so one-and-a half wavelengths fit in, and so on. These are shown in the image at the right. Explore these oscillations using the PhET sim to see how it works.

 

You can start your tied-down-at-the-ends system in any shape you want as long as the ends are both zero, but the only shape that will oscillate with a single clean frequency, maintaining its shape at all times, is one that starts like a pure normal mode. You can explore mixing of multiple modes and see what happens in the sim.

 

Normal or standing wave modes are of great importance. They represent the natural frequencies that something will oscillate at if struck. Not only are they responsible for the sounds produced by musical instruments, but the same phenomenon is what is responsible for the discrete quantum states we have discussed (see for example Quantum oscillators -- discrete states.

 

Of course these generalize to two and three dimensions as well. Watch this video link to see how to use standing waves to measure the speed of light with a microwave oven and a tray of peeps!

 

In this reading, we have only considered standing wave that arise when the ends of the oscillating medium are constrained to be 0 (nodes). In some physical situations (e.g., an open organ pipe), the ends are constrained to be extreme values (anti-nodes) rather than 0. There, the spatial derivative of what is oscillating is required to be 0 at the ends rather than the values. These end conditions also lead to standing waves but with different frequencies (as you can imagine by trying to fit waves in with maxima or minima rather than zeros.

 

 

Joe Redish 4/11/16 

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