Teaching Physics with the Physics Suite Edward F. Redish Problems Sorted by Type | Problems Sorted by Subject | Problems Sorted by Chapter in UP

Without missing a beat!

When two waves on an elastic string or spring they superpose to produce a combined result. When two sinusoidal wave of different frequencies overlap, they produce some interesting effects. In this problem you will explore what happens using the nice wave simulation by B. Surendranath Reddy. The control panel is shown in the figure below. (From an old version -- the colors have now changed.)

What is displayed her are two sine waves traveling on a string (with their amplitude greatly exaggerated for easy viewing). They are displayed in the top two lines separately in red and green. Their sum is then displayed on the third line in yellow. The position of a single point on the string is shown as a white dot. Familiarize yourself with the program by playing with the various controls and seeing what happens.

We will take the scale to be: vertical boxes = 1 mm; horizontal boxes = 10 cm.

A. Set both frequencies to be 16 Hz (= 16 sec-1) and choose the radio button "in phase". Start the simulation and watch what happens. Then press "Stop" and step the waves along a bit at a time using the left and right arrows on either side of the blue button under "Start". Note the behavior of the white dots. Now switch the radio button to "Out of Phase" and step the simulation. Describe what you observe and why the sum does what it does.

B. Using the scales specified above, calculate the wavelength (λ), the period (T), the wave number (k), and the angular frequency (ω) of the red wave. Can you find the velocity with which the waves are traveling on the string? If so, find it. If not, explain why not. (The simulation no longer has vertical bars. Take the total width of the simulation to be 240 cm and measure the widths on your screen or on a printout.)

C. Now change the frequency of the green wave to be 15 Hz and watch the result run. Stop it so that the large "bump" in the sum is all on the screen at once. Describe the resulting sum and explain why it looks like it does.

D. Set the red and green wave frequencies to 20 Hz. Reduce the green wave frequency a step at a time and fill in the following table for a single wave (say the green one):
 Frequency (Hz) Period (s) Wavelength (cm) k (cm-1) 20 19 18 17 16

E. The pattern that appears when you add two waves of nearby frequency is called beats. If you listen to sound waves of nearby frequencies, you can hear them beat against each other. Acoustical guitarists sometimes tune their guitars by this method, fretting one string so that it should be the same as the next and then tuning the open string until the beats disappear. Try this link to listen to beats on the web. As the frequencies get farther apart, notice that the beats get faster. One interesting relation is the relation between the spatial size of the group of waves and the difference between the wave numbers. Fill out the following table using Δk = kred - kgreen. (The "width of the bump" is not always perfectly clear. There is not necessarily a clean zero between successive bumps. When there is an ambiguity, figure out a way to define what you mean by the "width of the bump" in a way that someone else reading your description could reproduce your numbers.)
 fred (Hz) fgreen (Hz) Width of bump in the sum (cm) Δk (cm-1) 20 19 20 18 20 17 20 16 20 15

The product of the "width of the bump in the sum" and Δk is a pure number. Plot this product as a function of the frequency of the green wave.

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