9.2.P2

When a stretched string or spring that supports transverse waves is tied down at both ends, certain motions of the string are particularly simple -- standing waves. A standing wave is one in which all parts of the system oscillate with the same frequency. In this case there is no "traveling" component of the wave.*  In this problem, we'll explore some characteristics of standing waves using a "bead-and-spring" model as illustrated in the Normal Modes Simulation from the PhET group at Colorado.

In this simulation between one and 10 beads connected to walls and to each other by springs can be put in different shapes and then oscillate, their motion determined by the program from the forces felt by each bead and Newton's second law.   Download the program from the PhET website and run it (or run it at the PhET website). Select the following parameters: Sim. speed = normal Number of masses = 10 (slider all the way to the right) "Show springs" and "Show phases" boxes checked Polarization control = vertical (for transverse oscillations.   The screen should look like the figure shown at the right.

One way to create a standing wave is to move all the beads to a particular position and release them. This is what the program does. If the string has a length L and the beads starting shape is controlled by the function

y(x,t=0) = A sin(πnx/L)         n = 1, 2, 3, ...

then the starting shape will look like a half of a sine wave for n = 1, a full sine wave for n = 2, one-and-a half sine waves for n = 3, etc. Use the program sliders to adjust the amplitudes of the normal modes (one at a time) to see that this works.

A. Start the oscillation off in each of the modes, one at a time and run the program in order to confirm that if the shape of the beaded string is started off in a "pure" normal mode (only one of the amplitudes is non-zero) then the string will perform a periodic oscillation -- the same shape reappearing at regular intervals. Do and report the results of the following investigations.

A.1 The higher modes seem to move faster than the slower ones. Unfortunately, the program does not have a clock. Set the sim speed so that the first four modes complete one oscillation in between 5 and 30 seconds. Use a clock with a second timer to measure how long it takes each of the first 4 modes to complete an oscillation.

A.2 How does the period of the oscillation depend on the amplitude of the oscillation? Check at least three amplitudes for at least two different modes.

A.3 Explain your results in terms of the implications of the shapes of the starting positions on the forces on each bead. In particular: Why should the higher modes have shorter periods? Why should the periods not depend on the amplitudes?

B. When only one amplitude is set to be non-zero you should have found that the motion of the beads repeated themselves.

B.1 Now try setting all the sliders at non-zero values. Describe the motion of the string. Does it appear to repeat?

B.2 What do the "phases" sliders do?

B.3 Can you find settings for which more than one slider is significantly different from 0 in which the pattern appears to repeat? If you find one, indicate which sliders you had where and about how long a period it had compared to the period of the first normal mode.

C. The idealized theory of the beads and springs (massless springs, small angle approximation) predicts that the motion of the string as a function of position and time should be

y(x,t) = A sin(πnx/L) cos(ωt + φ)       n = 1, 2, 3, ...

If we think of πnx/L as a kind of "kx", find ω as a function of the tension and mass density of the string.

* Though a standing wave may be considered as the sum of two traveling waves of equal amplitude and having the same frequency that are traveling in opposite directions.

Joe Redish 4/13/13