V12.  Video Demonstration:  Standing Waves on a String

Goals:  1) To review and apply what you've learned about standing waves to waves on a string, and 2) to investigate quantitatively the affect of tension on waves on a string

Introduction: In this demonstration, two relationships involving wave speed are needed. One of them comes from the formula v = Dd/Dt and is v = f l.  The other relates to the rigidity and inertia of the medium. This relationship is:

Wave speed = (Tension/Linear density)½

Linear density is the ratio of the mass of a unit length of the string to that length.  In terms of the symbols FT for tension and µ for linear density,

v = (FT/µ)½.

It's important to realize that the latter relationship is what determines the speed of waves on a string.  v = f l doesn't determine the speed of waves on a string; v = f l simply describes the motion without addressing the causes of the motion.  An analogous situation has to do with the relationships a = Fnet/m and a = Dv/Dt.  Newton's 2nd law says the cause of the acceleration of an object is the sum of the forces on the object, while a = Dv/Dt describes how the velocity changes with time without saying why it changes.

The apparatus used in this demonstration produces standing waves on a horizontal string. There are nodes at each end of the spring. One end is attached to a driver that oscillates at a constant frequency. The other end passes over the end of a pulley and has weight hanging on it to produce tension. By adjusting the frequency of the oscillator or the tension in the string, standing waves with various numbers of antinodes can be produced. The apparatus is shown below.

Note some similarities between this situation and that of the Standing Waves on a Spring experiment.  In that experiment, the hand of the person producing the standing waves is like the oscillator in the string setup above.  The person holding the fixed end of the spring is like the fixed pulley in the string setup.  In both experiments, the string and spring are held at constant lengths.

View the video clip now.

Questions and activities:

Write answers to the following for practice and review.

  1. How could you accurately determine the linear density of the string?

  2. How can you determine the tension in the string?

  3. Given the weight hanging from the string and the linear density, how can you calculate the wave speed in the string?

  4. Draw the standing wave pattern of the fundamental frequency.

  5. If the length of the string is L, what is the wavelength of the fundamental?

  6. What frequency do we need to set for the oscillator to generate the fundamental?

  7. What if we wanted to generate the 2nd harmonic?  What frequency would be needed?

  8. Will this change in the frequency change the wave speed? Explain.

  9. How could we change the wave speed to make it, say, 25% greater than its present value?

  10. If we changed the wave speed, how would the wavelength have to change to produce 2 antinodes?

  11. For the greater wave speed, how would the frequency have to change to produce 2 antinodes?

  12. Suppose one did an experiment to measure fn vs. n for n = 1 to 4 with the string under constant tension. If one then did a linear fit to a plot of fn vs. n, how would one use the slope of the fit to determine the wave speed?