PhysicsQuest

WAVE BEHAVIOR
ACTIVITY III.

STANDING WAVES

Introduction
Part 1. Both Sides Closed
Part 2. One Side Open

INTRODUCTION

If you shake the end of a stretched rope, waves travel down it to the fixed end and are reflected back. The waves going down and back interfere. In most cases, the combined waveforms have a changing, jumbled appearance. But if the rope is shaken at just the right frequency, a steady waveform, or series of uniform loops, appears to stand in place along the rope.

Appropriately this phenomenon is called a Standing Wave. It arises because of interference with the reflected waves, which have the same wavelength, amplitude and speed. Since the two identical waves travel in opposite directions, the net energy flow down the rope is zero. The energy is “standing” in the loops.

1. Some points on the rope remain stationary at all times and are called nodes. At these points, the displacements of the interfering waves are always equal and opposite. If you apply the principle of superposition what kind of interference occurs at these points?

2. At other points, the rope oscillates back and forth at the same frequency. The points of maximum amplitude are called antinodes. What kind of interference must occur at these points?

Standing waves can be generated in a rope by more than one driving frequency. The frequencies at which standing waves are produced are called natural frequencies or resonant frequencies. The lowest natural frequency is called the fundamental frequency or first harmonic. All the other frequencies are called harmonic series (the first overtone is the second harmonic and so on.)
PART II. ONE SIDE OPEN

- Click on the LOWER button to the fundamental frequency.
- Click on ONE SIDE OPEN
- Record the wavelength, frequency, number of nodes and antinodes,
  and fraction of a wave that fill the tube.

QUESTIONS:

1. What pattern do you notice in terms of increasing the harmonic mode and the
   number of waves to fill the one-meter long tube?

2. What pattern is seen in the numerator value of the fraction of waves to fill the
   tube?

3. What pattern is seen regarding the change in frequency as the harmonic mode
   is increased?

4. In general, for a tube with one side open, the fundamental harmonic wavelength
   is  equal to _________________ the length of the tube.
CONCLUSION

1. Derive an equation in terms of v, L and the number of harmonic (n) to calculate
   the natural frequencies for standing waves in an open tube with both sides
   closed. Clearly show all the steps in your derivation.

2. Derive an equation in terms of v, L and the number of harmonic (n) to calculate
   the natural frequencies for standing waves in an tube with one side open.
   Clearly show all the steps in your derivation.
QUESTIONS:

1. What pattern do you notice in terms of increasing the harmonic mode and the
   number of waves to fill the one-meter long tube?

2. What pattern is seen in the numerator value of the fraction of waves to fill the
   tube?

3. What pattern is seen regarding the change in frequency as the harmonic mode
   is increased?

4. In general, for a an open tube with both sides closed, the fundamental harmonic
   wavelength is equal to _________________ the length of the tube.
PART I. BOTH SIDES CLOSED

- Open the STANDING LONGITUDINAL WAVES simulation. This simulation
  shows the formation of standing waves for longitudinal waves.
- Click on BOTH SIDES CLOSED
-  Record the wavelength, frequency, number of nodes antinodes,
  and fraction of a wave that fill the tube.
- To switch to the next harmonic press the HIGHER button.

If you have difficulty visualizing the number of waves take a look
at the diagram below.