# Section1Waves on a Spring¶ permalink

In order to perform useful and revealing experiments with waves, we must be able to generate waves of good shape. The hand-and-foot method works very well. Two people (labeled #1 and #2 in Figure 1.1) stretch out a spring and hold its ends at rest (one option is to stand on the end of the spring). Person #3 uses one hand and one foot to create the wave, as seen in Figure 1.1. In this way, we can produce waves of any desired length ($d$ in Figure 1.1) and amplitude ($A$). Person #3 releases the wave by letting go of the spring with their hand. The foot should remain in place. The wave will then flip over and travel down the spring, as shown in Figure 1.2.

Start with the long, small-diameter steel spring. Attach a flag of masking tape roughly in the middle of the spring. Watch the motion of the tape as the wave passes by. The tape moves perpendicular to the motion of the wave pulse. The reason that it makes sense to call this type of wave a transverse wave is that the prefix “trans” means other or different implying that a particle on the wave would move in a different direction than the wave pulse.

Using a slinky it is possible to generate a different kind of wave, called a longitudinal wave. We can use the technique shown in Figure 1.3 to create such a longitudinal compression wave.

Contrary to the transverse wave, a piece of tape affixed in the middle of the slinky moved in the same direction as the wave pulse, which moves within the same direction as the space it occupies through compression and subsequent decompression as the pulse passes.

# Subsection1.3What does the speed of the transverse wave depend upon?¶ permalink

It is possible to ascertain what the speed of a transverse wave depends upon by using a stop watch to measure the speed of waves while varying different variables.

We will examine the effects (or lack thereof) of wave amplitude, wave length, and the mass-density/tension of the spring on the speed of the wave. To do this we will vary the amplitude and wave length by altering the setup in the hand-and-foot method of wave generation; to vary mass-density and tension (which will change simultaneously) we will alter the length to which the spring has been stretched. When we stretch out the spring, we are actually changing two variables at once: we are putting the spring under greater tension and decreasing the density of the spring (the number of grams per centimeter of length).

From our data in Figures 1.4, 1.5, and 1.6, we can see that neither amplitude nor the width of the pulse affect the speed of the wave, while the spring length (which accounts for both mass density and tension) is linearly related to the speed of the wave.

# Subsection1.4Experiments with Crossing Waves¶ permalink

Four people (or two especially nimble ones) can produce nice crossing waves using the technique shown in Figure 1.7. In this way, you can produce waves on the same side or on opposite sides of the spring. Persons #3 and #4 should chant together, “One, two, three, four,” and release on “four.”

First we sending two waves on the same side of the spring; watch both waves as they travel down the spring in opposite directions and cross over one another. Notice that each of the waves arrives at the far end of the spring unaffected. That is, after the waves have crossed over one another, each continues on its way just as it would have if the other wave had not existed. This is one of the remarkable properties of waves that Huygens alludes to near the beginning of his Treatise on Light. Assuming that light is a wave, it explains why two people can gaze into one another's eyes. Now consider sending two waves at one another on opposite sides of the spring, the same is true in that they will arrive at the other end essentially unaltered, however what occurs between the ends is different.

# Subsection1.5Constructive and destructive interference¶ permalink

At the moment that the two waves are passing through one another, the actual shape of the spring depends on the fact that both waves are present. The wave forms can add together, or subtract, depending on whether the waves are on the same side or on opposite sides of the spring. The following experiments make this dramatically clear.

# Subsubsection1.5.1A Case of Constructive Interference

Let two people stretch out a spring on the floor. Arrange upside-down Styrofoam coffee cups as shown, about 12 inches from the spring.

Generate waves on the same side of the spring, each 10 inches in amplitude.

The two wave-makers should release in unison, using the chanting technique.

As can be seen in the Constructive Video, the two waves which could not independently disturb the cups, when experiencing constructive interference from both being on the same side of the spring exhibit an increase in amplitude allowing them to knock over the center cup(s).

# Subsubsection1.5.2A case of Destructive Interference

Now place the lines of coffee cups about 6 inches from the spring. Generate waves of 12-inch amplitude on opposite sides of the spring. Let fly in unison! (Using the chanting technique, of course.)

As can be seen in the Destructive Video, while the independent waves knocked over the outer cups, the inner cups where undisturbed due to the cancellation of amplitude.

# Subsubsection1.6.1Reflection from a Fixed End¶ permalink

Put a spring on the floor and stabilize one end (one option is to stand on it). Send in a wave and watch the wave while it is reflected from the fixed end of the spring. The reflected wave comes back on the opposite side of the spring as the incident wave

You can make this dramatically clear in the following way. Place an upside-down coffee cup a few inches from the spring near the middle of the spring. Then send a one-foot high wave down the other side of the spring, as shown. The wave will pass right by the cup without disturbing it. However, after reflection, the wave will come back on the cup's side of the spring and knock it over.

# Subsubsection1.6.2Reflection from a Free End¶ permalink

Tie a long piece of string to the spring, and stabilize the end of the string. This leaves the end of the spring free to wiggle back and forth.

Now send in a wave and watch to see that the wave is reflected on the same side.

# Subsubsection1.6.3Displacement of the Free End During Reflection¶ permalink

During the process of reflection, there is a brief time interval in which part of the wave has already been reflected and is traveling to the left, while part of the wave has not yet been reflected and is still traveling to the right. These two partial waves cross through one another and can add up, or interfere constructively.

Set up the arrangement shown in Figure 1.13. The two cups are placed one foot, four inches from the spring. The wave pulse is one foot high.

We see that the constructive interference allows the string to strike the cup the initial wave would not.

If there were no friction (and thus no loss of wave amplitude), the free end would experience a displacement doubling the amplitude of the initial wave.

Fasten the light brass spring to the small-diameter steel spring. Stretch them out (but be careful not to overstretch the brass spring, which is easily damaged).

# Subsubsection1.7.1Wave Speed

Try making a rectangular wave pulse on the steel spring as shown in Figure 1.14.

The speed of the wave as it crosses into the brass is decreased. Due to the tension being equal in both springs (as unequal tension would exert a force that would cause the spring to move towards equilibrium), we can conclude that the density of the brass spring is much lower than the density of the steel spring. This shows the dramatic dependence of the wave speed on a single variable (the density).

Try making a rectangular pulse on the brass spring and sending it towards the steel spring. As can be seen in the Double Spring Video, the speed of the wave increases dramatically as it undergoes a “phase” change.

# Subsection1.7.1Wavelength

Performing the same experiments, we can see that the length of the rectangular wave is decreased. This illustrates the intimate relationship between speed and wavelength. This relationship is an inverse relationship meaning that an increase in speed implies a decrease in wavelength and vice versa.

# Subsection1.8Partial Reflection and Transmission¶ permalink

Try sending a wave from the light brass spring towards the heavy steel spring, as in the Double Spring Video. Consider the small wave reflected from the interface between the two springs: Just as in optical reflection there is a portion of the light which is reflected while there is also a portion that is “refracted”. This is analogous to the optical phenomena of the partial reflection of light rays. This partially reflected wave flips over the spring while the “refracted” portion which continues on remains on the same side of the spring. This makes sense because in terms of rigidity the steel spring acts as a fixed end for the brass spring, albeit one that absorbs a portion of the wave allowing it to pass onward.

Now send a wave from the heavy towards the light spring. The partially reflected wave will travel on the same side of the spring as the light brass spring functions like a free end as in earlier experiments.

Stabilize one end of a long steel spring. Generate a continuous wave train by shaking the other end of the spring back and forth in a steady oscillation. You should find that for most shaking frequencies, the resulting motion of the spring is a messy jumble; however, at certain specific frequencies, you can produce a standing wave pattern. There are a number of different shaking frequencies that give rise to nice standing wave patters; these frequencies are such that the length of the spring is split evenly by the frequency.

The standing wave pattern is formed by aligning the waves so that the reflected waves coming back constructively interfere with the incident waves. This is also the reason that only particular shaking frequencies give a decent standing wave pattern as most frequencies would cause irregular interference resulting in a “jumble” of interference. With this knowledge, we know that the number of “bumps” in the standing wave pattern is related to the frequency as we for different frequencies will be “dividing” the spring into lengths and as this frequency increases so will the number of bumps.

# Subsection1.10Videos of Transverse Waves on Springs¶ permalink

These three videos are a good representation of what is seen in transverse waves along springs.