Wave Speed on a Stretched String

We can calculate the speed of a transverse wave on a stretched string as follows:

v² = F_t / μ, μ = m/L

• v is speed of the transverse wave
• F_t is tension of the stretched string
• m is mass of the string
• L is length of the string
• μ is linear mass density of the string

Fixed End Reflection

Consider an elastic rope stretched from end to end. One end will be securely attached to a pole on a lab bench while the other end will be held in the hand in order to introduce pulses into the medium. Because the right end of the rope is attached to a pole, the last particle of the rope will be unable to move when a disturbance reaches it. This end of the rope is referred to as a fixed end.

If a pulse is introduced at the left end of the rope, it will travel through the rope towards the right end of the medium. This pulse is called the incident pulse since it is incident towards (i.e., approaching) the boundary with the pole. When the incident pulse reaches the boundary, two things occur:

• A portion of the energy carried by the pulse is reflected and returns towards the left end of the rope. The disturbance that returns to the left after bouncing off the pole is known as the reflected pulse.
• A portion of the energy carried by the pulse is transmitted to the pole, causing the pole to vibrate.

Notable characteristics of the reflected pulse include:

• The reflected pulse is inverted. That is, if an upward displaced pulse is incident towards a fixed end boundary, it will reflect and return as a downward displaced pulse. Similarly, if a downward displaced pulse is incident towards a fixed end boundary, it will reflect and return as an upward displaced pulse. (Explanation: As the last particle of rope pulls upwards on the first particle of pole, the first particle of pole pulls downwards on the last particle of rope. This is merely Newton's third law of action-reaction. For every action, there is an equal and opposite reaction. The upward pull on the first particle of pole has little effect upon this particle due to the large mass of the pole and the lab bench to which it is attached. The effect of the downward pull on the last particle of rope - a pull that is in turn transmitted to the other particles - results in causing the upward displacement to become a downward displacement.)

• The speed of the reflected pulse is the same as the speed of the incident pulse(Why? Same medium).
• The frequency and wavelength of the reflected pulse are the same as the frequency and wavelength of the incident pulse (Why? Same wave-producing source and same medium).
• The amplitude of the reflected pulse is less than the amplitude of the incident pulse (Why? Energy loss).

Free End Reflection

Now consider what would happen if the end of the rope were free to move. Instead of being securely attached to a lab pole, suppose it is attached to a ring that is loosely fit around the pole. Because the right end of the rope is no longer secured to the pole, the last particle of the rope will be able to move when a disturbance reaches it. This end of the rope is referred to as a free end.

Once more if a pulse is introduced at the left end of the rope, it will travel through the rope towards the right end of the medium. When the incident pulse reaches the end of the medium, the last particle of the rope can no longer interact with the first particle of the pole. Since the rope and pole are no longer attached and interconnected, they will slide past each other. So when a crest reaches the end of the rope, the last particle of the rope receives the same upward displacement; only now there is no adjoining particle to pull downward upon the last particle of the rope to cause it to be inverted. The result is that the reflected pulse is not inverted. When an upward displaced pulse is incident upon a free end, it returns as an upward displaced pulse after reflection. And when a downward displaced pulse is incident upon a free end, it returns as a downward displaced pulse after reflection. Inversion is not observed in free end reflection.

Transmission of a Pulse Across a Boundary from Less to More Dense

Let's consider a thin rope attached to a thick rope, with each rope held at opposite ends by people. And suppose that a pulse is introduced by the person holding the end of the thin rope. If this is the case, there will be an incident pulse traveling in the less dense medium (the thin rope) towards the boundary with a more dense medium (the thick rope).

Upon reaching the boundary, the usual two behaviors will occur.

• A portion of the energy carried by the incident pulse is reflected and returns towards the left end of the thin rope. The disturbance that returns to the left after bouncing off the boundary is known as the reflected pulse.
• A portion of the energy carried by the incident pulse is transmitted into the thick rope. The disturbance that continues moving to the right is known as the transmitted pulse.

The reflected pulse will be found to be inverted in situations such as this. During the interaction between the two media at the boundary, the first particle of the more dense medium overpowers the smaller mass of the last particle of the less dense medium. This causes an upward displaced pulse to become a downward displaced pulse.

The transmitted pulse is not inverted. The more dense medium was at rest prior to the interaction. The first particle of this medium receives an upward pull when the incident pulse reaches the boundary. Since the more dense medium was originally at rest, an upward pull can do nothing but cause an upward displacement. For this reason, the transmitted pulse is not inverted. In fact, transmitted pulses can never be inverted. Since the particles in this medium are originally at rest, any change in their state of motion would be in the same direction as the displacement of the particles of the incident pulse.

The Before and After snapshots of the two media are shown in the diagram below.

Comparisons can also be made between the characteristics of the transmitted pulse and those of the reflected pulse.

• The frequency of the incident pulse, transmitted pulse, and reflected pulse will all be the same.
• The speed of transmitted pulse (in the more dense medium) is slower than the reflected pulse (in the less dense medium).
• The wavelength of the transmitted pulse (in the more dense medium) is smaller than that of the reflected pulse (in the less dense medium).
• The speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse.

Explanation:

The speed of a wave is dependent upon the properties of the medium. In this case, the transmitted and reflected pulses are traveling in two distinctly different media. Waves always travel fastest in the least dense medium. Thus, the reflected pulse will be traveling faster than the transmitted pulse.

Second, particles in the more dense medium will be vibrating with the same frequency as particles in the less dense medium. Since the transmitted pulse was introduced into the more dense medium by the vibrations of particles in the less dense medium, they must be vibrating at the same frequency. So the reflected and transmitted pulses have the different speeds but the same frequency. Since the wavelength of a wave depends upon the frequency and the speed, the wave with the greatest speed must also have the greatest wavelength.

Finally, the incident and the reflected pulse share the same medium. Since the two pulses are in the same medium, they will have the same speed. Since the reflected pulse was created by the vibrations of the incident pulse, they will have the same frequency. And two waves with the same speed and the same frequency must also have the same wavelength.

Transmission of a Pulse Across a Boundary from More to Less Dense

Consider a thick rope attached to a thin rope, with the incident pulse originating in the thick rope. If this is the case, there will be an incident pulse traveling in the more dense medium (thick rope) towards the boundary with a less dense medium (thin rope). Once again there will be partial reflection and partial transmission at the boundary. The reflected pulse in this situation will not be inverted. Similarly, the transmitted pulse is not inverted (as is always the case). Since the incident pulse is in a heavier medium, when it reaches the boundary, the first particle of the less dense medium does not have sufficient mass to overpower the last particle of the more dense medium. The result is that an upward displaced pulse incident towards the boundary will reflect as an upward displaced pulse. For the same reasons, a downward displaced pulse incident towards the boundary will reflect as a downward displaced pulse.

The Before and After snapshots of the two media are shown in the diagram below.

Comparisons between the characteristics of the transmitted pulse and the reflected pulse lead to the following observations.

• The frequency of the incident pulse, transmitted pulse, and reflected pulse will all be the same.
• The speed of the transmitted pulse (in the less dense medium) is faster than the reflected pulse (in the more dense medium).
• The wavelength of the transmitted pulse (in the less dense medium) is larger than the reflected pulse (in the more dense medium).
• The speed and the wavelength of the reflected pulse are the same as the speed and the wavelength of the incident pulse.