Characteristics of a Transmitted Pulse
A Less Dense to a More Dense Medium
A pulse (and a wave) carries energy through a medium from one location to another. But when the pulse reaches the end of the medium, where does the energy go? Does the energy disappear? Or does the energy pass into the new medium? The phenomenon which occurs when a wave reaches the end of the medium through which it travels is often termed boundary behavior. There are a variety of observations that can be made of the boundary behavior of a pulse. Such observations pertain to the changes (or lack of changes) in the frequency, wavelength, speed, amplitude, and phase of the pulse.
The animation below depicts the boundary behavior of a pulse which is moving along a less dense medium and incident towards (i.e. approaching) a more dense medium. Note that when the pulse reaches the end of the medium, a portion of its energy is transmitted into the more dense medium (in the form of a transmitted pulse), a portion of its energy remains in the less dense medium (in the form of a reflected pulse).
Rather than disappearing (and thus violating energy conservation), the energy carried to the boundary is divided up into a reflected pulse (which remains in the less dense medium) and a transmitted pulse (which passes across the boundary into the new medium). The reflected pulse has several noteworthy characteristics. First observe that the reflected pulse is inverted. Reflected pulses will always be inverted for boundary situations in which a pulse in a less dense medium reflects off the boundary with a more dense medium. Second, observe that the reflected pulse has a smaller amplitude than the incident pulse. The amplitude is representative of the energy carried by a wave. Since the total energy which is carried by the incident pulse is divided two ways at the boundary, the reflected pulse must have less energy than the transmitted pulse. This is the reason for why the energy of the reflected pulse (and thus its amplitude) would always be less than the energy of the incident pulse. Finally, observe that the speed and the wavelength of the incident pulse are the same as the speed and the wavelength of the reflected pulse. Wave speed depends upon the properties of the medium; and if the reflected pulse and incident pulse are in the same medium, then they must have the same speed.
Comparisons can also be made between the characteristics of the transmitted pulse and those of the incident pulse. Once more there are several noteworthy characteristics. First, observe that the transmitted pulse is not inverted. In fact inversion only occurs for the reflected pulse (if it occurs at all). Second, observe that the transmitted pulse has a smaller speed and a smaller wavelength than the incident pulse. This is always the case for boundary situations in which a pulse in a less dense medium reflects off the boundary with a more dense medium. Since wave speeds and wavelengths in strings are always greatest in a least dense medium, it would be expected that there is a decrease in wave speed and wavelength as the pulse crosses the boundary. Finally, when waves cross boundaries the frequency of the incident pulse is the same as the frequency of the transmitted pulse (though it is not evident from the above animation). The fact is that the vibration of the last particle in the incident medium creates the vibration of the first particle on the opposite side of the boundary. These two particles are joined in such a manner that the frequency at which one particle vibrates is equal to the frequency at which the other particle vibrates. Like two hands shaking with each other, the frequency at which one hand shakes can never be any different that the frequency at which the other hand shakes (assuming they remain adjoined to each other). It is this handshake principle that explains why the frequency of the incident pulse and the transmitted pulse must be the same.
In conclusion, the boundary behavior of waves is best summarized by the following statements:
- the wave speed is always greatest in the least dense medium,
- the wavelength is always greatest in the least dense medium,
- the frequency of a wave is not altered by crossing a boundary,
- the reflected pulse becomes inverted when a wave in a less dense medium is heading towards a boundary with a more dense medium,
- the amplitude of the incident pulse is always greater than the amplitude of the reflected pulse.
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