An example of wave propagation on a string composed of two segments of
different properties: the left with densityL = 1.0,
velocityL = 3.0, and the right with densityR =
4.0, velocityR = 1.5. The arrow marks the position of the
source (distance 6.5) which plucked the string at time 0. The vertical
dashed line indicates the position of the junction. Both ends of the
string are fixed.
The coefficients of reflection and transmission (shown above) depend
not only on the string properties, but on the direction that the wave
is travelling. As the wave travels from the right to the left, the
density of the string decreases, and the velocity increases so that the
reflection coefficient is strictly positive and the transmission
coefficient is greater than 1. This means that the reflected wave will
have the same sign as the incident, and that the transmitted wave
will have an amplitude greater than the incident.
This animation illustrates several important facets of wave propagation:
When a wave encounters a fixed boundary, only a reflected
pulse is generated
The polarity of the reflected pulse is opposite that of the
incident pulse for a fixed boundary
When a wave encounters the middle junction, both a
transmitted and a reflected pulse are generated
The polarity of the reflected pulse depends on the sign of
the reflection coefficient at the middle junction
When crossing the middle junction from right to left, the
transmitted wave has a larger amplitude than the incident
wave because the coefficient of transmission from the right to the left
is greater than one. Though this effect seems counter-intuitive, it
works because the total energy in the waves is a constant,
rather than the amplitude of the waves.
The wavelength of a pulse changes when going across the middle
junction, because the two halves have different velocities. The lower
velocity on the right-hand side gives a shorter wavelength.
Finally, since this is a linear process, the waves can add both
constructively and destructively. However, the waves have
no lasting effect on each other; after passing through each other, they
are unchanged. This idea is the basis for Fourier
series methods, which represent any piecewise continuous function as a
series of many harmonic waves.
Even though the example of the string is simple, the concepts it
illustrates are used everyday by scientists and engineers. By examining
seismic waves, we have learned much about the structure of the Earth.
Oil and mining companies regularly use this technique to locate and