Deduce the expressions fo the reflection and transmission amplitude and energy coefficients when a transverse wave travelling in +x direction in the string of impedance Z1 meets the junction of a string of impedance Z2.
Answers
A non-dispersive system has the property that all waves travel with the same speed,
independent of the wavelength and frequency. These waves are the subject of this and
the following chapter (broken up into longitudinal and transverse waves, respectively). A
dispersive system has the property that the speed of a wave does depend on the wavelength
and frequency. These waves are the subject of Chapter 6. They’re a bit harder to wrap your
brain around, the main reason being the appearance of the so-called group velocity. As we’ll
see in Chapter 6, the difference between non-dispersive and dispersive waves boils down to
the fact that for non-dispersive waves, the frequency ω and wavelength k are related by a
simple proportionality constant, whereas this is not the case for dispersive waves.
The outline of this chapter is as follows. In section 4.1 we derive the wave equation for
transverse waves on a string. This equation will take exactly the same form as the wave
equation we derived for the spring/mass system in Section 2.4, with the only difference
being the change of a few letters. In Section 4.2 we discuss the reflection and transmission
of a wave from a boundary. We will see that various things can happen, depending on
exactly what the boundary looks like. In Section 4.3 we introduce the important concept
of impedance and show how our previous results can be written in terms of it. In Section
4.4 we talk about the energy and power carried by a wave. In Section 4.5 we calculate the
form of standing waves on a string that has boundary conditions that fall into the extremes
(a fixed end or a “free” end). In Section 4.6 we introduce damping, and we see how the
amplitude of a wave decreases with distance in a scenario where one end of the string is
wiggled with a constant amplitude.