: Unit 8 Reading Notes –
PHYS 302: Vibrations and Waves – Unit 8 Reading Notes
French, A. P.
Vibrations and Waves
. New York: W.W. Norton & Company, 1971.
Chapter 7: Progressive Waves (pp. 201–216, up to “Wave Pulses”).
Chapter 6: Normal Modes of Continuous Systems. Fourier Analysis (pp. 167–170, up to “Longitudinal
Vibrations of a Rod”).
Chapter 7: Progressive Waves (pp. 228–230, up to “Dispersion…”).
Chapter 8: Boundary Effects and Interference (pp. 253–259, up to “Impedences…”).
Chapter 8: Boundary Effects and Interference (p. 264, bottom, and p. 265, up to “Waves in Two
Chapter 7: Progressive Waves
According to the video lectures, we regard a wave as anything that obeys the wave equation,
function of the form
) will solve this equation. The text
describes ocean waves and stresses the
idea that waves carry energy from one place to another. When we think of ocean waves, we can think of a tsunami
pulse in which only one peak would come across the ocean and that will obey the wave equation. But it is more
usual to think of waves with many crests, which are actually periodic travelling waves. The video lectures, which
take a more direct approach than the textbook, show that periodic travelling waves are well described by a function
of the form
, where we choose the sign (+ or –) depending on the direction of propagation
of the wave. The textbook approaches this differently, through normal modes. The textbook approach is circular:
one can relate standing waves to travelling waves by carefully applying boundary conditions. We will now look at
how normal modes lead to travelling waves. As the end of this section mentions, water waves are actually
complicated to analyze and are not a significant part of
Normal Modes and Travelling Waves
Fig. 7-1 shows details of a process demonstrated in the video lecture demo (with little commentary). One can
imagine moving the end of a stretched rope up and down (as in (a)) to generate a travelling wave. If the rope was
very long (imagine the length is infinite) the waves would continue towards the right forever. However, if there are
fixed ends, there will be (negative) reflections, and if the wavelength is chosen correctly, these reflections will cause
standing waves as other waves return from the reflection point. Much as the original wave is generated by the forces