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Resonance
Of all the types of waves we study, we are most familiar with water waves
as seen in oceans, lakes, rivers, and bathtubs. We’re also familiar with
waves created by air currents through fields of grasses or wheat. In reality,
we constantly experience waves of various types. Sound, light, radio, and
other forms of electromagnetic radiation surround us every moment of our
lives and although we do not directly “see” their waves, aside from visible
light, these phenomena can all be understood in terms of waves.
Furthermore, we show later that matter also behaves as a wave and that our
current quantum physics picture of the world is intimately connected with
a mathematical description known as the wave function. Waves are thus
the key to our understanding of nature on a fundamental level. In this
chapter we first return to the type of motion known as simple harmonic
motion that we used to describe a mass on a spring in Chapter 3. Here we
extend our previous discussions to include the frictional loss of energy,
known as damping, and the effects of a “driving force” used to sustain the
motion. With the addition of energy by this external force comes the
possibility of a resonance phenomenon in which the amplitude of
oscillation can grow rapidly. This is an extremely important idea in physics
that we will see often throughout the rest of our studies. We then introduce
some fundamental concepts concerning waves and consider traveling
waves along a string and along a coiled spring as mechanical examples of
the two basic forms of waves, transverse and longitudinal. As waves travel
along or through a medium, they meet and interact with boundaries or
obstacles, and different interactions possible at a boundary are considered,
including reflection and refraction. We also discuss one possible result
from such boundary conditions, the creation of standing waves. These are
important in such diverse areas as musical instrument
WAVE CONCEPTS
Mechanical waves are vibrational disturbances that travel through a
material medium (in this section we assume no energy dissipation).
Examples include water waves, sound waves traveling in a medium such
as air or water, waves along a string (as in a musical instrument) or along
a steel beam, or seismic waves traveling through the Earth. A general
characteristic of all waves is that they travel through a material medium
(except for electromagnetic waves which can travel through a vacuum) at
characteristic speeds over extended distances; in contrast, the actual
molecules of the material medium vibrate about equilibrium positions at
different characteristic speeds, and do not translate along the wave
direction. Mechanical waves on a stretched string can be directly
visualized. Imagine that we tie one end of a string to a fixed point and
stretch it tightly. We can send a wave pulse down the string by giving the
held end a single rapid up and down oscillation (Figure 10.7). The motion
of the string is vertical whereas the pulse travels horizontally along the
string. The vertical forces acting from one region of the string to the next
near the leading edge of the pulse are what sustain the pulse and cause it to
move along the string. If we continue to oscillate the held end at a fixed
frequency f, then we set up a series of identical oscillations, or a periodic
wave, that travels down the string (Figure 10.8). Such waves are called
transverse, because the medium oscillates in a plane perpendicular to the
direction in which the wave travels. Suppose we replace the string by a
stretched spring tied at one end. If we oscillate the free end of the spring
either once, or continuously, along the horizontal direction (along its axis),
we set up a longitudinal pulse, or periodic wave, in which the motion of
the material medium is an oscillation along the direction of propagation of
the wave (Figure 10.8). From a flash photo at some instant of time of the
string undergoing continuous oscillations, we can see that the wave
consists of a repeating series of positive (above axis, where the axis is the
unperturbed string) and negative (below axis) pulses. The distance between
corresponding points of one pulse and the next is called the wavelength, .
Because the waveform, or shape, is repetitive, or periodic, corresponding
points can be neighboring maxima, crests, of the wave, or minima, troughs,
of the wave, or any set of neighboring corresponding points (Figure 10.9).
254 WAV E S A N D RESONANCE FIGURE 10.6 One result of the 1989
earthquake near San Francisco, CA. The earthquake vibrations overlapped
with the suspended highway resonant frequencies causing large amplitude
vibrations leading to its collapse. FIGURE 10.7 Transverse wave pulse on
a string. A similar analysis applies to the longitudinal waves of the spring,
where now positive and negative refer to the compression or extension of
the spring compared to its unperturbed configuration. In this case it is easier
to see the wave variation with time clearly by performing the intermediate
step of graphing the longitudinal displacement as a function of time to
obtain a curve similar to Figure 10.9. As a wave moves along the string,
we can ask with what speed it is traveling. If we look at an arbitrary point
along the string, we will see exactly one wave move by in a period, the
time T 1/f required for one oscillation. The distance the wave travels in this
time is exactly