17.3 - Reflected waves and resonance
We have considered standing waves formed by two waves generated by separate
sources, but they can also be formed by a single wave reflecting off a fixed point. This is
the basis behind the sound production of many musical instruments. We will discuss
this topic using the example of a piece of string, with one end connected to a wave-
making machine that vibrates sinusoidally, moving the string up and down, and the
other end fixed to a wall.
To start, consider what happens when the wave machine generates a single pulse, as
illustrated above. We show the pulse at three positions over time. When the pulse
reaches the wall, it “yanks” on the wall. Newton’s third law dictates that there will be an
equal “yank” in the opposite direction, which sends a reflected pulse back down the
string. The reflected pulse is inverted from the original, so when a peak reaches a wall,
a trough returns in the opposite direction.
A wave is a continuous series of pulses. When the wave machine vibrates continuously,
each pulse will reflect off the wall, resulting in an inverted wave moving at the same
speed in the direction opposite to the original wave. The reflected wave travels down
the string toward the wave machine. We also consider the wave machine as fixed,
which is reasonable if its vibrations are small in amplitude. The wave then reflects off that piece of machinery just as it reflected off the wall.
The wave machine continues to vibrate, sending wave pulses down the string. If it vibrates in synchronization with the reflected wave, the result
will be two traveling waves of equal amplitude, frequency and speed, moving in opposite directions on the same string. This system has
created the condition for a standing wave: two identical traveling waves moving in opposite directions. You see this illustrated in Concept 2.
On the other hand, if the wave machine is not in sync with the reflected wave, it will continue to add “new” waves to the string that will combine
in more complicated ways, and there may be no obvious pattern of movement on the string.
If the wave machine works in synchronization with the reflected waves to create a standing wave, we say that it is working in resonance. Its
motion reinforces the waves, and the amplitude of the resulting standing wave will be greater than the amplitude of the vibrations of the wave
machine. This is akin to you pushing a friend on a swing. If you time the frequency of your “pushes” correctly, you will send your friend higher.
We will discuss next how this frequency is determined.
Pulse on string
Reflection is initial pulse inverted
Wave on string
Reflects at wall
Creates standing wave17.4 - Harmonics
Fundamental frequency: The
frequency of a standing wave
in a vibrating string that has
two nodes.
Harmonic: A frequency of a
standing wave in a string that
has more than two nodes.
We have considered vibrating strings fairly abstractly. However, strings connected to two fixed points are the basis for musical instruments
such as violins, cellos and so forth. To put it another way: Orchestras have “string” sections.
In this section, we want to put your knowledge of standing waves into practice. To do so, we ask a question: When a musician plays a note,
what determines the frequency at which the string will vibrate? To put the question another way, what are the possible frequencies of a
standing wave on a string fixed at both ends?
A tuning peg is used to change the frequency of a guitar string.