Answer:
8 feet.
The wavelength is the peak to peak distance, but can also be measured
in the following way: start at the left end, move past the first crest
to the node (1/3 of the way along), move past the trough to the next node
(2/3 of the length). The wavelength includes two consecutive antinodes
and it is 2/3 of the entire string length, or 2/3 of 12 = 8 ft. The wave
velocity is irrelevant to the question.
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2C.
Standing wave. Which mode? What frequency?
Again
consider the standing wave in the above figure. The length is still 12
ft and the wave velocity 16 ft/s. Which mode is this? What is the frequency
of this mode?
Answer:
This
is the 3rd harmonic or the 2nd overtone. The fundamental (or 1st harmonic)
has just one antinode fitting between the two walls (and two nodes—at
the ends). The first overtone (2nd harmonic) has two antinodes (and three
nodes); and the second overtone (3rd harmonic) has three antinodes (four
nodes) as does the figure.
The frequency
is wave velocity divided by wave length = (16 ft/s)/(8 ft) = 2 Hz.
___________________________
2C.
Standing wave. What can you change?
For a
standing wave on a given string
a) you can have any
and any f so long as the relationship v = f
holds.
b) only certain wavelengths and frequencies are allowed.
c) the wave velocity is totally determined by the medium.
d) the wave velocity depends on the dynamic and inertial parameters.
e) a and c only.
f) b, c and d only.
g) a, c and d only.
h) none of the above.
Answer:
f) When
you tie the ends of a string down, the wavelength must be such that there
are zeros (nodes) of the wave at the ends. This means that we cannot have
just any wavelength; the wavelength must be adjusted to fit into the space
properly. Thus only certain wavelengths, and hence frequencies, are allowed
and b) becomes true. c) and d) are also true: The wave velocity depends
on the properties of the medium such as the tension of a string or its
density (the dynamical and inertial properties mentioned).Thus the correct
answer is f).
