Ernst attaches a stretched string to a mass that oscillates up and down once every half
second, sending waves out across the string. He notices that each time the mass reaches the
maximum positive displacement of its oscillation, the last wave crest has just reached a bead
attached to the string 1.25 m away. What are the frequency, wavelength, and speed of the
waves?DETERMINING FREQUENCY:
The oscillation of the mass on the spring determines the oscillation of the string, so the
period and frequency of the mass’s oscillation are the same as those of the string. The
period of oscillation of the string is T = 0.5 s, since the string oscillates up and down once
every half second. The frequency is just the reciprocal of the period: f = 1 /T = 2 Hz.
DETERMINING WAVELENGTH:
The maximum positive displacement of the mass’s oscillation signifies a wave crest. Since
each crest is 1.25 m apart, the wavelength, , is 1.25 m.
DETERMINING WAVE SPEED:
Given the frequency and the wavelength, we can also calculate the wave speed:
m/s.
Phase
Imagine placing a floating cork in the sea so that it bobs up and down in the waves. The
up-and-down oscillation of the cork is just like that of a mass suspended from a spring: it
oscillates with a particular frequency and amplitude.
Now imagine extending this experiment by placing a second cork in the water a small
distance away from the first cork. The corks would both oscillate with the same frequency
and amplitude, but they would have different phases: that is, they would each reach the
highest points of their respective motions at different times. If, however, you separated
the two corks by an integer multiple of the wavelength—that is, if the two corks arrived at
their maximum and minimum displacements at the same time—they would oscillate up
and down in perfect synchrony. They would both have the same frequency and the same
phase.