Consider a wave traveling on a string. The transverse displacement of the string
ID: 1302940 • Letter: C
Question
Consider a wave traveling on a string. The transverse displacement of the string, y(x,t), as a function of position along the string, x, and time, t, is given by:
y(x,t) = A sin(2?x/? - 2?ft + ?)
1)
Find the values for the following quantities consistent with the functions shown in the graphs.
The amplitude: A =
mm
2)
The frequency: f =
Hz
3)
In order to calculate the wavelength, you must know the the speed as well as the frequency. The wave speed can be found from the two graphs: the peak that is at x = 0.0 m when t = 0 s is the same one that has moved to x = 0.03 m when t = 0.001 s. This tells us that the wave speed is 0.03 m / 0.001 s = 30 m/s. Now the wavelength can be calculated.
Note that there is an ambiguity. The peak at (x=0, t=0) might not have reached x = 0.03 m until t = 0.005 s. In that case, the speed is 6 m/s. We'll ignore this possibility. (This is the stroboscopic effect that makes the wagon wheels in old western movies appear to be rotating backwards.)
The wavelength: ? =
m
4)
By choosing the phase, ?, you can slide the graph to the left or right. Find the smallest positive value of ? that correctly describes the above wave.
?min =
Explanation / Answer
1) Amplitude is the max or min (usually the same) distance from the orgin that the wave travels. In this case, it is 1.5 mm.
2) Frequency is the number of cycles in a given time period. It takes the wave .004 seconds to travel 1 cycle (peak to peak). Thus, f = 1/.004 = 250 Hz.
3) Speed = f*wavelength. So, wavelength = speed/f. Therefore, 30/250 = .12 m.
4) 90 degrees as that is the phase shift needed to shift the graph so that it starts at y = 0.