Homework on Waves, 2000 version (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material. You should show all of your calculations (neatly) and justify all of your answers for full credit.

The voltage from a signal generator is described by V(x,t) = (3.1 V) cos[(2.45 rad/m)x + (0.325 rad/s)t + 3.14]

1. Identify and/or calculate each of the following parameters of the wave: amplitude, wave number, wavelength, angular frequency, frequency, period, velocity, and phase constant. Show all calculations.
amplitude: 3.1 Volts
wave number (k): 2.45 rad/m
wavelength (l): l = 2p/k = 2.56 m
angular frequency (w): .325 rad/s
frequency (f): f = w/2p = .05 Hz
period (T): T = 1/f = 19.3 s
velocity (v): v = fl = .13 m/s
phase constant (f): 3.14

2. Sketch this wave as a function of distance for t = 0, from x=0 to x = 5.0 m.

One knows that the wavelength is ~2.5. Therefore, about 2 full cycles will fit in the window between x = 0 and x = 5.0. Cos (0) is equal to 1. However, because of the phase constant, the argument of the cosine function at x = 0 is actually 3.14. This shifts the starting point one half cycle so that the wave begins at its minima rather than maxima.

3. In which direction is the wave moving? How do you know this?
The wave is moving left. One knows this because the sign proceeding the wt term is positive. At larger values of t the argument of the cosine is increased- shifting the starting point of the graph to the right.

4. Sketch the wave at a slightly later time, t > 0, again from x = 0 to x = 5.0 m.

For t > 0 the argument of the cosine function becomes greater. The result of this is that the plot shows an area on the cosine curve which is shifted to the right relative to the first plot. Since our window in x is fixed (x = 0 to x = 5.0 m) this can be understood to be the wave moving by us toward the left.

1.	Identify and/or calculate each of the following parameters of the wave: amplitude, wave number, wavelength, angular frequency, frequency, period, velocity, and phase constant. Show all calculations. amplitude: 3.1 Volts wave number (k): 2.45 rad/m wavelength (l): l = 2p/k = 2.56 m angular frequency (w): .325 rad/s frequency (f): f = w/2p = .05 Hz period (T): T = 1/f = 19.3 s velocity (v): v = fl = .13 m/s phase constant (f): 3.14
2.	Sketch this wave as a function of distance for t = 0, from x=0 to x = 5.0 m. One knows that the wavelength is ~2.5. Therefore, about 2 full cycles will fit in the window between x = 0 and x = 5.0. Cos (0) is equal to 1. However, because of the phase constant, the argument of the cosine function at x = 0 is actually 3.14. This shifts the starting point one half cycle so that the wave begins at its minima rather than maxima.
3.	In which direction is the wave moving? How do you know this? The wave is moving left. One knows this because the sign proceeding the wt term is positive. At larger values of t the argument of the cosine is increased- shifting the starting point of the graph to the right.
4.	Sketch the wave at a slightly later time, t > 0, again from x = 0 to x = 5.0 m. For t > 0 the argument of the cosine function becomes greater. The result of this is that the plot shows an area on the cosine curve which is shifted to the right relative to the first plot. Since our window in x is fixed (x = 0 to x = 5.0 m) this can be understood to be the wave moving by us toward the left.