Homework on Waves (solutions)

A photograph of a wave on a rope is taken at time t = 0 is represented below.

This wave can be described by the expression

y = y_m sin (kx + wt + f)

1.	What is the wavelength of this wave? What is the wave number? Show your work.
	The wavelength is 8 m (from the graph). The wave number is equal to 2p divided by the wavelength: p/4, or 0.785, with units of 1/m.
2.	What is the amplitude of this wave?
	3 m. Amplitude is measured from the center of the wave to a peak or trough.
3.	Will the wave be moving to the right or to the left? How do you know this?
	To the left. To get the same value with an increased time, you must decrease x.
4.	What is the phase constant of this wave? You can check your sign by setting t = x= 0 and seeing whether your answer gives the right value for the displacement at the y-axis.
	Looking at the wave at the y-axis, it is 1/8 of a cycle past (or 7/8 of a cycle behind) where a sine wave with phase constant zero would be at that point. Thus, the phase shift of this wave is 2p times 1/8 (or -7/8). Valid answers are p/4 (0.785), -7p/4 (-5.498), or either of these with any multiple of 2p added to them.
5.	What effect will increasing the MAGNITUDE of the phase constant (making it more negative or more positive) have on the wave as it is displayed on the graph?
	*A more negative phase constant will move the wave to the right. A more positive phase constant will move the wave to the left.*
6.	If the wave is moving at a speed of 100 m/s, what is the period of the wave? What is the angular frequency? Show your work.
	velocity = wavelength / period, or period = wavelength / velocity. (8 m) / (100 m/s) = 0.08 s w = 2p / T, so w = 25p Hz (1/s), or 78.5 Hz (1/s).
7.	What effect will increasing the period of the wave have on the wave as it is displayed on the graph?
	None. The graph displays displacement vs. x-position, and changing the period (a time variable) will not affect this.