16.1 Traveling waves  (Page 5/14)

Characteristics of a wave

A transverse mechanical wave propagates in the positive x -direction through a spring (as shown in [link] (a)) with a constant wave speed, and the medium oscillates between $\text{+}A$ and $\text{−}A$ around an equilibrium position. The graph in [link] shows the height of the spring ( y ) versus the position ( x ), where the x -axis points in the direction of propagation. The figure shows the height of the spring versus the x -position at $t=0.00\text{s}$ as a dotted line and the wave at $t=3.00\text{s}$ as a solid line. (a) Determine the wavelength and amplitude of the wave. (b) Find the propagation velocity of the wave. (c) Calculate the period and frequency of the wave.

Strategy

1. The amplitude and wavelength can be determined from the graph.
2. Since the velocity is constant, the velocity of the wave can be found by dividing the distance traveled by the wave by the time it took the wave to travel the distance.
3. The period can be found from $v=\frac{\lambda }{T}$ and the frequency from $f=\frac{1}{T}.$

Solution

1. Read the wavelength from the graph, looking at the purple arrow in [link] . Read the amplitude by looking at the green arrow. The wavelength is $\lambda =8.00\text{cm}$ and the amplitude is $A=6.00\text{cm}.$
   

   

2. The distance the wave traveled from time $t=0.00\text{s}$ to time $t=3.00\text{s}$ can be seen in the graph. Consider the red arrow, which shows the distance the crest has moved in 3 s. The distance is $8.00\text{cm}-2.00\text{cm}=6.00\text{cm}.$ The velocity is
   
   $v=\frac{\text{Δ}x}{\text{Δ}t}=\frac{8.00\text{cm}-2.00\text{cm}}{3.00\text{s}-0.00\text{s}}=2.00\text{cm/s}.$
3. The period is $T=\frac{\lambda }{v}=\frac{8.00\text{cm}}{2.00\text{cm/s}}=4.00\text{s}$ and the frequency is $f=\frac{1}{T}=\frac{1}{4.00\text{s}}=0.25\text{Hz}.$

Significance

Note that the wavelength can be found using any two successive identical points that repeat, having the same height and slope. You should choose two points that are most convenient. The displacement can also be found using any convenient point.

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