Solve ALL
Guided Project 66: Traveling waves Topics and skills: Functions of two variables, partial derivative, graphing The word wave elicits images of water waves rolling in beach. However, wave is used to describe any disturbance that propagates through a medium. So the word includes water waves, sound waves, electromagnetic waves (light waves and cell phone signals), seismic waves, and waves in traffic on a crowded highway, In this project, we examine some of the fundamental mathematical properties of waves. 1. Consider the function z= f(x, t) = 3 sin (pi x- 2 pi t). It is easy to graph this function (Figure 1) and indeed we waves. But this graph really doesn't explain how and why waves propagate. To interpret the graph suppose that x and t represent distance (measured in meters) and time (measured in seconds), respectively. Specifically, the x-axis is the direction in which the wave propagates and z represents the height of the surface of the wave at a particular location and time. The maximum values of z correspond to crests of the wave and the minimum values of z correspond to troughs of the wave. What are the minimum and maximum values of z? 2. Now imagine that you are sitting on the Jt-axis, say at x = 2, watching the wave pass. Set x = 2 in z = f(x, t) and graph z = z(2, t) on the interval 0 t 2. Explain how z varies on 0 t 2. How many times do you observe a crest of the wave pass? The time that elapses between passing crests is the period of the wave. How many wave crests pass per second? The number of wave crests that pass per second passes is the frequency of the wave, usually measured in cycles per second. Instead of freezing x as we did in Step 2, we can also freeze t. Suppose you take a snapshot of the wave at t = 1. Set t = 1 in z = f(x, t) and graph z =f(x, 1) on the interval 0 x 4. Explain how z varies on 0 x 4. What is the distance between crests? The distance between crests is called the wavelength of the wave. How far does the crest of the wave travel in one second of time? What is the speed of the wave?