Imagine a perfect, smooth wave
out on the ocean far enough from shore so that it has not started to
break (complications involved in describing real waves will be
discussed later in this tutorial). If we take a snapshot picture of
this wave at a single instant
in time and measure the distance in meters from one peak to the next we
have measured the wavelength, l of
the wave.
If instead we watch a floating cork at a single location in
space and measure the time in seconds between arriving peaks we have
measured the period, T of the wave. We could also measure the number of times the cork bobs up, down and back up per second which would be the frequency in Hertz or cycles per second. The period and frequency are inverses of each other: f = 1/T. The height of the wave at any location and time, measured from the middle, or equilibrium position is the amplitude.
As a first approximation,
water waves and many other kinds of waves can be modeled by the mathematical functions sine or cosine or some
combination of them. For a wave traveling along the x axis the mathematical description of the amplitude (or magnitude) of a wave at
location x and time t can be written as
where A is the maximum amplitude (maximum height measured from the middle of the wave). Here we have used the wavenumber k = 2p/l, the angular frequency, w = 2pf and f , the phase angle in radians which are often easier to use mathematically.
With the following physlet you can explore
different values of amplitude, wavenumber, wavelength, angular frequency, frequency, period and
phase. Make any changes you wish in the y(x,t)= window, click 'set' and then 'play' to see the wave in action.
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1.1. Double the maximum amplitude (from 1.2 to 2.4) and click 'play'. What effect does maximum amplitude, A, have on the wave?
1.2. Double the wave number, k, (from 2.0 to 4.0)and click 'play'. What effect does the wave number have on the wave?
1.3. Double the angular frequency, w, (from 0.8 to 1.6)and click 'play'. What effect does the angular frequency have on the wave?
1.4. Stop the wave with 'pause' and then double the phase, f , (from 10.0 to 20.0) and click 'set'. Try several different values for the phase. What effect does the phase have on the wave?
1.5. Go back to the original wave by clicking 'reload initial'. Pause the wave and measure the wavelength, l, on the graph (hold the left mouse down to get the x and y locations of any point on the graph, the wavelength is the x distance between peaks or troughs). Calculate the wavenumber, k , from this wavelength. How does your value for wavenumber compare with the wavenumber in the equation?
1.6. Now start the original wave in
motion by clicking 'play'.
Use the time numbers in the lower left corner to find the period, T, of the wave (the time from when one peak passes a point until the next peak passes the
same point). To get an accurate number you can use the 'step' buttons.
From the period you measure, calculate the angular frequency, w. How does
your value
for angular frequency compare with the angular frequency in the
equation?
1.7. Go back to the original wave by clicking 'reload initial'. Change the minus sign in the equation between kx and w t to a plus sign and click 'play'. What does changing this sign do to the wave?
1.8. Now change the plus sign in front
of the phase to a minus sign and click 'play'. Try several values of
phase (you may want to use the 'pause' button to be sure you can tell
what is happening). What does changing this sign do to the wave?