Tutorial 2: Sine Waves

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, λ 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 displacement. The maximum displacement is called the amplitude.

As a first approximation, water waves, electromagnetic 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 displacement of a wave at location x and time t can be written as y(x,t) = A sin (k x - ω t + φ) where A is the amplitude (maximum height measured from the middle of the wave). Here we have used the wavenumber k = 2 π/λ (measured in radians per meter), the angular frequency, ω = 2 π f (measured in radians per sec), and φ, the phase angle (in radians) which are often easier to use mathematically.

In this and all of the following exercises the angles will be in radians (Hint: Set your calculator to radian mode to avoid problems later on!)

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, hit enter or return and then the play button, Play , to see the wave in action. The intial values are amplitude, A = 1.2 m; wavenumber, k = 2.0 rad/m; angular frequency, ω = 0.8 rad/s; and phase angle, φ = 10.0 radians. Clicking in on the graph shows the coorinates of the cursor in a yellow box in the lower left corner.