Sound is generated by compression waves in the air. The tone is determined by the frequency (which is defined as 1/T with T the period of the function), while the amplitude of the wave determines the loudness of the sound. A simplified model for a sound wave can be given by the cosine function. In this problem you will explore what happens when two sound waves of equal magnitude (loudness) are superimposed or added together. When two waves are superimposed they often interfere with each other. The result is a much more complicated behavior. This problem examines this phenomenon, and in particular studies what is known as beats or slowly oscillating changes in loudness.

a. Suppose that one sound is modeled by f(t) = cos(6t) and another is given by g(t) = cos(7t). Use Maple or Excel to graph each of these functions for t [0, 2p]. Find the period and amplitude of f(t) and g(t). For amplitude use the largest in absolute value of the function from your graph.

b. Now we add the two sound waves together, and observe what happens. Note that the amplitude of the resulting function is the loudness of the sound heard. Use Maple or Excel to graph F(t) = f(t) + g(t) for t [-2p, 2p], and find the period of F(t).

c. From the past we have seen that the derivative can be used to find maxima and minima. Let Maple take the derivative (and write this in your Lab report), then find the absolute maximum and absolute minimum for t [0, 2p]. Note there are several local extrema (where the derivative is zero), but you will use Maple's fsolve routine restricted to the appropriate interval to find the largest maximum and smallest minimum. Give both the t and F(t) values at these extrema.

d. Now define h(t) = cos(4t) and take H(t) = f(t) + h(t). Rework the questions asked in Parts b. and c. for H(t).