Question #5 please (Sound Waves and Beats)
Posted: Mon Jul 18, 2022 5:38 am
Question #5 please (Sound Waves and Beats)
5.) Suppose you are adding two sound waves with equal amplitudes A and slightly different frequencies fi and f2. Let us write the equations for the time dependence of these waves (at a fixed position x) as AP(t) = Acos(27 fit) AP2(t) = Acos(21 f2t) (a) Using the trigonometric identities a - (a+b cosa + cosb = 2cos (67") cos (676) cos a + cos b = 2 cos COS 2 la- sin a + sin b = 2 cos sin (a+b ( 2 ) Write the sum of your two sound waves APtot = AP(t) + AP2(t) as a product of two trigonometric functions. (b) The two trigonometric functions which you found in the previous part have two different oscillation frequencies which depend on fi and f2. Call these ffast and fslow. Write equations for ffast and fslow in terms of fi and f2. (c) Suppose fı = 19Hz and f2 = 21Hz. What are the first two times where the slowly oscillating trig function has a value of zero? These will be two times where the sound intensity is at a minimum.
(d) Suppose fi = 19Hz, and f2 = 21Hz, Sketch the resulting wave if these two sounds interfere with each other. The x-axis of your sketch should be at least 1.0 second long. Hint: the product of two trig functions can be thought of as a "cosine within a cosine” (or a “sine” within a cosine”, etc...). For example, the trig function with the longer period can be thought of as amplitude of the faster oscillating function. Thus, you have a fast oscillation whose amplitude varies slowly in a sinusoidal manner.
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(d) Suppose fi = 19Hz, and f2 = 21Hz, Sketch the resulting wave if these two sounds interfere with each other. The x-axis of your sketch should be at least 1.0 second long. Hint: the product of two trig functions can be thought of as a "cosine within a cosine” (or a “sine” within a cosine”, etc...). For example, the trig function with the longer period can be thought of as amplitude of the faster oscillating function. Thus, you have a fast oscillation whose amplitude varies slowly in a sinusoidal manner.