1.5. Exercises#

Exercise 1.1

Write an equation describing a continuous sinusoid \(x(t)\) oscillating at 5 cycles per second.

Exercise 1.2

For each of the following waves, convert them into an equivalent expression in standard form (as in (1.3)). What is the amplitude (\(A\)), frequency (\(f\)) and initial phase (\(\phi\)) for each one?

  1. \(x(t) = \sin(2\pi \cdot 3 \cdot t)\)

  2. \(x(t) = 2 \cdot \cos\left(\pi \cdot t - \frac{\pi}{2}\right)\)

  3. \(x(t) = -\sin(4\pi \cdot t + \pi)\)

Exercise 1.3

A waveform is described to you in terms of non-standard units:

  • \(f = 600 \left[\frac{\text{half-cycles}}{\text{minute}}\right]\)

  • \(\phi = 120^\circ\)

Convert this into a standard form with \(f\) expressed in Hz and \(\phi\) in radians. Use dimensional analysis to check your conversion.

Exercise 1.4

Imagine we have a sound source of constant power, and a microphone at distance \(r\) observes sound pressure \(p\).

  1. At what distance should we place the microphone to observe sound pressure \(p/2\)?

  2. If the sound pressure level is \(L\) [dB] at distance \(r\), what distance would produce a pressure level of \(L-5\) [dB]?

Exercise 1.5

If we assume the average human can hear frequencies between 20 and 20000 Hz, approximately how many octaves is this range?