Chapter 6.3, Problem 57E

Pressure of a Plucked String If a string with a fundamental frequency of 110 Hz is plucked in the middle, it will vibrate at the odd harmonics of 110. 330. 550,... Hz but not at the even harmonics of 220, 440, 660,... Hz. The resulting pressure P caused by the string is graphed below and can be modeled by the following equation.

$P=0.003\sin 220\pi t+\frac{0.003}{3}\sin 660\pi t+\frac{0.003}{5}\sin 1100\pi t+\frac{0.003}{7}\sin 1540\pi t$

(Source: Benade, A., Fundamentals of Musical Acoustics. Dover Publications. Roederer, J., Introduction to the Physics and Psychophysics of Music. Second Edition. Springer-Verlag.)

(a) Duplicate the graph shown here.

(b) Describe the shape of the sound wave that is produced.

(c) At lower frequencies, the inner ear will hear a tone only when the eardrum is moving outward. This occurs when P is negative. Determine the times over the interval [0.0.03 ] when this will occur.