Question

12) The diddley bow is an instrument which consists of a single-string tensioned between two points on a board, often using a glass bottle (see figure) for a bridge and a way to amplify the instrument's sound. The speed of sound in air is v = 343 m/s. a) What tension (in N) is required in a string with mass-per-unit-length u = 1 g/m to produce a fundamental frequency of G, or 49 Hz, if the length is L = 1 m? b) If our instrument is a point source that emits its power uniformly in all directions, what power is required to produce a sound level of 70 dB a distance 10 m away? (reference intensity 1. = 10-12 W/m2) c) We experience sound as the tympanic membrane, or eardrum, vibrates from being hit with sound waves. The diameter of the tympanic membrane for the average human is 8 mm. How much energy is delivered to the eardrum each second someone plays this 70 dß sound?

Answer 1

a) The formula for the fundamental frequency (f) of a standing wave in a string of length L, mass unit length $$$\mu $$$ , and tention T is,

T 1 2 V

For tension T we get,

T = 4Lºf T = 4(1m) (49Hz)? (1g/m 100kes) T = 9.6 N

b) The formula for the intensity level of a soud wave is,

$$$\beta {\rm{ = 10log}}\left( {\frac{I}{{{I_o}}}} \right)$$$

From this the intensity of the given sound wave becomes,

$$$\begin{array}{l}70\;{\rm{dB}} = 10\log \left( {\frac{I}{{{{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}}}} \right)\\I = \left( {{{10}^7}} \right)\left( {{{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}} \right)\\I = {10^{ - 5}}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}\end{array}$$$

The relation between intensity of a point source and power (P) at a point r distance from the source is given as,

$$$\begin{array}{l}P = \frac{I}{{4\pi {r^2}}}\\P = \frac{{{{10}^{ - 5}}}}{{4\pi {{\left( {10\;{\rm{m}}} \right)}^2}}}\\P = 0.8 \times {10^{ - 8}}\;{\rm{W}}\end{array}$$$

c) We know that power is given by the product of area and intensity. And, energy (E) is given by the product of power and time. Therefore,

E = PAt = LAAT E = (10-5W/m2) (41(9) 2) (1 s) E = (10-5W/m²) ( 41 (8x1074 m)) (15) E = 2.01 x 10-9 J