Help with questions on Sound Waves and Harmoics

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SUMMARY

The discussion focuses on the physics of sound waves and harmonics, specifically addressing wave speed, linear density, and tension in strings. It establishes that increasing the tension in a string by a factor of four results in a wave speed increase by a factor of two, as per the wave speed formula. The linear density of a wire with a cross-sectional area of 4.2 x 10-8 m2 and density of 7900 kg/m3 is calculated to be 0.00033 kg/m. Additionally, it explores the fundamental frequencies of musical notes, calculating the next harmonics for a C string and the tension ratio between A and C strings.

PREREQUISITES
  • Understanding of wave mechanics and wave speed equations
  • Knowledge of linear density calculations
  • Familiarity with fundamental frequency and harmonics in music theory
  • Basic principles of tension in strings and its effect on wave propagation
NEXT STEPS
  • Study the wave speed formula and its applications in different mediums
  • Learn about linear density and its significance in wave mechanics
  • Explore harmonic series and their calculations in string instruments
  • Investigate the relationship between tension, length, and frequency in vibrating strings
USEFUL FOR

Physics students, music theorists, string instrument makers, and anyone interested in the principles of sound waves and harmonics.

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1.) A transverse wave is traveling with a speed of 300m/s on a horizontal string. If the tension in the string is increased by a factor of four, what is the speed of the wave?

2.) A wire has a cross-sectional area of 4.2 X 10-8 m2 and is made from a material whose density is 7900 kg/m3. Determine the wire's linear density (m/L).

3.) A uniform cord has a mass of 0.300 kg and a total length of 6.00 m. Tension is maintained in the cord by suspending a 2.00 kg mass from one end. Find the speed of a pulse on this cord. Assume the tension is not affected by the mass of the cord.

4.) A middle C string of the C-major scale on a piano has a fundamental frequency of 262 Hz. And the A note has a fundamental frequency of 440 Hz.

a.) Calculate the frequencies of the next two harmonics of the C string.

b.) If the strings for the A and C notes are assumed to have the same mass per unit length and the same length, determine the ratio of tensions in the two strings.

c.) In a real piano, the assumption we made in (b) is only half true. The string densities are equal, but the A string is 64% as long as the C string. What is the ratio of their tensions?
 
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