Download - Standing waves
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STANDING WAVESONVIOLIN STRINGS
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STANDING WAVES• Are stationary (as opposed to travelling waves)
• Vs
STANDING WAVE TRAVELLING WAVE
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STANDING WAVES
• Are the superposition of two harmonic waves with equal amplitude, frequency and wavelengths but moving in opposite direction
v
v
Resulting Standing Wave from adding the two harmonic waves
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STANDING WAVES• Can be generated by plucking a string with
both ends fixed• Nodes are points with zero amplitudes• Antinodes are points with maximum
amplitudes
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STANDING WAVES ON STRINGS• Strings with two fixed ends can only produce
standing waves with an integral number of half wavelength called normal modes• =
where L = string length n = number of antinodes = 1, 2, 3, 4, …
• The fundamental frequency (1st harmonic) is the lowest frequency (longest wavelength)• =
where T = tension in the string = linear mass density of the string =
• The allowed frequencies are called harmonics• = n n = 1, 2, 3, 4, …
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QUESTION PART 1Tom wants to make a violin for his sister as a birthday present. Violins usually make sound frequencies ranging from 200~3000Hz. He has a few 30 cm long strings with linear mass densities:A 2.8 kg/m B 4.0 kg/mC 0.62 g/mWhich string should he use to make the violin in order to get a fundamental frequency of 700Hz if the tension in the string is kept at 70 N?
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Hints
• What variables are given in the question?• The fundamental frequency (), tension (T), and string length (L)
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Hints
• What variables are given in the question?• The fundamental frequency (), tension (T), and string length (L)
• Which equation to use when solving for linear mass density?• =
where T = tension in the string = linear mass density of the string =
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Solution — Tom should use string B = 700 Hz T = 70 N L = 30 cm = 0.30 m = Solve for
= = = = 3.97 kg/ m 4.0 kg/m
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QUESTION PART 2
The violin string broke after a few weeks, but Tom doesn’t have anymore of the same string. If he uses a string with linear mass density of 4.7 kg/m, what should the tension be in the string in order to produce the same sound frequency (700 Hz)?
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Hints
•What variables are given in the question?• The fundamental frequency (), linear mass density (), and
string length (L)
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Hints
•What variables are given in the question?• The fundamental frequency (), linear mass density (), and
string length (L)•Which equation to use when solving for tension?• =
where T = tension in the string = linear mass density of the string =
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Solution = 700 Hz = 4.7 kg/m L = 30 cm = 0.30 m = Solve for T
= T = = 4.7 kg/m = 82.9 N 83 N
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