standing waves

STANDING WAVESONVIOLIN STRINGS

STANDING WAVES• Are stationary (as opposed to travelling waves)

• Vs

STANDING WAVE TRAVELLING WAVE

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

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

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, …

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?

Hints

• What variables are given in the question?• The fundamental frequency (), tension (T), and string length (L)

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 =

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

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)?

Hints

•What variables are given in the question?• The fundamental frequency (), linear mass density (), and

string length (L)

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 =

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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