Lecture 9 - Standing Waves (Ch 14.6-14.7)

# Lecture 9 - Standing Waves (Ch 14.6-14.7) - Lecture 9...

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Lecture 9 Standing Waves

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Boundary Conditions and Quantization o Because the ends of the string are fixed, they form a “boundary condition” Whatever else happens to the string, the points x=0 and x=L must be fixed (zero displacement) o This limits the possible vibration patterns to specific wavelengths We will see the same (important) principle at work in quantum mechanics and the quantum model of the atom! string Air column
The Fundamental Mode o The vibration with one antinode (in the middle of the string) and two nodes (at the ends) is called the fundamental mode, or the first harmonic o For this mode: A x=0 x = L First harmonic

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Wavelength of the Second Harmonic o The next possible mode of vibration has two anti-nodes and a node in the middle: Antinodes: x = L/4, 3L/4 Nodes: x=0, L/2, L o This “second harmonic” has L = λ (or λ = L) N A N A N Second harmonic:
Wavelength of the n’th Harmonic o There are an infinite number of harmonics, with the nodes spaced closer and closer as n increases o In general for the n’th harmonic: L = n λ n /2 , or λ n = 2L/n o Drawing a picture helps a lot!

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Frequency of Harmonics o The frequency of a given harmonic is determined by its wavelength and the wave speed on the string: o The fundamental mode (n=1, first harmonic) has the lowest possible frequency for the string Higher harmonics have higher frequencies
Tension, Density and Frequencies o The tension T and mass density μ of a string determine the wave speed on it, so the natural frequencies on a string are:

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Lecture 9 - Standing Waves (Ch 14.6-14.7) - Lecture 9...

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