8.4: Standing Waves in Strings

Question 1

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Answer 1

• At the node, the amplitude and vibration will always be zero as the string is fixed
• Halfway between the two nodes, the amplitude of the wave will be a maximum (antinode)
• In a standing wave, the nodes and antinodes remain stationary, which is why it is called a standing wave
• Standng waves are not stationary, it is the relative position of the nodes and antinodes that remains unchanged

Question 2

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Standing Waves

Answer 2

• Standing waves are not a natural consequence of every wave reflection
• They are only produced by the superposition of two waves of equal amplitude and frequency, travelling in opposite directions
• They are the result of resonance and occur only at the natural frequencies of vibration of the string

Question 3

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Harmonics

Answer 3

• Resonant frequencies produced in the complex vibration of standing waves in a string instrument are called harmonics
• The simplest mode of vibration has only one antinode, called the fundamental frequency
• Higher level harmonics are called overtones

Question 4

Answer 4

Question 5

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Timbre of instruments

Answer 5

Different instruments will have different sounds due to the different quality or timbre. Each sound is made of a strong fundamental frequency, but other weaker frequencies are mixed in, called overtones, which differ between instruments; harmonics are produced in a string simultaneously, the instrument and the air around also vibrate, creating a complex mixture of frequencies we hear as an instrumental note

Question 6

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Harmonics and Frequency

Answer 6

• The fundamental frequency usually has the greatest amplitude, and therefore the greatest influence on the sound of the note
• For each subsquent harmonic, the amplitude decreases, the harmonics represent the resonant frequencies for the string; they can be calculated from the relationship between the length of the string L and the wavelength of the corresponding standing wave

Question 7

Formula for harmonics

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Answer 7

For a fixed string at both ends: The first harmonic or fundamental frequency has one antinode in the centre of the string
* first harmonic: λ = 2l
* second harmonic (has two antinodes): λ = 2l/2=l
* third harmonic has three antinodes: λ = 2l/3
* Essentially λ=2l/n
* λ is the wavelength (m)
* l is the length of the string (m)
* n is the number of the harmonic (number of antinodes)

Question 8

Answer 8

Question 9

Frequency equation for harmonics

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Answer 9

Question 10

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Standing Waves that are fixed at one end and free at the other

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Answer 10

• Only odd numbered harmonics are possible as these only satisfy the condition of having a node at the fixed end and an antinode at the free end
• A node will always form at the fixed end, and an antinode will always form at the free end of the string
• The first harmonic/fundamental frequency will have a wavelength four times the length of the string

Question 11

Formula for waves with one fixed end

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Answer 11

Question 12

• Since ffundamental = v/2L then it is true to say that …

Answer 12

fn= n x ffundamental