Standing Waves

Unit 9: Waves and Sound

Standing Waves

• Sometimes when you vibrate a string, cord, or chain, it’s possible to get it to vibrate in a manner such that you’re generating a wave, but the wave doesn’t propagate (travel).
• It just sits there vibrating up and down.
• This is called a standing wave.

Standing Waves

• Waves which appear to be vibrating vertically without traveling horizontally are called standing waves
• Created from identical waves moving in opposite directions
• Individual points on a string oscillate up and down

Types of �Standing Waves

• Two fixed points – adjacent points oscillate out of phase
• Ex. a guitar string

• One fixed point – the wave is reflected with the same amplitude in the opposite direction

Anatomy of a Standing Wave

• Points that never move are called nodes and are located λ/2 (half a wavelength) apart
• Antinodes are halfway between the nodes, where particles oscillate with maximum displacement

Resonance

• The frequency at which an object tends to vibrate when hit/struck/plucked/strummed is known as its natural frequency or resonant frequency
• When a force is applied at the natural frequency, an object will resonate
• A force applied to a guitar string will resonate and be heard, but always at its natural frequency
• During an earthquake, shock waves can cause a building to oscillate at its natural frequency, causing damage

Harmonics

• In a standing wave, resonance occurs at specific frequencies called harmonics
• The pattern of each harmonic can be used to determine the wavelength
• Harmonic patterns will differ depending on how the sound is created: string, open-open pipe, or open-closed pipe

Wavelength

½ λ = L 🡪 λ = 2L

λ = 2L/2 = L

λ = 2L/3 = L/1.5

λ = 2L/4 = L/2

(fundamental)

Standing Waves on a String

Example: guitar, violin, piano

Note: each harmonic is the addition of half a wavelength

Example

A guitar string is 1.5 meters long and is vibrating as the first (fundamental) harmonic. The string vibrates up and down with 33 complete vibrational cycles in 10 seconds. Determine the speed of the wave (by first finding frequency and wavelength).

Givens

L = 1.5 m

33 cycles in 10 seconds

1^st Harmonic

Frequency

f = cycles/second

f = 33/10

f = 3.3 Hz

Wavelength

1^st Harmonic: λ = 2L

λ = 2(1.5)

λ = 3 m

Speed

v = f λ

v = (3.3)(3)

v = 9.9 m/s

(fundamental)

Standing Waves in a Pipe

Open-Open Pipe

Example: flute, xylophone

Open-Closed Pipe

Example: clarinet, bottle

Harmonic Pattern Wavelength

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1^st

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2^nd

​

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3^rd

Harmonic Pattern Wavelength

​

1^st

​

​

3^rd

​

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5^th

Example

A clarinet (an open-closed instrument) is 66 cm long. The speed of sound in warm air is 350 m/s. What the fundamental frequency of the clarinet?

Givens

L = 0.66 m

v = 350 m/s

1^st Harmonic

Wavelength

1^st Harmonic: λ = 4L

λ = 4(0.66)

λ = 2.64 m

Frequency

v = f λ

350 = f (2.64)

f = 132.6 Hz

Note:

• There is always a node when the end is closed and an antinode when the end is open