A standing wave occurs when two waves of equal amplitude, wavelength, and frequency traveling in opposite directions interfere with each other. When waves reflect off boundaries like the ends of a guitar string or inside a tube, they can form standing waves where the wave appears to stay in place rather than propagate. A mathematical model of a standing wave on a guitar string shows that the string can only oscillate at certain wavelengths, called modes, that correspond to different frequencies. The first mode is the fundamental frequency, and higher modes are called harmonics. Many musical instruments produce standing waves at harmonic frequencies of the fundamental, and the combination of these harmonics creates sounds that the human brain perceives as music.

1. What is a standing wave?
When two waves with EQUAL amplitude wavelength and frequency but moving in opposed
directions interfere (combine), they form a STANDING WAVE
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WAVE’S WORLD:
HOW STANDING WAVES ARE MUSIC TO OUR EARS
Standing waves occur on media such as strings or hollow
tubes because when the waves hit the boundaries, they
are reflected at the same frequency but in opposite
directions which forms standing waves
+
A good model for the mathematical properties
of standing waves is a guitar string
D(x,t) = Asin(kx-ωt) + A(sin(kx+ωt)
through trigonometric identities becomes
D(x,t) = 2Asin(kx)cos(ωt)
Taking the aptitude with respect to position we get:
A(x) = 2Asin(kx) = 2Asin(2𝛑x )
λ
Or graphed:
2A
2A
0
λ
4
λ
2
λ 2λ
• Regions that reach maximum amplitude +2A are known as antinodes
• Regions that never move are known as nodes
• Nodes are half a wavelength apart
• Nodes are a quarter of a wavelength from antinodes
Guitar strings are fixed on both ends,
which makes them perfect candidates for
standing waves
Because of the equation above, a string can only oscillate
with certain wavelengths, called modes
λ = 2L m = 1,2,3..
m
!m = mode
Each wavelength corresponds to a different frequency
(from v = f λ) the frequency for m=1 is called the
fundamental frequency. All others: m=1,2,34… are
called harmonics
m= 1
m= 2
m= 3
Many musical instruments (including the human voice
produce standing waves at a variety of frequencies
However, all frequencies produced are harmonics of
the fundamental frequency. The combination of these
harmonics produces a sound pleasing to the
human brain, a sound we call music.
MATH