Standing waves occur when two harmonic waves of equal amplitude, frequency, and wavelength move in opposite directions in a medium. The resulting wave function is the sum of the individual wave functions. This results in a wave that oscillates in amplitude but whose nodes and antinodes remain stationary. Nodes are points of zero amplitude, while antinodes are points of maximum amplitude. The distance between nodes and between antinodes is half the wavelength.
This Learning Object 6 is based on Standing Waves. It gives a detailed description about what a standing wave is and how standing waves are formed. As well, the equations necessary to break down standing waves to get the amplitude, nodes, antinodes, etc are all included. Furthermore, a practice question is included along with a detailed solution to solve the problem with ease.
This Learning Object 6 is based on Standing Waves. It gives a detailed description about what a standing wave is and how standing waves are formed. As well, the equations necessary to break down standing waves to get the amplitude, nodes, antinodes, etc are all included. Furthermore, a practice question is included along with a detailed solution to solve the problem with ease.
The learning object focuses on the basics of standing waves and the formulas associated with the concept. There is a simple example at the end of the powerpoint and many formulas and descriptions along the way.
CBSE Physics/ Lakshmikanta Satapathy/ Wave Motion Theory/ Reflection of waves/ Traveling and stationary waves/ Nodes and anti-nodes/ Stationary waves in strings/ Laws of transverse vibration of stretched strings
A presentation covering some of the basics of standing waves and the differences between standing and travelling waves. Practice questions are included at the end.
Please download the presentation in order to see the gifs.
The learning object focuses on the basics of standing waves and the formulas associated with the concept. There is a simple example at the end of the powerpoint and many formulas and descriptions along the way.
CBSE Physics/ Lakshmikanta Satapathy/ Wave Motion Theory/ Reflection of waves/ Traveling and stationary waves/ Nodes and anti-nodes/ Stationary waves in strings/ Laws of transverse vibration of stretched strings
A presentation covering some of the basics of standing waves and the differences between standing and travelling waves. Practice questions are included at the end.
Please download the presentation in order to see the gifs.
Physics 101 LO6 which explains the components of standing waves, generates its equation, and tests the understanding of students by creating a practice problem with a worked solution in the end.
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Harmonic Waves (up to but not including Travelling waves) and shows the formulas associated with the topic, as well as definition and examples. In the form of a pdf.
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2. Standing Wave - Definition
• When two harmonic waves that have equal amplitude,
frequency, and wavelength are moving in opposite
directions along the same medium, the resulting wave is
defined as a standing wave.
*Assume in this case that both waves have a phase
constant of zero.*
3. Standing Wave – Wave Function
• The wave moving in a direction of increasing position (x)
has a wave function of D1(x,t) = A sin(kx-ωt).
• The wave moving in a direction of decreasing position
has a wave function of D2(x,t) = A sin(kx+ωt).
• The resultant standing wave function is:
D(x,t) = D1(x,t) + D2(x,t)
= A sin(kx-ωt) + A sin(kx+ωt)
= A [sin(kx-ωt) + sin(kx+ωt)]
= 2A sin(kx) cos(ωt)
Note: wave function obtained from a
trigonometric identity.
4. Standing Wave – Wave Function (cont’d)
• To make the amplitude of a standing wave dependent on
the position, x, we can rewrite the wave function of a
standing wave,
D(x,t) = 2A sin(kx) cos(ωt)
as D(x,t) = A(x)cos(ωt)
where Amplitude = A(x) = 2A sin(kx)
5. Standing Wave – Nodes and Antinodes
• Nodes are points on a standing wave that have an
amplitude of zero.
• Antinodes are points on a standing wave that have the
maximum amplitude possible (2A/-2A).
• All standing waves have nodes and antinodes.
This is because amplitude, A(x), is a sine function, meaning
that certain points have an amplitude of zero, certain points have the
maximum amplitude possible, and the remaining points have an
amplitude that is between 0 and 2A or 0 and -2A.
6. Standing Wave – Nodes and Antinodes (cont’d)
• Nodes are points that seam to be standing still (they are
at rest). Nodes are sometimes referred to as being points
of no displacement.
• Antinodes are points that undergo the maximum
displacement possible.
• A standing wave alternates between nodes and antinodes
node followed by antinode followed by node,
and so on.
7. Standing Wave – Nodes and Antinodes (cont’d)
• Nodes are points where the amplitude is equal to zero.
A(x) = 2A sin(kx) = 0
sin(kx) = 0
sin(2πx/λ) = 0
Therefore the distance between two successive nodes is half
a wavelength.
• Antinodes are point where the amplitude is equal to +2A.
A(x) = 2A sin(kx) = +2A
sin(kx) = +2A
sin(2πx/λ) = +2A
Therefore the distance between two successive antinodes is
also half a wavelength.
• This also results in a distance of a quarter of a wavelength
between adjacent nodes and antinodes.
8. Practice Questions
1) When does a standing wave form?
a) When interference occurs between two waves
moving in the same direction.
b) When a wave moves from one medium to
another.
c) When interference occurs between two waves
moving in different directions.
2) The amplitude of a standing wave is:
A(x) = (0.5m) sin(3.00x). What is the location, x, of the first
point where the amplitude is equal to 0.17m?
9. Solutions
Answers:
1) c) When interference occurs between two waves moving in
different directions.
2) A(x) = (0.5m) sin(3.00x)
0.17m = (0.5m) sin(3.00x)
0.17/0.5 = sin(3.00x)
0.34 = sin(3.00x)
sin-1(0.34) = 3.00x
sin-1(0.34) = x
3.00
0.1156… = x
0.12m = x
