A standing wave is formed by two traveling waves moving in opposite directions with the same amplitude, wavelength, and frequency. The superposition of these waves results in locations of maximum and minimum displacement known as anti-nodes and nodes. The equation for a standing wave is the sum of the equations for the two traveling waves. This creates a stationary wave pattern with anti-nodes at locations of maximum displacement and nodes at locations of no displacement.

1. Standing Waves
Rachel Sunderland

2.  A standing wave is two waves that are moving in
opposite directions with the same amplitude,
wavelength, and frequency.
 They result from the superstition of waves when a
wave reflects from a surface and interferes with the
wave travelling in the original direction
What is a Standing Wave?

3.  D1(x,t) = A sin(kx + ωt) <- occurs at increasing x
 D2(x,t) = A sin(kx - ωt) <- occurs at decreasing x
 Together, the equation of a standing wave becomes:
 D(x,t) = D1(x,t) + D2(x,t)
 = A sin(kx – ωt) + A sin(kx + ωt)
 = A sin(kx – wt) + sin(kx – wt)
 = 2A sin(kx) cos (ωt)
Standing Wave Equations

4.  In this photo, the wave on the top is moving to the
right and the wave to the bottom is moving to the left
 They combine together in order to form one wave
with an amplitude of 2A at the anti-node and 0 at the
node
Standing Waves Visual Example

5.  A node is the point where the amplitude is zero and
therefore, there is no displacement from equilibrium
 The antinode is the point where the maximum
displacement occurs from equilibrium and thus, the
amplitude is also at its maximum point
 Since a standing wave is two waves moving in
opposite directions, the maximum amplitude of a
standing wave is twice the amplitude of one wave
component of the travelling wave
What is a Node?

6. Picture of Nodes/Anti-nodes
 A node is where the displacement is zero and an anti-
node is where the displacement is 2A as seen in the
photograph below

7.  Nodes occur when the displacement is zero
 Therefore, sin (2πx/𝜆) = 0
 Since is zero at every pi interval so we can substitute
it in for 0
 2πx/𝜆 = mπ where m = positive or negative (1,2,3…)
 In order to simplify, we bring x to one side, and cancel
out the π’s in order to get the equation:
 X = m𝜆/2
Calculating the Position of a Node

8.  At what position does the wave function:
0.75m sin(17.2x) have its first node?
 We use the equation: X = m𝜆/2
 1st we need to solve for the wavelength
 We know that k = 2π/𝜆
 Therefore, 17.2 = 2π/𝜆 and 𝜆 = 0.365
 We can then plug it into the equation x = 1(0.365/2) =
0.1825
Example

9.  Anti-Nodes occur when the displacement is (+/-) 2A
 Therefore, sin (2πx/𝜆) = (+/-) 1
 Since (+/-) occurs at every pi/2 interval so we can
substitute it in for (+/-) 1
 2πx/𝜆 = (m+ 1/2)π where m = positive or negative
(1,2,3…) which also equals (m+1/2) *𝜆/2
 In order to simplify, we bring x to one side in order to
get the equation:
 X = (+/-) 𝜆/4, 3𝜆/4, 5𝜆/4 …
Calculating the Position of an
Anti-Node

10.  At what position does the wave function:
0.50m sin(5.9x) have its first anti-node?
 We use the equation: X = (+/-) 𝜆/4, 3𝜆/4
 1st we need to solve for the wavelength
 We know that k = 2π/𝜆
 Therefore, 5.9 = 2π/𝜆 and 𝜆 = 1.07
 We can then plug it into the equation x = 1(1.07/2) =
0.535
Example

11.  What is the amplitude of a wave when its at the
location x = 0.35m, and its wavelength is 2.13m
 1st we need to determine what the value of K is
 We can use the equation K = 2π/𝜆
 Therefore, k = 2π/2.13 = 2.95 rad/m
 We can then use this information to complete the
formula
Integrated Example

12.  A(x) = 2A sin (2π/𝜆)
 Plugging in values gives us
 0.35m = 2(x) sin (2.95)
 0.35m = 2x (0.0514)
 6.81 = 2x
 x = 3.41
 Therefore, at the location x = 0.35m and with a wavelength
of 2.13m, the amplitude of the wave is 3.41m
Integrated Example Part 2

13.  "Standing Waves." Mr Magarey's Official Year 11 Physics
Class Wikispace -. NHS, n.d. Web. 09 Mar. 2015.
 "S-cool, the Revision Website." Standing Waves. N.p.,
n.d. Web. 09 Mar. 2015.
Sources Cited