A standing wave is composed of two harmonic waves of equal amplitude, wavelength, and frequency that move in opposite directions. Nodes occur where the amplitude is zero and do not move. Antinodes occur at points of maximum amplitude. For a standing wave with wavelengths of 0.80m and frequencies of 95Hz, the equation is D(x,t) = 6sin(7.85x)cos(596.9t). The nodes occur at 0m, 0.40m, and 0.80m, and the antinodes occur at 0.20m, 0.60m, and 1.0m. The amplitude at any point x is given by A(x)=6sin(7.

1. Standing Waves
By: Priya Suresh

2. Let’s Start With A Definition…
 What is a standing wave?
 A standing wave is composed of 2 harmonic waves with
EQUAL amplitude, wavelength, and frequency but the
waves move in OPPOSITE directions
*for simplicity we may assume that the phase constants are
0 for both waves

3. Some Other Key Terms…
 NODE: points on the wave that have zero amplitude
and remain at rest
 ANTINODE: points on the wave that have maximum
amplitude
*since nodes do not move, energy is conserved in a
standing wave

4. Amplitude Of A Standing Wave
 The amplitude of the wave at a specific point can be
described with the following equation
 A(x) = 2Asin(kx) = 2Asin(2π*x/λ)
 Antinodes have maximum amplitude at 2A
 Nodes have zero amplitude
 All other points have amplitude between 0 and 2A

5. Equation For Position Of Wave
As A Function of Time D(x,t)
D(x,t) = 2Asin(kx)cos(ωt)
Or another form is…
 D(x,t) = A(x)cos(ωt)

6. Nodes And Antinodes
 Nodes occur at x= …-λ,-λ/2, 0, λ/2, λ…
 Antinodes occur at x=…-3λ/4, -λ/4, λ/4, 3λ/4…
 The distance between 2 consecutive nodes/antinodes
is λ/2
 An antinode is λ/4 apart from a node
 Speed of a wave is greatest at antinodes
 Speed of a wave is zero at nodes

7. Now Lets Use What We
Know To A Solve A Problem
Two sinusoidal waves with wavelengths 0.80m, amplitudes 3.0m and
frequencies 95Hz are moving in opposite directions to produce a
standing wave. (Assume the phase constants are 0).
a) Determine the equation of the standing wave.
b) If the wave starts at x=0, at what values of x will there be nodes?
(Give the first 3 locations)
c) At what values of x will there be antinodes? (Give the first 3
locations)
d) Determine the equation to find the amplitude at a given x value
e) What is the amplitude at x=5

8. Solving The Problem Step by Step
a) The equation for a standing wave is D(x,t) =
2Asin(kx)cos(ωt)
 k = 2π/λ= 2π/0.80 = 7.85 rad/m
 ω = 2πf = 2π(95) = 596.90 rad/s
 A = 3  2A = 6m
Therefore the equation is…
D(x,t) = 6sin(7.85x)cos(596.9t)

9. b) The first 3 nodes occur at x=0, λ/2, λ
x=0m,
x=0.80/2=0.40m
x=0.80m
Therefore the nodes occur at: 0m, 0.40m, 0.80m
c) The first 3 antinodes occur at x= λ/4, 3λ/4, 5λ/4
x=0.80/4=0.20m
x=3(0.80)/4=0.60m
x=5(0.80)/4=1.0m
Therefore the antinodes occur at: 0.20m, 0.40m, 1.0m

10. d) The equation to determine the amplitude is:
A(x) = 2Asin(kx)
A= 3.0m
k= 7.85rad/m
Therefore the amplitude is given by…
A(x)= 6sin(7.85x)
e) At x=5 the amplitude is…
A(5)= 6sin(7.85*5) =5.99m

11. Thank You