![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.](https://image.slidesharecdn.com/learningobject6-150309003724-conversion-gate01/85/Physics-101-Learning-Object-6-3-320.jpg)
Physics 101 - Learning Object 6
- 1. PHYSICS 101- LEARNING OBJECT 6 Standing Waves
- 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