![Standing Wave
Mathematical Description:
D(x,t)=D1(x.t)+D2(x,t)
=A sin(kx-wt)+A sin(kx+wt)
=A[sin(kx-wt)+sin(kx+wt)]
- (Using the trigonometric identity:
sin(a-b)+sin(a+b)=2sin(a)cos(b),
with a=kx and b=wt)
- We can get:D(x,t)=2A sin(kx) cos(wt) or,
D(x,t)=2A sin(2π/λ*x) cos(2π/T*t)
Therefore, the wave is not a travelling wave,
we can call it a standing wave.](https://image.slidesharecdn.com/learningobjectlo6-150308044748-conversion-gate01/85/Learning-object-lo6-4-320.jpg)
Learning object lo6
- 2. Standing Waves • In physics, a standing wave, also known as a stationary wave, is a wave in a medium in which each point on the axis of the wave has an associated constant amplitude. The locations at which the amplitude is minimum are called nodes, and the locations where the amplitude is maximum are called antinodes.
- 3. • For standing waves, we can consider two harmonic waves of equal amplitude, wavelength, and frequency that are moving in opposite directions. • In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. • For example, a wave traveling to the right along a string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave.
- 4. Standing Wave Mathematical Description: D(x,t)=D1(x.t)+D2(x,t) =A sin(kx-wt)+A sin(kx+wt) =A[sin(kx-wt)+sin(kx+wt)] - (Using the trigonometric identity: sin(a-b)+sin(a+b)=2sin(a)cos(b), with a=kx and b=wt) - We can get:D(x,t)=2A sin(kx) cos(wt) or, D(x,t)=2A sin(2π/λ*x) cos(2π/T*t) Therefore, the wave is not a travelling wave, we can call it a standing wave.
- 5. • A position-dependent amplitude, A(x) can be written as: • A(x)=aA sin(kx)=2A sin(2π * x/λ) • At the nodes of a standing wave, A(x)=0. Nodes occur when: sin(2π/λ * x)=0, therefore, x=0, +/- λ/2, +/-λ, +/- 3λ/2, +/-2λ… • The distance between two consecutive nodes is half a wavelength. • At the antinodes of standing wave, A(x)= +/- 2A, which occurs when: sin(2π/λ * x)= +/-1
- 6. Sample Question • The amplitude of a standing wave is given by: A(x)=(30cm) sin(2x). Determine the amplitude and the wavelength, and the location of the first node and the first antinode. • Solution: • 2A=30cm, 2π/λ=2 rad/s. Therefore, A=15cm, λ=πm=3.14m • The first node occurs when x=0m • The first antinode occurs when x=λ/4=3.14m/4=0.78m