Boundary conditions for an antinode?

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In summary, the net displacement of a standing wave with two sinusoidal waves of the same frequency, wavelength, and amplitude traveling in opposite directions is given by the equation D(x,t) = 2a*sin(kx)*cos(ωt). The boundary conditions for standing waves on a string from x=0 to x=L are D(0,t) = 0 and D(L,t) = 0, which satisfies the equation when L = 0.5mλ, m=1,2,3... (or λ = 2L/m). For antinodes at x = 0 and x = L, the equation for disturbance is D(x,t) = 2a*cos(kx)*cos(ω
  • #1
BomboshMan
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Hi,

If I have two sinusoidal waves with the same frequency, wavelength and amplitude, traveling along the same line in opposite directions, the net displacement of the resulting standing wave wave is given by

D(x,t) = 2a*sin(kx)*cos(ωt)

the boundary conditions for standing waves on a string from x=0 to x=L are

D(0,t) = 0 (which satisfies the above equation at all times), and
D(L,t) = 0, which satisfies the equation when L = 0.5mλ , m=1,2,3... (or λ = 2L/m)

This I understand because the displacement of a node at any time must = 0.

Now I've been told by my physics lecturer (who's not very good), that for antinodes at x = 0 and x = L, e.g. standing waves in a pool, the equation for disturbance is

D(x,t) = 2a*cos(kx)*cos(ωt),

and that the boundary conditions for antinodes at x = 0 and x = L are

D(0,t) = ±2a
D(L,t) = ±2a

which I get where this is coming from (same sort of thing as boundary conditions for nodes), but wouldn't this suggest that the displacement of x = 0 and x = L are ±2a at all times? Surely it only makes sense to say the max displacement of x = 0 and x = L is ±2a

I may just be getting confused, but if someone could shed some light on this it would help a lot!

Thanks,

Matt
 
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  • #2
BomboshMan said:
and that the boundary conditions for antinodes at x = 0 and x = L are

D(0,t) = ±2a
D(L,t) = ±2a

That looks wrong. I think it should be
##\partial D(0,t)/\partial x## = 0
##\partial D(1,t)/\partial x## = 0

which gives ##D(0,t) = D(1,t) = 2a \cos(\omega t)##.
 

1. What is an antinode?

An antinode is a point on a standing wave where the amplitude of the wave is at its maximum. This means that the particles at an antinode are moving with the greatest displacement.

2. How are antinodes formed?

Antinodes are formed when two waves with the same frequency and amplitude are superimposed, resulting in a standing wave. The points where the waves are in phase and have maximum amplitude become the antinodes.

3. What conditions are necessary for an antinode to exist?

An antinode will only exist if there is a standing wave present. This requires two waves with the same frequency and amplitude to be superimposed, as well as a medium for the waves to travel through.

4. How do antinodes differ from nodes?

Nodes are points on a standing wave where the amplitude of the wave is at its minimum, and the particles at a node do not experience any displacement. In contrast, antinodes have maximum amplitude and particles at an antinode experience the greatest displacement.

5. What are some real-life examples of antinodes?

Some examples of antinodes include the vibrating strings of a guitar or other stringed instruments, sound waves in a tube, and the vibrations on the surface of a drumhead. Antinodes can also be seen in nature, such as in standing waves in water or in vibrating air columns in wind instruments.

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