Standing waves, find distance

In summary, the string of length 1.5m with both ends clamped is vibrating in fundamental mode with an amplitude of 4mm at the center. To find the distance between points with an amplitude of 2mm, the equation y=Asinkxcoswt is used with k=2π/λ and λ=3. The amplitude being half the maximum gives the equation 0.5=sin(2πx/λ), leading to the solution of x=0.25m. The distance between the points is 1m. The given answer of 50cm is incorrect, as this is a standing wave with nodes at the two ends.
  • #1
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Homework Statement


A string of length 1.5m with its two ends clamped is vibrating in fundamental mode. Amplitude at the center of the string is 4mm. Find the distance between the points who have amplitude 2mm.


Homework Equations


y=Asinkxcoswt
λ=2*L

The Attempt at a Solution



First of all i was trying to get the distances of the points from one of the nodes.
so putting t=0 i get y=Asinkx

k=2π/λ
λ=2*L = 2*1.5 = 3

so y= Asin[itex]\frac{2π}{λ}[/itex]x
i know the amplitude is 2mm which is half the max amplitude.
[itex]\frac{A}{2}[/itex]=Asin[itex]\frac{2π}{λ}[/itex]x

So that gives
0.5=sin[itex]\frac{2π}{λ}[/itex]x
[itex]\frac{π}{6}[/itex]=[itex]\frac{2πx}{λ}[/itex]

0.25=x
Now the other particle will be 1.25m from the node.

so the difference is 1m

but the answer given is 50cm
Where i have gone wrong??
 
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  • #2
I believe your solution is correct, the answer given is wrong. for a normal traveling wave the answer would be would be 50cm but this is a standing wave (n=1) with nodes at its two ends so I believe your answer is correct!
 

1. What are standing waves?

Standing waves are a type of wave that occurs when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a pattern of nodes and antinodes that appear to be standing still.

2. How do you find the distance between nodes in a standing wave?

The distance between nodes in a standing wave can be found using the formula: d = λ/2, where d represents the distance between nodes and λ represents the wavelength of the wave.

3. Can standing waves occur in any medium?

Yes, standing waves can occur in any medium as long as two waves with the same frequency and amplitude are traveling in opposite directions and can interfere with each other.

4. How do standing waves differ from traveling waves?

Standing waves differ from traveling waves in that they do not transfer energy from one point to another. Instead, the energy remains in one location and is simply oscillating back and forth.

5. What applications do standing waves have in science and technology?

Standing waves have various applications in fields such as acoustics, optics, and electronics. They are used in musical instruments, laser technology, and in the design of antennas and resonators.

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