1-D wave resonance in the case of an Open-Ended String

In summary, the conversation discusses the conditions for a fixed and free string at both ends, and the solution for standing waves in strings and pipes. The general solution is given by the equation y = A sin(kx+ωt)+ B cos(kx+ωt), and for a fixed string, the conditions are y(0,t) = 0 and y(L,t) = 0. The condition for a free string at both ends is given by the equations ∂y/∂x|L = 0 and ∂y/∂x|0 = 0.
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Homework Statement
I want to know the conditions in the case of Open-End.
Relevant Equations
## \frac{1}{v^2} \frac{∂^2y}{{∂t}^2} = \frac{∂^2y}{{∂x}^2} ## and general solution ## y = A sin(kx+ωt)+ B cos(kx+ωt) ##
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  • #2
Try $$\left. \frac{\partial y}{\partial x} \right|_{x=L} =0.$$
 
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  • #3
kuruman said:
Try $$\left. \frac{\partial y}{\partial x} \right|_{x=L} =0.$$
242686


if free string both ends. The condition is ##\left. \frac{\partial y}{\partial x} \right|_{x=L} =0.## and ## \left. \frac{\partial y}{\partial x} \right|_{x=0} =0.## ?
 
  • #4
Another said:
View attachment 242686

if free string both ends. The condition is ##\left. \frac{\partial y}{\partial x} \right|_{x=L} =0.## and ## \left. \frac{\partial y}{\partial x} \right|_{x=0} =0.## ?
Yes.
 
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Related to 1-D wave resonance in the case of an Open-Ended String

1. What is 1-D wave resonance?

1-D wave resonance is a phenomenon that occurs when a wave is reflected back and forth between two fixed points, causing the wave to amplify and create a standing wave pattern.

2. What is an open-ended string?

An open-ended string is a string that is not attached to anything on one or both ends, allowing for free movement of the string.

3. How does 1-D wave resonance occur in an open-ended string?

In an open-ended string, when a wave is reflected back and forth between the two ends, the reflected waves interfere with each other, creating areas of constructive and destructive interference. This results in the formation of a standing wave pattern, where certain points on the string remain stationary while others vibrate with maximum amplitude.

4. What factors affect 1-D wave resonance in open-ended strings?

The main factors that affect 1-D wave resonance in open-ended strings are the length of the string, the tension applied to the string, and the frequency of the wave. All of these factors influence the wavelength and speed of the wave, which in turn affects the resonance pattern.

5. What are some real-life applications of 1-D wave resonance in open-ended strings?

1-D wave resonance in open-ended strings can be seen in musical instruments, such as guitars and violins, where the strings are free to vibrate and create standing waves. It is also utilized in engineering and architecture, where resonance is used to design structures that can withstand vibrations and minimize potential damage.

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