Waves on a Cylinder - Solutions & Analysis

In summary, you are trying to create a superposition of waves that will create a "node" at θ = π/3. You want to limit the value of n such that you don't repeat the same positions on the cylinder.
  • #1
unscientific
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Homework Statement



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Homework Equations





The Attempt at a Solution



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Not sure what they mean by general superposition of solutions...Do i use D'alembert's solution whereby I assume initial velocity = 0, and therefore:

z = z(θ-ct) + z(θ+ct)
 
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  • #2
bumpp
 
  • #3
Why are you restricting k to positive integers only?


unscientific said:
Not sure what they mean by general superposition of solutions...Do i use D'alembert's solution whereby I assume initial velocity = 0, and therefore:

z = z(θ-ct) + z(θ+ct)

I don't understand the assumption of zero initial velocity.
You want to construct a specific superposition of solutions for ω2 that will create a "node" at θ = π/3.
 
  • #4
TSny said:
Why are you restricting k to positive integers only?




I don't understand the assumption of zero initial velocity.
You want to construct a specific superposition of solutions for ω2 that will create a "node" at θ = π/3.

i'm not sure how to create that superposition do i use A1*exp(kθ-ωt) + A2*exp(k+ωt) or something?
 
  • #5
What are the two values of k corresponding to ω2?

Superimpose two waves with those values of k. Note that you do not want to change the sign of the ωt term in the exponential. After forming the superposition, you will be able to factor out a common factor of e-iωt.

You might want to review the concept of "standing waves"on a string.
 
  • #6
TSny said:
What are the two values of k corresponding to ω2?

Superimpose two waves with those values of k. Note that you do not want to change the sign of the ωt term in the exponential. After forming the superposition, you will be able to factor out a common factor of e-iωt.

You might want to review the concept of "standing waves"on a string.

The '-ωt' term involves the velocity of the wave, so you must superimpose:

A1*ei(-kx-ωt) + A2*ei(kx-ωt)

so that the waves destructively superimpose.
 
  • #7
v60z9e.png


Is this correct??
 
  • #8
That looks good to me. You want to limit the value of n such that you don't repeat the same positions on the cylinder.
 

1. What is the purpose of studying waves on a cylinder?

Studying waves on a cylinder helps us understand the behavior of waves in different contexts and how they interact with different surfaces. This knowledge can be applied in various fields such as acoustics, fluid dynamics, and structural engineering.

2. How do waves behave on a cylinder?

Waves on a cylinder exhibit both longitudinal and transverse motion, which is a combination of radial and circumferential displacement. The interaction between these two types of motion produces complex wave patterns that are dependent on factors such as the cylinder's shape, size, and material properties.

3. What are the solutions to the wave equation for a cylinder?

The solutions to the wave equation for a cylinder are known as Bessel functions. These functions describe the amplitude and phase of the wave at different points on the cylinder's surface. The specific form of the Bessel function depends on the type of boundary conditions imposed on the cylinder.

4. How are waves on a cylinder analyzed and measured?

Waves on a cylinder can be analyzed through mathematical models and numerical simulations. The amplitude and frequency of the waves can be measured using various experimental techniques such as laser Doppler vibrometry, strain gauges, or accelerometers.

5. What are some practical applications of understanding waves on a cylinder?

The knowledge gained from studying waves on a cylinder has practical applications in various fields, including noise control, underwater acoustics, and the design of cylindrical structures such as pipes, chimneys, and wind turbines. It also has implications in the development of medical devices, such as stents and catheters, which involve interactions with cylindrical surfaces in the human body.

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