Waves on a Cylinder - Solutions & Analysis

[unscientific]

Homework Statement

Homework Equations

The Attempt at a Solution

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)

 

[unscientific]

bumpp

 

[TSny]

Why are you restricting k to positive integers only?

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.

 

[unscientific]

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?

 

[TSny]

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.

 

[unscientific]

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*e^i(-kx-ωt) + A₂*e^i(kx-ωt)

so that the waves destructively superimpose.

 

[unscientific]

Is this correct??

 

[TSny]

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.