- #1

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- TL;DR Summary
- exact or approximated calculation of the sinusoidal length in one period [0,2π] or corresponding to [0,λ] including the amplitude of the wave A as well as the wave number k.

The known expression of the wave function is

where A is the amplitude, k the wave number and ω the angular velocity.

The mathematical definition of arc length for a generical function in an interval [a,b] is

where, in our sinusoidal case:

For our purpose (calculation of the length in one period [0,2π] corresponding to [0,λ]), I guess that we can remove the ωt term from the expression, so that

Now, I am really struggling to calculate this integral. There are some instances in internet where mathematicians calculated the length of half period [0,π] of a sinusoidal wave as

which accordingly implies

The fact is that I want to include the amplitude of the wave and the wave number by means the term
as well as the wave number alone in the cos argument
in the main integral, as shown previously.

WolframAlpha and SymboLab calculates a solution involving complicated ellpitical integrals . Could you please help me in finding the right way to calculate the integral? An approximated solution would be also appreciated. The main requirement is the amplitude A and the wave number k are considered in the derivation and in the finalr result. Eventually, l has to be funtion of A and k in the final result.

Thank you in advance for your precious support.

Best,

E.

where A is the amplitude, k the wave number and ω the angular velocity.

The mathematical definition of arc length for a generical function in an interval [a,b] is

where, in our sinusoidal case:

For our purpose (calculation of the length in one period [0,2π] corresponding to [0,λ]), I guess that we can remove the ωt term from the expression, so that

Now, I am really struggling to calculate this integral. There are some instances in internet where mathematicians calculated the length of half period [0,π] of a sinusoidal wave as

which accordingly implies

The fact is that I want to include the amplitude of the wave and the wave number by means the term

WolframAlpha and SymboLab calculates a solution involving complicated ellpitical integrals . Could you please help me in finding the right way to calculate the integral? An approximated solution would be also appreciated. The main requirement is the amplitude A and the wave number k are considered in the derivation and in the finalr result. Eventually, l has to be funtion of A and k in the final result.

Thank you in advance for your precious support.

Best,

E.