Standing waves question

This is my first post on the forum, and I hope you guys can help me

My questions is this,
There are two strings which are connected with constant tension everywhere, but they have different linear mass densities. The second mass density is double that of the first (μ1= 2μ1). A wave propagates along the "lighter" string with an amplitude of 10 cm and a wavelength of 10 cm. I have to find the wavelength of the wave propagated through the second string, assuming frequency is the same everywhere, and also what is the amplitude of the transmitted wAve?

Relevant equations which I have used are

v= ω/k = λω/2π where k= 2π/λ

Also v^2 = T/μ

All I have done so far is to notice that if the second mass density is double the first then the second velocity is a factor of sqrt2/2 smaller than the first? So would λ then simply be the same factor larger than that of the first wavelength as frequency is constant?

If this is the case then my answer was 14 cm

I haven’t yet begun the second part as I wanted to do the first part first

Cheers!

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