A long string is composed of two different materials. The left sidehas a smaller

Question

A long string is composed of two different materials. The left sidehas a smaller linear mass density than the right side. Thecomposite string is placed under a uniform tension, then the leftside is vibrated with a constant frequency. It is observed that thewavelength on the light string is twice that of the heavystring. 1. what is the ratio of the linear mass densities? 2. what is the ratio of the amplitudes of vibration? 3. what is the ratio of the transverse speeds? 4. what is the ratio of the propagation speeds? 5. what is the fraction of incident energy that is transmitted tothe more massive string?
I promise to rate you as lifesavers if your answer iscorrect! A long string is composed of two different materials. The left sidehas a smaller linear mass density than the right side. Thecomposite string is placed under a uniform tension, then the leftside is vibrated with a constant frequency. It is observed that thewavelength on the light string is twice that of the heavystring. 1. what is the ratio of the linear mass densities? 2. what is the ratio of the amplitudes of vibration? 3. what is the ratio of the transverse speeds? 4. what is the ratio of the propagation speeds? 5. what is the fraction of incident energy that is transmitted tothe more massive string?
I promise to rate you as lifesavers if your answer iscorrect! I promise to rate you as lifesavers if your answer iscorrect!

Explanation / Answer

The important thing in this problem is that the wave speed isdefined by the linear mass density:v=(FT/) Of course, we also know that the product of wavelength andfrequency of a wave in a string is a constant which is the wavespeed f=v=(FT/) So if 1=22 and we want tofind out the ratio of the 's, rearrange for : =FT/(f)2 so 1/2=F1/(12f12).(22f22)/F2and the tensions and frequencies must be the same so they canceland we are left with 1/2=22/12=22/222=0.25 4. The wave propagation speed ratio isv1/v2=1f1/2f2,and the frequencies cancel out sov1/v2=1/2=22/2=2 Anyway, having the ratio of 's and v's is good, now we can workout the ratio of the impedances:Z1/Z2=1v1/2v2=0.5 So now we're in a position to look at the transmission of the wavefrom the light string to the heavy string: T1to2=ymax,2/ymax,1=2Z1/(Z1+Z2)=(subin above values)=2/3 2. the ratio of the amplitudes is simplyymax,1/ymax,2=3/2 3. Transverse speeds are given by the time derivation of theposition of bits of string: vtrans(t)=dy/dt=d/dtymaxsin(t+)=ymaxcos(t+) And we know the ratio of the ymax's, and also f doesn'tchange (so doesn't change) sovtrans,1(t)/vtrans,2(t)=ymax,1cos(t+)/ymax,2cos(t+)=ymax,1/ymax,2=3/2 5. I'm guessing at this one, hope my guess is intelligent... With simple harmonic oscillators, energy is proportional toamplitude squared. So the fraction of energy transmitted would be (2/3)^2=4/9.Maybe? 5.Reflection and transmission at a boundary between dissimilarmaterials -

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