# Sinusoidal Wave traveling on a Composite String

1. Mar 16, 2010

### golnat

Cosider a sinusoidal wave traveling along a composite string. The string is under constant tension, F, and consists of a light portion (x<0) with mass per unit length mu1 joined in a continuous manner to a heavier portion (x>0) with mass per unit length mu2. Let y_1i(x,t)=A_1i*sin(wt-k_1x) be the form of the incident wave.

When the wave meets the joint between the two segments, it will be partially reflected and partially transmitted into the heavier segment. Let y_1r(x,t) be the reflected wave and y_2(x,t) be the transmitted wave.

a) Will the trasmitted wave have the same wavenumber as the incident wave? Same frequency? What about the reflected wave?

b) Write an expression for the total wave amplitude at x=0. In the limit of small amplitute waves, the vertical force on a portion of the string is F_y=F*dy/dx (partials). Write expressions for F_y on both sides of the boundary at x=0.

c) Use the results of part b) to solve for the reflected and transmitted wave amplitudes. Calculate the coefficients of reflection and transmission (or refraction), denoted R and T, and defined by ratios

R=A_1r/A_1i
T=A2/A_1i

Answers should depend only on mu1 and mu2. Comment on the behavior in some special cases:

mu2/mu1 -> 1 or -> infinity. What about -> 0?
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Okay, I first tried just thinking about it. I realized that the frequencies of the incident and transmitted waves ought to be the same because if they weren't the waves would get out of phase which isn't allowed because the string is one continuous string. Also, the for reflected wave, the wavenumber and frequency should be the same because if the mass of the heavier string were to be taken to infinity, the string would become just like a wall that would reflect the waves exactly as they came in but with opposite amplitude.

I can't figure out what to work with math-wise to show that the frequencies should be the same as well as for the wavenumbers.

Thanks for you time and help.