Height And Width of Reflected Pulses (through less dense medium) ?

[Ted1508]

Homework Statement

Consider the waveform shown below heading towards a boundary between two
strings. Let string 1 have mass per unit length of µ1 = 0.05 kg/m and let string 2
have a mass per unit length µ2 = 0.02 kg/m. Let the tension in both strings be T
= 100N.
-->
____/\____ _ _ _ _ _ _ _
string 1 ... string 2

Using the reflection and transmission coefficients, sketch the reflected and
transmitted pulses after the incident pulse has completely passed through the
boundary. How do the width and height of the reflected and transmitted pulses
compare to the incident pulse?

Homework Equations

(Ar + Ai) * cos(wt) = At * cos(wt)
Ai + Ar = At
k(1) * (Ar - At) = -k(2) * Atr

The Attempt at a Solution

r = (v(2) - v(1)) / (v(2) + v(1))
τ = 2*v(1) / (v(1) + v(2)) = 2 / (1 + v(1) / v(2))
v(2) = sqrt( (f * τ) / µ2 ) & vise versa
v(1) / v(2) = sqrt( µ2 /µ1 )
r = (1-0.63)/(1+0.63) = 0.225, t= 1.225

I know the pulse in the lighter string should look like ___ _ /\ _ _ (taller but less wide)

But how do I quantitatively illustrate that the width and height change, and by how much?

Help is definitely appreciaTed!

 

[member 553083]

The width and height of the reflected and transmitted pulses can be determined by using the reflection and transmission coefficients. The reflection coefficient, r, is given by: r = (v2 - v1)/(v2 + v1)where v1 and v2 are the wave velocities in strings 1 and 2 respectively. The transmission coefficient, t, is given by:t = 2v1/(v1 + v2)Using these coefficients, we can determine how the amplitude and width of the reflected and transmitted pulses will change. The amplitude of the reflected pulse will be equal to r times the amplitude of the incident pulse, while the amplitude of the transmitted pulse will be equal to t times the amplitude of the incident pulse. The width of the reflected pulse will be equal to r times the width of the incident pulse, while the width of the transmitted pulse will be equal to t times the width of the incident pulse. Thus, we can conclude that the reflected and transmitted pulses will be both narrower and less tall than the incident pulse.