A long string has characteristic impedance and carries a sinusoidal wave of angular…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Superposition, boundaries and interference · Superposition, boundaries and interference

Modes, Packets and Media, Ch Reflection, transmission, matching and tunnelling at boundaries

Problem

A long string has characteristic impedance

Z = 0.40\,\mathrm{kg\,s^{-1}}

and carries a sinusoidal wave of angular frequency

\omega = 10\,\mathrm{rad\,s^{-1}}

. Its far end is attached to a load. Take the load impedance of a spring of stiffness

\kappa

to be

Z_{\rm spring} = \kappa/(i\omega)

and of a dashpot of coefficient

b

to be

Z_{\rm dashpot} = b

, and of a mass

m

to be

Z_{\rm mass} = i\omega m

, all in the sign convention of a forward wave

e^{i(\omega t-kx)}

. ((a)) The load is a spring of stiffness

\kappa = 2.0\,\mathrm{N\,m^{-1}}

and a dashpot of coefficient

b = 0.30\,\mathrm{kg\,s^{-1}}

connected in parallel (same velocity, forces add, so impedances add). Find the complex reflection coefficient

r

and the fraction of the incident power absorbed. ((b)) The load is a mass

m = 0.050\,\mathrm{kg}

and a spring of stiffness

\kappa = 2.0\,\mathrm{N\,m^{-1}}

connected in series (same force, velocities add, so admittances add). Find

r

and the fraction of the incident power absorbed, and explain the result in one sentence.

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