Characteristic impedance. For a transverse sinusoidal mechanical wave propagatin

Question

Characteristic impedance. For a transverse sinusoidal mechanical wave propagating through a medium, we can define a characteristic impedance Z=F_y/v_y, where v_y = v_y0 e^i(kx - omega t + phi) is the transverse particle velocity and F_y = F_y0 e^i(kx - omega t) is the applied oscillating force in the y direction that drives particle motion. Show that the characteristic impedance for a transverse wave on a string is Z = tau/v = mu v. Imagine cutting the string, and applying an oscillating force F_y that is exactly the same as would have been applied by the piece of string you've cut away. The resulting y velocity? Use the pulse equation to express it in an alternate form. (We'll see that the characteristic impedance governs wave reflection and transmission at interfaces between different media.)

Explanation / Answer

given, Fy = Fyo*e^(i(kx - wt))
and Vy = Vo(e^(i(kx - wt + phi)))

now, if we cut the string at some point, then is the tension in the string is T
then Tsin(theta) is the y component of force
and y component of velocity will be v
but y = A*cos(kx - wt) [ for no phase difference ]
dy/dt = Awsin(kx - wt)
so, Vy = Awsin(kx - wt)
also, tan(theta) = dy/dx = -Ak*sin(kx - wt)

for small angles, Tsin(theta) = Ttan(theta)
hence Ty/Vy = T*tan(theta)/Vy = T*Ak*sin(kx - wt)/Aw*sin(kx - wt)
Ty/Vy = Z = T*k/w
but w/k = v ( wave speed)
so, Z = T/v

now for a string of mass density mu, v = sqroot(T/mu)
or T = v^2*mu
hence
Z = T/v = v^2*mu/v = mu*v

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---