Mechanical impedance at ends of bar

[phyziks]

Assuming very small losses, find a condition relating the mechanical impedances of the supports at the ends of a bar of length L and longitudinal wave speed c if the bar is to have an integral number of wavelengths between its ends when it is vibrating longitudinally.


mechanical impedance
-Zmo/PLC = [r + j(x + tan kL)]/[(1 - x tan kL) + jr tan kL]


at the ends there is possibility of the nodes only .
The possible wavelengths
λ 1 = 2L , λ2 = L , λ3=2L/3 , λ4=L/2…….λn= 2L/n
the possible frequencies
n1=c/2L , n2=c/L , n3=3c/2L ………………
so the only above waves of frequencies and wavelengths can be produced in the above condition

 

[NateD]

. Therefore, the condition for an integral number of wavelengths between the two ends of the bar is: x + tan kL = 0.