Waves and vibrations on a string

[Diku Khanikar]

Homework Statement

Can anyone please help me doing these questions?

Relevant Equations

E=(1/2)T(∂y∂x)^2

Q.1. The length of a stretched string fixed at both ends has a length of l=10 cm, mass per unit length ρ= 0.01 gm/cm. If the tension ' T ' is produced by hanging a 11 kg weight at both ends of the string, then calculate,

a) The wavelength of the first two harmonics,

b) The speed of the wave

c) The frequency of the third and fourth harmonics

Q. 2. Prove that the potential energy density of a vibrating string is given by
E=(1/2)T(∂y∂x)^2
where T is the tension in the string and y=y(x,t) is the transverse displacement of the string.

 

[etotheipi]

That question took a turn! The first three should follow from consideration of ##v_p = f\lambda##, whilst the last is a bit more difficult.

It's hard to give a hint without giving the whole thing away (and the derivation I know is pretty hand-wavy anyway); maybe try considering the EPE of small 'arc' of string?

 

[ORF]

Hi,

Here it is the solution to the last question:
https://oyc.yale.edu/sites/default/files/Erratum-Wave energy_0.pdf

I hope it helps.

Regards,
ORF