Question

A wave on a string, mass per unit length of 0.2 g/cm, travels in the z direction with the string displaced in the x and y directions as described by the equations

x = 5 sin(kx - ωt) cm, and

y = 5 cos(kx - ωt) cm.

The wavelength and the wave velocity are λ=10 cm and v =9 cm/s respectively. What average power is transmitted along the string?

Answer 1

This is a picture of waves travelling in the z direction simplified into plane waves. I believe the formula I use here is P = I/a.

Answer 2

If your wave travels in the z direction, then its equations must be:

x = 5 sin(kz - ωt) cm
y = 5 cos(kz - ωt) cm

Your wave shows circular polarisation.

Answer 3

Kinetic energy lineic density is:

\(\large e_K=\frac 12 \mu v^2\), where v(z,t) is the velocity of a string's point.

As displacement is 5 cm on a circle, velocity is 5ω, independent of z and t.
With wave celerity and lineic mass of the string, you can find
\(e_K=8.10^{-4}J.m^{-1}\)

Answer 4

I'm still unsure how to compute the average power transmitted from here.

Answer 5

also how can \[e_{K}\] come out with units J/m

Answer 6

a) 0.2 × 10-4 W

b) 0.1 × 10-3 W

c) 0.4 × 10-5 W

d) 1.4361×104 W

Answer 7

I do not know what concepts you know and are able to use.

It seems the easiest way is to computer lineic kinetic energy in the first place;

Then lineic kinetic energy and elastic potential energy are in a ratio 1:1.

So the overall lineic energy is

\(\large e=2e_K=1.6\times 10^{-3}J.m^{-1}\)

As energy is carried by the wave at the same celerity as the wave itself, power transmitted the product of lineic energy by celerity.

Answer is P = e.v = 0.144 mW

Answer 8

*is the product