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## Homework Statement

A horizontally oriented string with mass per unit length ##\mu=0.1kgm^{-1}## is under tension T=1.0N. The left end of the string moves up and down in a simple harmonic motion with an amplitude A=0.10m and frequency f=3.0Hz. This sets up a sinusoidal wave along the string with the wave function ##y(x,t)=Acos(kx-\omega t+\phi)##.

i.Find the wave velocity (v) and the wavelength for the wave (##\lambda##).

ii.y(0,0)=0.071m, find ##\phi##.

iii.After traveling for some distance the wave arrives at the right end of the string and is reflected. The reflected wave is described by ##y_r=-Acos(kx+\omega t +\phi)##. Show that the superposition of the incoming and reflected waves gives the resultant standing wave: ##y_s=2Asin(kx+ \phi)sin(\omega t)##.

iv. Using the information available from (i) to (iii) above, determine the length of the string.

## Homework Equations

i. ##v=\sqrt{\frac{T}{\mu}}##

##\lambda=\frac{v}{f}##

ii. ##\phi=cos^{-1}(0.71)##

## The Attempt at a Solution

I think I've done parts (i) to (iii) but I'm struggling with part iv. We have what the wave equation for the standing wave is so I think this should just be a case of using a boundary condition for the ends of the string. The one I've been using is that ##y_s(x,t)=0## when ##x=L \forall t## where L is the length of the string. This boundary condition being taken from the wave equation of the reflected wave being inverted and traveling in the opposite direction to the incident wave so it has a fixed end.However, this gives me that $$sin(kL+\phi)=0 \implies L=\frac{n\pi-\phi}{k}$$ where n is a positive integer.

Firstly, is this correct? And secondly is there a way for me to figure out which value of n to use? I recall that for a string with one free end and a fixed end we should only have odd numbered harmonics but i see nothing here excluding the even values of n. I did consider using a second boundary condition for x=0 but it seems like as we have fixed values of A,##\phi## and k we already know what the amplitude at x=0 has to be.