Boundary conditions – strings

• You may be wondering how a situation could ever develop in which two identical waves come from opposite directions. Well, wonder no more.
• When you pluck a stringed instrument, waves travel to the ends of the string and reflect at each end, and return to interfere under precisely the conditions needed for a standing wave.
• 
• Note that there are two nodes and one antinode.
• Why must there be a node at each end of the string?

Sketching and interpreting standing wave patterns

• Observe that precisely half a wavelength fits along the length of the string or $\frac{1}{2}\lambda = L$.
• Thus we see that $\lambda = 2L$.
• Since $v = f\lambda$ we see that $f = \frac{v}{2L}$ for a string.
• This is the lowest frequency you can possibly get from this string configuration, so we call it the fundamental frequency $f_1$.
• The fundamental frequency of any system is also called the first harmonic.

•  The next higher frequency has another node and another antinode.
• We now see that λ = L.
• Since v = λf we see that f = v/L.
• This is the second lowest frequency you can possibly get and since we called the fundamental frequency f₁, we’ll name this one f₂.
• This frequency is also called the second harmonic.