Wave equation fundamentals

Figure 1: String plucked at its center to a distance $y$.

A standing wave on a string of mass $m$ fixed at both ends ($x = 0$ and $x = L$) is described by

$$y(x,t) = A \sin(kx) \sin(\omega t)\;.$$ (1)

Express all answers below in terms of the fundamental quantities $L$, $m$, $A$, $k$, and $\omega$.

(a)

What is the $x$-component of the force due to the string on the fixed point $x = 0$? (Remember, the string is under tension so it pulls on whatever is holding it.)

(b)

What is the $y$-component of the force due to the string on the fixed point $x = 0$ at any time $t$.

(c)

Consider a tiny chunk of string of length $dx$ between $x$ and $x + dx$. Find the $x$- and $y$-components of the force on the left side of this chunk (at $x$) due to the rest of the string.

(d)

Find the $x$- and $y$-components of the force on the right side of this chunk (at $x + dx$) due to the rest of the string.

(e)

Find the net force on the chunk.

(f)

Verify that $\sum \vec{F} = m\vec{a}$ works for the chunk in the limit $dx \rightarrow 0$. (Note that you should be able to do better than saying $0=0$.)

(g)

The net force $F_y$ on the chunk in the $y$-direction is proportional is to its displacement $y$ from equilibrium. Use this to compute an effective spring constant $K=-F_y/y$. Compute the frequency you would expect from an object of mass equal to the mass of the chunk tied to a spring of constant $K$, and compare to $\omega$.