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In reading about the Fender Rhodes piano, which uses metal rods instead of strings, you come across the following equation:

$$\rho \frac { { \partial }^{ 2 }f }{ \partial { x }^{ 2 } } = Y\frac { { \partial }^{ 2 }f }{ \partial { t }^{ 2 } }$$
…where $\rho$, the rod density, and $Y$, the Young’s modulus of the rod material, are constants. Which statement below most completely describes what you can conclude by examining this formula?

A

This equation describes a simple harmonic oscillation of the rod with angular frequency $\omega = \sqrt{Y\rho}$. The function $f(t) = A \sin{(\omega t})$

B

The solution of this equation is a wave of the form $f(x,t) = A \sin{(k x - \omega t)}$. The wave speed is given by $v = \sqrt{ \cfrac {\rho}{Y}}$.

C

The solution of this equation is a function $f(x,t)$ that describes a wave traveling in the $+x$ direction with speed $v = \sqrt{ \cfrac {\rho}{Y}}$. The function $f(x,t)$ can be any continuous function.

D

The solution of this equation is a function $f(x,t)$ that describes a wave traveling in the $+x$ or $-x$ directions with speed $v = \sqrt{ \cfrac {\rho}{Y}}$. The function $f(x,t)$ can be any continuous function.

E

The solution of this equation is a function $f(x,t)$ that describes a wave traveling in the $+x$ or $-x$ direction with speed $v = \sqrt{ \cfrac {\rho}{Y}}$. However, the function $f(x,t)$ can only be a superposition of a sinusoidal waves whose frequencies are integer multiples of $\sqrt{ \cfrac {\rho}{Y}}$.

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