Chapter 7.FOM, Problem 7P

Vibrating String When a violin string vibrates, the sound produced results from a combination of standing waves that have evenly placed nodes. The figure $\left(1\right)$ illustrates some of the possible standing waves. Let’s assume that the string has length $\pi$.

a. For fixed $t$, the string has the shape of a $\mathrm{sine}$ curve $y=A\sin \alpha x$. Find the appropriate value of $\alpha$ for each of the illustrated standing waves.

b. Do you notice a pattern in the values of $\alpha$ that you found in part (a)? What would the next two values of $\alpha$ be? Sketch rough graphs of the standing waves associated with these new values of $\alpha$.

c. Suppose that for fixed $t$, each point on the string that is not a node vibrates with frequency $440\text{ Hz}$. Find the value of $\beta$ for which an equation of the form $y=A\cos \beta t$ would model this motion.

d. Combine your answers for parts (a) and (c) to find functions of the form $y\left(x,t\right)=A\sin \alpha x\cos \beta t$ that model each of the standing waves in the figure. (Assume that $A=1$.)

Figure $\left(1\right)$