Question

When a violin string vibrates, the sound produced results from a combination of standing waves that have evenly placed nodes. The figure 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 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 Hz [Find out how this relates to the graph]. Find the value of \beta for which an equation of the form $y = A cos \beta t$ would model this motion.
d) Combine you answers for parts (a) and (c) to find functions of the form $y(x,t) = A sin \alpha x cos \beta t$ that model each of the standing waves in the figure. (Assume that A = 1.)

Answer 1

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