7. Vibrating String When a violin string vibrates, the sound produced results from a com- bination 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 7. (a) For fixed 1, the string has the shape of a sine curve y = A sin ax. Find the appropriate value of a for each of the illustrated standing waves. (b) Do you notice a pattern in the values of a that you found in part (a)? What would the next two values of a be? Sketch rough graphs of the standing waves associated with these new values of a. (c) Suppose that for fixed 1, each point on the string that is not a node vibrates with frequency 440 Hz. Find the value of B for which an equation of the form y = A cos Bt would model this motion. (d) Combine your answers for parts (a) and (c) to find functions of the form (x, t) = A sin ax cos ßt that model each of the standing waves in the figure. (Assume that A = 1.)

Question
7. Vibrating String When a violin string vibrates, the sound produced results from a com-
bination 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 7.
(a) For fixed 1, the string has the shape of a sine curve y = A sin ax. Find the appropriate
value of a for each of the illustrated standing waves.
(b) Do you notice a pattern in the values of a that you found in part (a)? What would the
next two values of a be? Sketch rough graphs of the standing waves associated with
these new values of a.
(c) Suppose that for fixed 1, each point on the string that is not a node vibrates
with frequency 440 Hz. Find the value of B for which an equation of the form
y = A cos Bt would model this motion.
(d) Combine your answers for parts (a) and (c) to find functions of the form
(x, t) = A sin ax cos ßt that model each of the standing waves in the figure.
(Assume that A = 1.)
Expand
Transcribed Image Text

7. Vibrating String When a violin string vibrates, the sound produced results from a com- bination 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 7. (a) For fixed 1, the string has the shape of a sine curve y = A sin ax. Find the appropriate value of a for each of the illustrated standing waves. (b) Do you notice a pattern in the values of a that you found in part (a)? What would the next two values of a be? Sketch rough graphs of the standing waves associated with these new values of a. (c) Suppose that for fixed 1, each point on the string that is not a node vibrates with frequency 440 Hz. Find the value of B for which an equation of the form y = A cos Bt would model this motion. (d) Combine your answers for parts (a) and (c) to find functions of the form (x, t) = A sin ax cos ßt that model each of the standing waves in the figure. (Assume that A = 1.)

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