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Let's use the above result to derive the slope of the ideal
vibrating string
From Eq.(C.11), we have the string displacement given by
By linearity of differentiation, the string slope is given by
Consider only the right-going component, and define
with
.
By the chain rule,
The left-going component is similar, but with
.
Thus, the string slope in terms of traveling velocity-wave components can be written as
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