ispeakmath
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Post by ispeakmath on Jul 14, 2016 0:48:43 GMT
Hi all,
I suck at fundamental frequencies of tightened strings and whatnot, so I'm just asking if there are any good books, websites, etc. that you find explain fundamental frequencies in a friendly way.
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Post by Vincent on Jul 25, 2016 22:30:54 GMT
I don't know about any good resources that explain the whole thing in full (I'm not the best in waves). However, I can give a quick explanation with specific things to look up.
1. Nature of waves. Waves are perturbations that propagate spacially and temporally. (Specifically, the second spacial derivative is proportional to the second temporal derivative via the wave equation, but that may be too advanced for you now).
2. Speed of a transverse wave. All you need to know right now is the general formula for the speed of a transverse wave, sqrt(T/mu), where T is the tension in the string and mu is the mass per unit length. The proof lies in investigating a transverse perturbation and deriving the wave equation for a transverse wave, if you're interested. Why is the speed important? It relates frequency, what you're looking for, and wavelength.
3. The physics of plucking a string fixed at both ends. Here's where it gets interesting.
When you pluck a string in any which manner, you're basically releasing pulses moving to the left and right. These pulses can have any which shape, but a concept called Fourier decomposition states that these pulses can be decomposed into a sum of sinusoidal waves with different frequencies and wavelengths that travel independently of each other. Now, let's follow these waves.
The wave moves along until it hits one of the ends. What happens? In this case, the end needs to stay fixed. If a perturbation reaches the end, the end needs to create an opposite perturbation so it can stay fixed. The created perturbation creates a wave on the string that travels away from the end because it is the only way to go. So at any point in time, the end is creating the exact opposite perturbation, the exact opposite wave, traveling in the opposite direction. This is called a "hard" reflection, where a wave reflects with a pi phase shift.
As I said before, these frequencies travel independently. For the vast majority of frequencies, what happens is that the wave gets reflected from both ends, causing a complicated interference pattern that changes with time. The string looks like it's jiggling randomly, and due to the pattern changing with time, any given point on the string will not even have a frequency because it is not undergoing simple oscillation. Because perturbations cause sound waves, this will be heard as noise.
However, for a thin slice of frequencies denoted as the harmonics (or frequencies very near them), this doesn't happen. The waves are still reflected, but they are reflected "just right" so that the wave shape becomes stable (i.e. they interfere to create a stable pattern called a standing wave where every given point on a string oscillates with the same amplitude and the same frequency.) This is the familiar picture of the standing wave. The frequencies that these waves vibrate in are called the harmonics. Because everything is oscillating with the same frequency, you can hear that frequency distinctly and not as noise.
Finding the frequencies with the prior information is the easy part. In this case, the waves interfere as a perpetual sinusoid (I'm too lazy to prove this rigorously, but it's very intuitive). Just draw a picture where the string is fixed at both ends like the following:
Remembering that two bumps are a wavelength, you can deduce the wavelength from the length of the string. Just use distance=rate*time or wavelength=speed*period to solve for the period/frequency.
Hope this helps!
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