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1. When properly tuned, the 6 strings of a guitar at their full length of L = 0.650 m are intended to produce the following frequencies: String Wave speed Frequency (Hz) 82.4 Note (m/s) 1 E2 A2 2 110.0 3 D3 146.8 4 G3 196.0 B3 246.9 6. E4 329.6 Use the frequency equation to calculate the wave speed v in each string.

1. When properly tuned, the 6 strings of a guitar at their full length of L = 0.650 m are intended to produce the following frequencies: String Wave speed Frequency (Hz) 82.4 Note (m/s) 1 E2 A2 2 110.0 3 D3 146.8 4 G3 196.0 B3 246.9 6. E4 329.6 Use the frequency equation to calculate the wave speed v in each string.

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1. When properly tuned, the 6 strings of a guitar at their full length of L = 0.650 m are intended to produce the following frequencies: String Note Frequency Wave speed (Hz) (m/s) 1 E2 82.4 2 A2 110.0 3 D3 146.8 4 G3 196.0 B3 246.9 6 E4 329.6 Use the frequency equation to calculate the wave speed v in each string. 2. Say that string 2 is improperly tuned, so that its wave speed is 11 m/s faster than the ideal speed you calculated in Question 1. What frequency would it produce? Is that frequency too high (“sharp") or too low ("flat")? By how many Hz is the note out of tune? 3. If string 4 is improperly tuned, so that it plays 5.2 Hz too low, what is the actual wave speed in the string? Do you need to increase or decrease the speed to get the note in tune? By how much must v change? Now let's consider the placement of the frets. We will use String 3, though the same pattern works for all strings. At its full length of L = 0.650 m, String 3 produces a frequency of 146.8 Hz (a D). To play other notes, we need to put frets at the right positions to produces notes in half-step increments (D#, E, F, etc.). There are the first few higher frequencies we want String 3 to produce.
Transcribed Image Text:1. When properly tuned, the 6 strings of a guitar at their full length of L = 0.650 m are intended to produce the following frequencies: String Note Frequency Wave speed (Hz) (m/s) 1 E2 82.4 2 A2 110.0 3 D3 146.8 4 G3 196.0 B3 246.9 6 E4 329.6 Use the frequency equation to calculate the wave speed v in each string. 2. Say that string 2 is improperly tuned, so that its wave speed is 11 m/s faster than the ideal speed you calculated in Question 1. What frequency would it produce? Is that frequency too high (“sharp") or too low ("flat")? By how many Hz is the note out of tune? 3. If string 4 is improperly tuned, so that it plays 5.2 Hz too low, what is the actual wave speed in the string? Do you need to increase or decrease the speed to get the note in tune? By how much must v change? Now let's consider the placement of the frets. We will use String 3, though the same pattern works for all strings. At its full length of L = 0.650 m, String 3 produces a frequency of 146.8 Hz (a D). To play other notes, we need to put frets at the right positions to produces notes in half-step increments (D#, E, F, etc.). There are the first few higher frequencies we want String 3 to produce.
Guitars use standing waves in their strings to create sound waves at specific frequencies. When you pluck a string, it creates a standing wave with 1 loop, the fundamental frequency f; of the string, given by f1 2L fi = fundamental frequency, in Hz v = speed of the wave in the string, in m/s L = Length of the string, in m According to the equation, there are only two ways to change the frequency the string produces. You can change the speed v of the waves in the string, or you can change the length of the string L. How is this accomplished on a guitar? Guitars have two ways to change the speed of the wave, v. First, they can vary the density of the string. Waves travel faster through thin strings, producing higher frequencies, and slower through fat strings, producing lower frequencies. Most guitars have 6 strings of different density, arranged from fattest (low notes) to thinnest (high notes). The second way to change the wave speed is to change the tension in the string. Waves travel faster through tight strings, producing higher frequencies, and slower through loose strings, producing lower frequencies. Each string on the guitar is wrapped around a "tuning knob" at the top, which can be turned to tighten or loosen the string. These knobs are used to make slight adjustments in the string's frequency, to bring it "in tune" with the other strings. You can also produce different frequencies by changing the length L of the string. While the strings themselves have a fixed length, you can put your finger on the string to shorten the oscillating part of the string. There are a number of metal bars or "frets" built into the neck of the guitar. When you press down on the string, it hits the closest fret; that becomes one end or node of the standing wave, shortening the length, which increases the frequency. fret New, shorter L In this exercise, you will calculate some of the wave speeds and frequencies of a guitar, and determine the correct placement of the frets.
Transcribed Image Text:Guitars use standing waves in their strings to create sound waves at specific frequencies. When you pluck a string, it creates a standing wave with 1 loop, the fundamental frequency f; of the string, given by f1 2L fi = fundamental frequency, in Hz v = speed of the wave in the string, in m/s L = Length of the string, in m According to the equation, there are only two ways to change the frequency the string produces. You can change the speed v of the waves in the string, or you can change the length of the string L. How is this accomplished on a guitar? Guitars have two ways to change the speed of the wave, v. First, they can vary the density of the string. Waves travel faster through thin strings, producing higher frequencies, and slower through fat strings, producing lower frequencies. Most guitars have 6 strings of different density, arranged from fattest (low notes) to thinnest (high notes). The second way to change the wave speed is to change the tension in the string. Waves travel faster through tight strings, producing higher frequencies, and slower through loose strings, producing lower frequencies. Each string on the guitar is wrapped around a "tuning knob" at the top, which can be turned to tighten or loosen the string. These knobs are used to make slight adjustments in the string's frequency, to bring it "in tune" with the other strings. You can also produce different frequencies by changing the length L of the string. While the strings themselves have a fixed length, you can put your finger on the string to shorten the oscillating part of the string. There are a number of metal bars or "frets" built into the neck of the guitar. When you press down on the string, it hits the closest fret; that becomes one end or node of the standing wave, shortening the length, which increases the frequency. fret New, shorter L In this exercise, you will calculate some of the wave speeds and frequencies of a guitar, and determine the correct placement of the frets.
Expert Solution
Step 1

The fundamental frequency produced on a string producing standing waves is given as

f=v2Lv is the speed of the waves on the stringL is the length of the string

In the given problem, the guitar strings produce different notes, for the fixed length of each string of 0.65 m

These different notes have different frequencies, and thus different wave speeds as the standing waves travel on these strings

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