Tension required to tune a string to the note of ##E_2##

In summary, the boundary condition for a wave on a guitar string is that the displacement, represented by the variable psi, must be 0 at the points where the string is fixed. The real standing wave solution is given by psi = psi_0cos(kx+phi_x)cos(omega t+phi_t), and the allowed oscillation frequencies can be found by setting cos(kx+phi_t) = 0 at the points x=0 and x=L. This yields the expression f = (n/2L)sqrt(W/M) for the allowed frequencies, where n is the wavenumber. To calculate the tension required to tune the string to the note E2, it is necessary to determine whether E2 is
  • #1
thomas19981

Homework Statement


State the boundary condition which must be met at a point where the string of question 2 is fixed.

Hence find the real standing wave solutions to the wave equation, and determine the allowed oscillation frequencies, when such a string of length ##L## is fixed at its ends.

If the bottom string of a guitar has a mass of ##5.4 g/m## and its length is determined by the distance ##0.648 m## from the bridge to the nut, find the tension required to tune the string to the note known as ##E_2## (a frequency of ##82.4 Hz##)."

Question 2 describes a wave on a guitar string with the wave equation:

##\frac {\partial^2 \psi}{\partial t^2}=\frac WM \frac {\partial^2 \psi}{\partial x^2}##

Where $W$ is the tension, ##M## is the mass per unit length and ##\psi## is the displacement. It has been shown that the velocity of the wave is equal to ##\sqrt{W/M}##

Homework Equations

The Attempt at a Solution



So for the first part I stated that ##\psi## must be ##0## at the points where the sting is fixed as it has no displacement.

Next I stated that the real standing wave solution is given as follows: ##\psi=\psi_0cos(kx+\phi_x)cos(\omega t+\phi_t)##.

For the allowed oscillation frequencies I did the following:

##\psi=0## at ##x=0## and ##x=L##

Hence, ##cos(kx+\phi_t)=0## at these two points.

Setting ##x=0## yields that ##\phi_x=\pi/2##

Setting ##x=L## gives ##k=n\pi/L##

Since ##k## is the wavenumber ##\lambda=2\pi/k## so ##\lambda=2L/n##

##v=f\lambda## gives the allowed oscillation frequencies as ##f= \frac{n}{2L}\sqrt{W/M}##.

When I come to try to calculate the tension however I rearrange f above to get: ##W=\frac{4L^2f^2}{n^2}M## and I've been given all the numbers to calculate ##W## except I don't know what to pick for the value of ##n##. If I had to guess I would say ##n=2## just because in the question it asks for ##E_2## but I have no clue.

Any explanation or help would be very much appreciated.

 
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  • #2
You need to determine whether the frequency known as E2 is the fundamental or lowest frequency for that guitar string. If it is, then ##n=1##. I suggest that you search "guitar tuning" on the web for clues to the answer.
 

Related to Tension required to tune a string to the note of ##E_2##

1. What is the formula for calculating the tension required to tune a string to the note of ##E_2##?

The formula for calculating tension on a string is T = (F * L) / (2 * L), where T is tension, F is frequency, and L is the length of the string.

2. What factors affect the tension required to tune a string to ##E_2##?

The tension required to tune a string to a specific note is affected by the string's material, thickness, and length, as well as the string's tensioning mechanism and the desired pitch of the note.

3. How does the tension on a string affect the sound produced?

The tension on a string directly affects the pitch of the note produced. Higher tension results in a higher pitch, while lower tension results in a lower pitch. The tension also affects the tone and resonance of the sound.

4. What is the typical tension range for tuning a string to ##E_2##?

The tension required to tune a string to ##E_2## can vary depending on the factors mentioned above, but typically falls within the range of 50-100 pounds.

5. Can tuning a string to ##E_2## with too much tension damage the string or instrument?

Yes, tuning a string to ##E_2## with too much tension can cause the string to break and potentially damage the instrument. It is important to follow recommended tension guidelines for your specific string and instrument to prevent damage.

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