Question

A certain violin string is \(30 \mathrm{~cm}\) long between its fixed ends and has a mass of \(2.0 \mathrm{~g}\). The string sounds an \(\mathrm{A}\) note \((440 \mathrm{~Hz})\) when played without fingering. Where must one put one's finger to play a \(\mathrm{C}(528 \mathrm{~Hz})\) ?

Short answer

One should put the finger 5 cm from one end to play a C note (528 Hz).

Step-by-step solution

Calculate the Speed of the Wave

To find the speed of the wave, one will first use the given frequency of the \(A\) note and the length of the string. From the formula \(f_1 = \frac{v}{2L}\), it follows that \(v = 2f_1L\). Plugging in the given values, \(v = 2(440 \, \mathrm{Hz})(0.30\, m) = 264 \, \mathrm{m/s}\).

Calculate the Length of the Vibrating Segment for the C Note

Next, we need to calculate the length of the string for the \(C\) note. Since the speed of the wave is determined by the properties of the string and remains constant, we can use the fact that the frequency is inversely proportional to the length. That is, \(f_2 = \frac{v}{2L_2}\), and solving for \(L_2\), \(L_2 = \frac{v}{2f_2}\). Substituting the speed found in Step 1 and the given frequency of the \(C\) note, \(L_2 = \frac{264 \, \mathrm{m/s}}{2(528 \, \mathrm{Hz})} = 0.25 \, m = 25\, cm\).

Find the Position of the Finger

By placing a finger on the string, one changes the length of the string that can vibrate, thereby changing the frequency of the produced sound. For the \(C\) note, the finger should be placed such that the length of the string from the finger to the end of the violin string is 25 cm. We are assuming the finger is placed closer to one of the fixed ends, so the finger position from the end where the finger is closer would be \(30\, cm - 25\, cm = 5\, cm\).

Key concepts

Frequency of a Vibrating String

When a string vibrates, it produces sound waves that we perceive as musical notes. The frequency of these notes depends on several factors, which include the length of the string, the tension in it, and the mass per unit length. The frequency (\(f\)) of a vibrating string is given by the formula: \[ f = \frac{v}{2L} \]where - \(v\) is the speed of the wave on the string, and - \(L\) is the length of the vibrating part of the string.
It's important to note that higher frequency notes are produced by vibrating a shorter length of the string, all else being the same. In the context of our exercise, to change the note from A (440 Hz) to C (528 Hz), one needs to alter the effective length of the string by placing a finger at the right position.

Standing Waves

Standing waves are a fascinating phenomenon and crucial in understanding how musical instruments work. They occur when waves reflect back and forth along a medium like a string and interfere with each other. This results in a pattern where certain points remain stationary, called nodes, while points in between vibrate with maximum amplitude, called antinodes.
In a string fixed at both ends, standing waves form at specific frequencies, which are called harmonics. The fundamental frequency, also the first harmonic, is when there is one antinode in the middle and nodes at the ends.
Think of it as a jump rope being shaken up and down, where certain parts of the rope don't move much. Instruments like violins exploit these standing waves to create different musical notes by changing the positions where nodes and antinodes occur.

String Mass and Length

The mass and length of a string play a significant role in defining the characteristics of the sound it produces. A string's mass per unit length, \(\mu\), affects the speed of the wave traveling through it, which in turn affects the frequency.
The speed of the wave (\(v\)) can be calculated using the formula:\[ v = \sqrt{\frac{T}{\mu}} \]Here, - \(T\) is the tension in the string.- \(\mu\) is the mass per unit length, calculated as the mass of the string divided by its total length.
Changing the length of the string that can vibrate will change the frequency of the wave. When you put your finger on a string instrument, you're effectively shortening the string, which raises the pitch. Therefore, the place you put your finger on a violin string determines the note that is played, by altering the length of the string that is free to vibrate.