Problem 73 - College Physics

Question

A violin string is tuned to a certain frequency (first harmonic or the fundamental frequency). (a) If a violinist wants a higher frequency, should the string be (1) lengthened, (2) kept the same length, or (3) shortened? Why? (b) If the string is tuned to 520 Hz and the violinist puts a finger down on the string one-eighth of the string length from the neck end, what is the frequency of the string when the instrument is played this way?

Step-by-Step Solution

Answer

(a) The string should be shortened for a higher frequency. (b) New frequency is approximately 594.3 Hz.

Step 1: Understanding String Frequency

The frequency of a string is determined by several factors, including the tension, length, and mass per unit length of the string. For a given string under constant tension, the frequency is inversely proportional to its length. When the length of the string is decreased, the frequency increases, and vice versa.

Step 2: Analyzing Part (a)

If a violinist wants a higher frequency, the length of the string must be decreased. Therefore, the string should be shortened. Shortening the string increases the frequency because the vibration occurs over a shorter length.

Step 3: Calculating Frequency Change

When a violinist places a finger down on the string one-eighth of the string length from the neck end, the vibrating length of the string becomes \( \frac{7}{8} \) of the original length. This change in length will affect the frequency of the string.

Step 4: Using the Frequency Formula

The formula for frequency is \( f = \frac{v}{2L} \), where \( v \) is the speed of sound in the string and \( L \) is the length of the string. When the length becomes \( \frac{7}{8}L \), the new frequency \( f' \) becomes \( f' = \frac{v}{2(\frac{7}{8}L)} \).

Step 5: Solving for New Frequency

Substitute the original frequency \( f = 520 \text{ Hz} \) into the formula. The new frequency is \( f' = 520 \times \frac{8}{7} \approx 594.3 \text{ Hz} \).

Key Concepts

Frequency and Wavelength

When studying the physics of musical instruments, understanding the relationship between frequency and wavelength is essential. Imagine a wave traveling along a string. The frequency refers to the number of wave cycles passing a point per second, usually measured in Hertz (Hz). The wavelength is the physical length of one cycle of the wave. These two properties are intrinsically linked to the speed of the wave. This relationship can be expressed by the equation:

• Speed = Frequency × Wavelength

If the wave is traveling along a string at a certain speed and we change the wavelength by changing the string length—as is done by fingering on a violin—the frequency will change accordingly. Shortening the string length increases the frequency because the waves fit into a shorter space, increasing the number of cycles per second. Conversely, lengthening the string decreases the frequency, as the waves have more space to stretch. Thus, by controlling the string length, musicians can control the pitch of the note produced.

Harmonics in Strings

Harmonics, or overtones, are fascinating elements of sound that occur naturally in stringed instruments and add complexity to the music you hear. When a string vibrates, it doesn't just vibrate at its fundamental frequency. It also vibrates at multiple frequencies that are integer multiples of the fundamental frequency. These are called harmonics.

• The first harmonic is the fundamental frequency itself.
• The second harmonic is twice the frequency of the fundamental.
• The third harmonic is three times the fundamental frequency, and so on.

These harmonics are important for the timbre or color of the sound produced by an instrument. Musicians can manipulate the presence of harmonics by changing how and where they play the instrument. On a violin, for example, placing a finger lightly on the string can isolate certain harmonics, creating a different sound than the fundamental frequency would produce alone. Understanding harmonics enriches both the playing and the listening experience of music.

Sound Waves and Vibrations

Sound waves are disturbances that travel through a medium, like air, water, or solid materials. In musical instruments, sound waves are created by vibrations—rapid back and forth movements that propagate through the medium. For example, when a violin string is plucked, it vibrates, and these vibrations travel through the air to our ears as sound waves. Every sound wave has properties like wavelength, frequency, amplitude, and speed.

• **Wavelength** is the distance between consecutive points of a wave.
• **Frequency** dictates how many times the wave oscillates back and forth in one second.
• **Amplitude** refers to the height of the wave, which correlates with the loudness of the sound.

The vibrations can also be visualized as wave patterns moving through the medium. This is evident in string instruments like the violin; you can feel these vibrations if you touch the instrument while playing. By altering string tension, length, and mass, different frequencies and amplitudes can be achieved, producing various pitches and volume levels. This interplay of vibrations and sound waves forms the foundation of how musical instruments produce sound.