A string is clamped at both ends and tensioned until its fundamental frequency is \(85 \mathrm{Hz}\). If the string is then held rigidly at its midpoint, what's the lowest frequency at which it will vibrate?

When a string is plucked or otherwise excited, it vibrates not just at its fundamental frequency but also at higher frequencies known as overtones or harmonics. Imagine the string as being divided into segments that vibrate at their own specific frequencies. The fundamental frequency, the lowest frequency at which the string vibrates, is called the first harmonic. The frequencies of subsequent harmonics are integer multiples of this fundamental frequency. Thus, if the fundamental frequency is denoted by the symbol \( f_1 \) and is 85 Hz, the second harmonic would be \(2 \times f_1 = 170 Hz\), the third harmonic would be \(3 \times f_1 = 255 Hz\), and so on, forming a series known as the harmonic series.

The presence of these harmonics enriches the sound of the string, giving it a more complex and pleasing tone. In the context of the exercise at hand, when the string is held at the midpoint and then plucked, it vibrates at a frequency that is the second harmonic of the original fundamental frequency since the vibrating length is halved.

String resonance refers to the phenomenon where a string vibrates at certain preferred frequencies that relate to its physical dimensions and tension. These preferred frequencies at which a string naturally resonates are called its natural frequencies. The most prominent of these is the fundamental frequency, and it is at this frequency that the string tends to vibrate most strongly when disturbed.

When a string is clamped at both ends, as in our exercise, the ends act as fixed points, creating standing waves. In the exercise scenario, initially, the string vibrates at its fundamental frequency. Once the string is held at its midpoint and plucked, the midpoint becomes another fixed point, creating a node. This change effectively halves the vibrating length of the string, leading it to vibrate at a frequency that is twice its original fundamental frequency, yielding a new resonance pattern and causing the string to emit a higher pitched note.

Frequency and pitch are intimately connected when it comes to sound waves and musical instruments. Frequency, measured in Hertz (Hz), refers to the number of complete vibrations that occur in one second. Pitch is our perception of the frequency of sound waves and is what allows us to distinguish a high note from a low one.

As the frequency of a vibrating string increases, so does the pitch of the sound it produces. This is exactly what's described in the exercise: by holding the string at its midpoint, the frequency of the string is doubled. This increase in frequency leads to a higher pitch being heard. Mathematically, you can double the original fundamental frequency of 85 Hz to find the new frequency, which would give you \(2 \times 85 Hz = 170 Hz\). This new frequency corresponds to a pitch one octave above the original pitch because in music, an octave is defined by a doubling of the frequency. It's important for students to grasp this connection to understand not just the exercise but also the broader principles of acoustics and music theory.