Question

The A string of a violin is a little too taut. Four beats per second are heard when it is sounded together with a tuning fork that is vibrating accurately at the pitch of concert \(\mathrm{A}(440 \mathrm{~Hz})\). What is the period of the violin string vibration?

Short answer

The period of the violin string vibration is 0.00225 seconds.

Step-by-step solution

Find the frequencies

The beating frequency (difference in frequencies) is given as 4Hz. Since we know the tuning fork's frequency is 440Hz, the two possible frequencies for the violin string are either \(440Hz + 4Hz = 444Hz\) or \(440Hz - 4Hz = 436Hz\). However, given that the A string is 'too taut', we can infer that the frequency is higher, which makes the violin string's frequency 444Hz.

Calculate the Period

We can now calculate the period of the violin string vibration using the formula \(T = 1/f\). If we substitute \(f = 444Hz\) in the formula, we get \(T = 1/ 444 s = 0.00225 s\).

Key concepts

Acoustics and Beat Phenomenon

Acoustics, the study of sound, delves into how sound waves behave in various environments. A fascinating aspect of acoustics is the creation of beats, an auditory illusion occurring when two sound waves of similar but not identical frequencies are played together. The sound level alternates between loud and soft, a phenomenon known as 'beating'.

This occurs because as the waves interact, they periodically reinforce and cancel each other, known as constructive and destructive interference, which leads us to our next concept.

Wave Interference

Wave interference is at the core of the beats phenomena. It describes how two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Interference can be constructive when waves are in phase and reinforce each other, or destructive when they are out of phase and cancel each other out. This principle applies to all types of waves, including sound.

In the case of the violin string and tuning fork, the slight difference in frequencies creates points where the sound waves interfere destructively (creating a beat or silence) and constructively (creating a louder sound), which we perceive as beats.

Sound Frequency

Sound frequency is measured in Hertz (Hz) and represents the number of cycles a sound wave completes in one second. When discussing a violin string or a tuning fork, frequency determines pitch. The more taut a string is, the higher the frequency and, consequently, the higher the pitch of the note produced.

In the given exercise, the frequency of the violin's A string was slightly higher than the concert pitch A (440 Hz), leading to a higher pitch and beats in the sound because its frequency was 444Hz, as we calculated.

Vibrations and Waves

Vibrations are the back and forth motions that create waves, which carry energy from one place to another. In sound waves, these vibrations are mechanical perturbations that travel through a medium, such as air, creating alternating compressions and rarefactions.

Relation Between Vibration and Sound

The frequency of these vibrations dictates the sound frequency. The A string of a violin, when plucked, vibrates at exactly the right frequency to produce the concert pitch A if it is tuned well. However, if too taut, it vibrates more rapidly, thus emitting a higher frequency and affecting the period of the wave, as highlighted in the exercise.