91Ó°ÊÓ

The harmonic frequency of a string refers to the specific frequencies at which the string vibrates. In a string fixed at both ends, like a violin string, the harmonics are determined by

• the length of the string
• the tension in the string
• the mass per unit length of the string

When a string oscillates, it produces standing waves with nodes at both ends.
In the fundamental mode, also known as the first harmonic (\( n = 1 \)), one half-wavelength (\( \lambda \)) fits along the length of the string.
This is why the wavelength in this mode is twice the length of the string, \( \lambda = 2L \), where \( L \) is the length of the string.
The frequency (\( f \)) of these oscillations is determined using the formula \( f = \frac{v}{\lambda} \), where \( v \) is the speed of the wave on the string.
For our violin string, plugging in \( v = 250\, \text{m/s} \), and \( \lambda = 0.3\, \text{m} \) gives the fundamental frequency as 833.33 Hz.

Sound waves are vibrations that travel through a medium, such as air, water, or solids. They are longitudinal waves, meaning that the oscillation of the particles of the medium is parallel to the direction of wave propagation.
Sound waves have several key properties:

• Frequency: This determines the pitch of the sound; higher frequencies equate to higher pitches.
• Wavelength: This is the distance between successive compressions or rarefactions in the wave.
• Speed: This is how fast the wave travels through the medium, depending on the medium's properties.

In the context of our violin string, the emitted sound wave's frequency is the same as the string's frequency at 833.33 Hz, since the vibrations are transferred directly to the air.
This direct transfer ensures that the pitch of the sound remains the same as the frequency of the string.
However, the speed of the sound wave in air differs from the wave speed on the string, which affects the wavelength.

When discussing sound waves travelling through air, a crucial property is their wavelength, which is defined as the distance over which the wave's shape repeats.
Unlike waves on the string, the speed of sound in air is much slower, specifically given as 348 m/s in this exercise.
To find the wavelength of the sound wave in air, we use the formula: \( \lambda = \frac{v}{f} \), where

• \( \lambda \) is the wavelength in air
• \( v \) is the speed of sound in air
• \( f \) is the frequency of the sound wave

Substituting our known values of \( v = 348\, \text{m/s} \) and \( f = 833.33\, \text{Hz} \), we compute \( \lambda \approx 0.417\, \text{m} \).
This means that the sound wave travels approximately 0.417 meters in air before repeating its cycle, which describes the distance between consecutive points of similar phase, such as two consecutive compressions.