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Calculating the wave speed in a vibrating string is a crucial part of understanding wave mechanics. Wave speed (\( v \)) is determined by the tension (\( T \)) applied to the string and the string's linear mass density (\( \mu \)). The relationship is given by the formula:

• \[ v = \sqrt{\frac{T}{\mu}} \]

In this context, the tension is the force applied along the string, which in the original exercise is 35 N. This force generates waves throughout the material. By determining the linear mass density (explained in the next section), one can use this formula to find out how fast waves travel along the string.
This is a vital step because knowing the wave speed allows further calculations concerning wave frequency and wavelength.

Linear mass density (\( \mu \)) is a measure of how much mass is distributed per unit length along the string. It is essential for calculating wave characteristics. To find the linear mass density, divide the total mass (\( m \)) of the string by its length (\( L \)):

• \[ \mu = \frac{m}{L} \]

In our exercise, the mass of the wire is converted to kilograms before finding the linear mass density — from 5.625 g to 0.005625 kg. The wire length is measured to be 0.75 m, so \( \mu \) is computed as 0.0075 kg/m.
This value is critical as it contributes directly to calculating the wave speed and indirectly affects the wave's frequency.

When a vibrating string is said to be in its second overtone, it means it is vibrating in the third harmonic mode. Understanding overtone and harmonics is essential in wave physics. In simple terms:

• The fundamental frequency corresponds to the first harmonic.
• The first overtone corresponds to the second harmonic.
• The second overtone corresponds to the third harmonic.

For the third harmonic, the string length (\( L \)) can hold three half-wavelengths. Therefore, the wavelength (\( \lambda \)) for the second overtone is given by:

• \[ \lambda = \frac{2L}{3} \]

This is a unique feature of standing waves in a fixed medium, where nodes and antinodes form a pattern. Each harmonic has a distinct wavelength and frequency, integral to solving vibrational problems accurately.

The frequency (\( f \)) of a wave signifies how often the wave oscillates per second. Calculating the frequency of a vibrating string involves using the wave speed (\( v \)) and wavelength (\( \lambda \)) with the formula:

• \[ f = \frac{v}{\lambda} \]

In our scenario, with a wave speed of approximately 68.06 m/s and a calculated wavelength of 0.5 m, the frequency is:

• \( f \approx 136.12 \, \text{Hz} \)

This frequency translates directly to the sound wave frequency in air since the frequency remains constant when the medium changes, even though the speed of sound in air is different. Understanding how to determine frequency is critical in applications like music production and acoustic engineering.