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The linear mass density, often denoted as \( \rho \), is a measure of how much mass is distributed along the length of a wire or string. It is calculated using the formula:
\[ \rho = \frac{m}{L} \]
where \( m \) is the mass and \( L \) is the length. In the exercise, the wire has a mass of 100 grams, which is equivalent to 0.1 kilograms, and a length of 10 meters:
\[ \rho = \frac{0.1 \text{ kg}}{10 \text{ m}} = 0.01 \text{ kg/m} \]
This means each meter of the wire has a mass of 0.01 kilograms. A lower linear mass density usually means the wire is lighter, which impacts the speed at which waves can travel through it.

The speed at which waves travel through a medium, such as a wire, depends on two factors: the tension force \( T \) applied to the wire and its linear mass density \( \rho \). The formula to calculate wave speed \( v \) is given by:
\[ v = \sqrt{\frac{T}{\rho}} \]
Substituting the given tension \( T = 250 \text{ N} \) and \( \rho = 0.01 \text{ kg/m} \), we find: \[ v = \sqrt{\frac{250}{0.01}} = \sqrt{25000} = 158.11 \text{ m/s} \]
The wave speed is critical because it directly affects the frequencies of standing waves that can form in the wire, with higher speeds resulting in higher frequencies.

Harmonic frequencies are specific standing wave frequencies that occur at natural resonance points within the wire. The fundamental frequency \( f_1 \) is the lowest frequency at which a standing wave can form and is calculated using:
\[ f_1 = \frac{v}{2L} \]
For a wave speed \( v = 158.11 \text{ m/s} \) and wire length \( L = 10 \text{ m} \), we get:
\[ f_1 = \frac{158.11}{2 \times 10} = 7.91 \text{ Hz} \]
The second harmonic \( f_2 \) is twice the fundamental frequency:
\[ f_2 = \frac{2v}{2L} = \frac{v}{L} = 15.81 \text{ Hz} \]
The third harmonic \( f_3 \) is three times the fundamental frequency:
\[ f_3 = \frac{3v}{2L} = 23.72 \text{ Hz} \]
Each harmonic frequency corresponds to a different pattern of standing waves on the wire, with increasing frequencies having more nodes and antinodes along the length of the wire.