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Wave speed is an important concept when analyzing waves. It tells us how fast a wave is traveling through a medium, like a stretched string. To find the speed of a wave, we use the relationship between the angular frequency and the wave number. The formula is:

• \( v = \frac{\omega}{k} \)

Here, \( v \) represents wave speed, \( \omega \) represents angular frequency, and \( k \) is the wave number. In our exercise, \( \omega = 30 \, \text{s}^{-1} \) and \( k = 1 \, \text{m}^{-1} \). By substituting these values into the formula, the wave speed \( v \) is calculated to be 30 meters per second. This means the wave travels at 30 m/s through the string. Understanding wave speed helps us predict how quickly and how far the wave will move along the string in a given time.

Linear mass density, often denoted as \( \mu \), is a measure of how much mass is distributed along a unit length of a string or other linear medium. It is expressed in kilograms per meter (kg/m). In the context of waves traveling on a string, linear mass density significantly affects the wave's properties.For this problem, the linear mass density \( \mu \) is given as \( 1.2 \times 10^{-4} \, \text{kg/m} \). This means every meter of string has a mass of \( 1.2 \times 10^{-4} \) kilograms. Linear mass density plays a crucial role in determining wave speed and tension. By understanding \( \mu \), we can better predict how the string will respond to the forces applied by the wave. The relationship between linear mass density, wave speed, and tension is vital for accurately calculating the tension in the string.

Tension in a string refers to the force exerted along the string, which impacts how waves propagate through it. To quantify this tension, we use the wave speed formula in conjunction with the linear mass density. The formula is:

• \( v = \sqrt{\frac{T}{\mu}} \)

Where \( v \) is the wave speed, \( T \) is the tension in the string, and \( \mu \) is the linear mass density. The rearranged formula to find tension, \( T \), is:

• \( T = v^2 \mu \)

In our exercise, we substitute the known wave speed (\( v = 30 \, \text{m/s} \)) and linear mass density (\( \mu = 1.2 \times 10^{-4} \, \text{kg/m} \)) into the formula. Calculating further, we find the tension \( T = 1.08 \, \text{N} \). This tension supports the wave as it moves, ensuring stability in its propagation through the medium. Understanding the tension is crucial because it helps predict how the string behaves under the influence of waves.