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When a cable is stretched taut, the speed at which a wave travels through it is a crucial concept. The wave speed, denoted by \(v\), can be calculated using the equation \(v = \sqrt{\frac{F}{\mu}}\). In this formula, \(F\) represents the tension in the cable, and \(\mu\) is the linear mass density of the cable. The wave speed determines how quickly a disturbance will travel along the length of the cable.
Intuitively, the greater the tension in the cable, the faster the wave will travel. This is because higher tension effectively "tightens" the medium, making it less compressible and more robust to disturbances.
The linear mass density plays the opposite role. A higher value of \(\mu\) means more mass per unit length, which slows down the wave, as more energy is required to move that mass along the cable. Together, these two variables control the speed at which transverse waves travel in a stretched cable.

The tension \(F\) in a cable is a force that stretches it, influencing the speed at which waves travel through it. This force is usually measured in newtons (N). When you apply tension to a cable, you're pulling it tight within its elastic limit. In our problem, the cable is under a tension of \(2000 \, \text{N}\).
Tension directly affects wave speed. Greater tension increases the speed, as the cable becomes more rigid and resistant to bending. Understanding this relationship helps in predicting how waves behave in different mediums and under various forces.
Real-life applications include musical instruments like guitars, where tuning involves adjusting the tension in the strings to get the desired pitch. Similarly, in engineering, tension control is essential when designing structures that experience wave-like forces, such as bridges or electrical cables.

Linear mass density, represented by \(\mu\), is the amount of mass per unit length of the cable. It is expressed in units such as kilograms per meter (kg/m). To find this value, you divide the total weight of the cable by its length. In our exercise, \(\mu = \frac{70 \, \text{N}}{30 \, \text{m}} = 2.33 \, \text{kg/m}\).
This parameter is crucial because it affects how quickly waves can travel along the cable. A denser cable has more mass in each segment, requiring more energy to move, thus reducing the wave speed. Consequently, linear mass density inversely influences the speed of transverse waves.
When designing systems that rely on wave propagation, such as transmission lines, this concept ensures that engineers choose materials that balance speed and efficiency. Proper understanding of \(\mu\) leads to more robust and effective designs.

The travel time \(t\) of a wave on a cable is the time it takes to reach from one end to another and back. It is calculated using the formula \(t = \frac{\text{Total Distance}}{v}\), where \(v\) is the wave speed. In the given exercise, the wave travels a distance of \(60 \, \text{m}\) (since it goes to the end and returns), with a calculated speed of \(29.4 \, \text{m/s}\).
Substituting these values, we find \(t = \frac{60}{29.4} \approx 2.04 \, \text{s}\). This means the entire journey takes approximately 2.04 seconds. This insight is critical in applications such as telecommunications and data transmission, where timing accuracy is crucial.
Understanding wave travel time allows for precise synchronization in various technological applications and is essential for students learning about wave mechanics and their practical implications.